Chapter 14 — Front Matter
Purpose and Overview
This chapter introduces geometry as the study of shape, space, measurement, and formula reasoning. Geometry is not a separate world from arithmetic. It is arithmetic applied to lines, angles, flat figures, and solid objects. Every formula in this chapter uses operations students have already practiced: addition, multiplication, division, fractions, decimals, and variables.
The goal of Chapter 14 is not to teach a complete geometry course. That comes later in the MGU Mathematics Series. The goal here is to give students the foundational vocabulary, measurement skills, and spatial reasoning they need before entering Algebra I, introductory physics, construction and design courses, and geometry proper. Students who finish this chapter will know how to measure, label, and compute the size of the most common geometric figures, and they will understand what those measurements actually mean.
Opening Scenario
A homeowner wants to tile a rectangular bathroom floor that measures 9 feet by 7 feet. She also wants to put baseboard trim around the entire edge of the room. She needs to order the right amount of tile for the floor and the right length of trim for the walls. These are two different measurements. One describes how much surface the floor covers. The other describes how long the edges are. A student who understands geometry can solve both problems quickly, check the answers for reasonableness, and explain the difference between the two measurements clearly. Both of those ideas — surface coverage and edge length — have names in geometry. They are area and perimeter. By the end of this chapter, students will handle both easily, along with volume, angles, coordinates, and several common shapes.
15Sections
12Objectives
40+Key Terms
20+Worked Examples
Learning Objectives
- 1Identify and describe basic geometric objects: points, lines, segments, rays, and planes.
- 2Name, classify, and measure angles; identify complementary and supplementary pairs.
- 3Identify and describe common two-dimensional shapes and their properties.
- 4Calculate the perimeter of polygons and the circumference of circles.
- 5Calculate the area of rectangles, triangles, parallelograms, trapezoids, and circles.
- 6Apply perimeter and area formulas correctly, including to composite figures.
- 7Calculate the volume of rectangular prisms, cubes, and other basic solid figures.
- 8Use correct units for length, area, and volume and convert between units when needed.
- 9Substitute values into geometry formulas using variables and solve for missing dimensions.
- 10Plot and read points on the coordinate plane in the context of geometric figures.
- 11Solve geometry word problems using the Read, Represent, Solve, Check, Explain method.
- 12Recognize and avoid the most common geometry mistakes.
Sections 14.1 – 14.3
The Language of Geometry: Building Blocks and Shapes
What Geometry Studies · Points, Lines, Segments, Rays · Angles and Basic Shapes
Section 14.1
What Geometry Studies
Geometry is the branch of mathematics that studies shape, size, position, and the properties of space. The word comes from Greek roots meaning "earth measurement." Ancient peoples used geometry to survey land, build structures, navigate by the stars, and design tools. Today geometry serves the same purposes, just in more complex settings.
This chapter focuses on two areas of geometry that connect most directly to arithmetic and pre-algebra.
Measurement geometry deals with sizes. It asks: How long is this edge? How much surface does this floor cover? How much space does this container hold? Students will compute perimeter, area, and volume.
Coordinate geometry deals with location. It asks: Where is this point? How far apart are these two points? What does this relationship look like when graphed? Students will use the coordinate plane to describe position and relationships.
Both branches rely on the same arithmetic skills. Computing an area requires multiplication. Comparing measurements requires the number line. Plotting a point requires understanding signed numbers. Students entering this chapter already have all the tools they need. Geometry simply gives those tools a new context.
Geometry as a Language
Geometry has its own vocabulary, notation, and symbols. A student who says "the long side" when they mean "the length" is being vague. A student who says "the area" when they mean "the perimeter" is making a conceptual error. Precision matters. When a new term appears, ask: What does this word describe? How is it different from other terms I know?
MGU Library Connection
- MGU Geometry Terms Dictionary — expanded definitions and diagrams of every term introduced in this chapter
- Appendix G: Geometry Formula Reference — all formulas from this chapter in one place
Section 14.2
Points, Lines, Segments, and Rays
The most basic objects in geometry are not shapes or figures. They are the building blocks from which all figures are made: points, lines, segments, rays, and planes.
Points
A point is an exact location in space. It has no size, no length, no width, and no thickness. It is simply a position. A point is represented by a dot and named with a capital letter. Every corner of a shape is a point. Every location on a graph is a point. Every intersection of two lines is a point.
Draw a small dot labeled A. Below it, write "Point A."
Lines
A line is a straight path that extends in both directions without end. A line has no beginning and no end. It continues infinitely in both directions. A line is named using any two points on it, with a double-headed arrow written above the letters.
Draw a horizontal line with arrows on both ends. Mark two points and label them B and C. Label the line with a two-headed arrow symbol over "BC."
Everyday vs. Geometric Meaning
In everyday language, people say "line" to mean a segment with a beginning and end. In geometry, a line is infinite. When students mean a finite piece of a line, they should say "segment."
Line Segments
A line segment is a portion of a line with two endpoints. It has a definite length that can be measured. A segment is named by its two endpoints with a flat bar over the letters, and its length is written without the bar.
Draw a horizontal segment with two endpoints labeled D and E. Show the overline notation for the segment, and just "DE" for the length.
Rays
A ray is a portion of a line that has one endpoint and extends infinitely in one direction. Think of a flashlight beam: it starts at the bulb and goes outward without stopping. A ray is named by its endpoint first, then any other point on the ray.
Draw a ray starting at point F, passing through point G, with an arrow continuing to the right. Label F as the endpoint.
Order Matters in Ray Names
The endpoint always comes first in the name of a ray. The ray from F through G is different from the ray from G through F because they start at different points.
| Object | Description | Key Property |
|---|
| Point | Exact location, no size | No dimensions |
| Line | Straight, infinite in both directions | Extends forever both ways |
| Segment | Part of a line, two endpoints | Has measurable length |
| Ray | Part of a line, one endpoint | Extends forever one way |
| Plane | Flat surface, infinite in two directions | No thickness |
Section 14.3
Angles and Basic Shapes
What an Angle Is
When two rays share a common endpoint, they form an angle. The shared endpoint is called the vertex of the angle. The two rays are called the sides of the angle. Angles are measured in degrees (°). A full rotation around a point measures 360°. A half rotation measures 180°. A quarter rotation measures 90°.
| Type | Measure | Description |
|---|
| Acute | Less than 90° | A sharp, narrow corner |
| Right | Exactly 90° | A perfect square corner; marked with a small square |
| Obtuse | Between 90° and 180° | A wide, open angle |
| Straight | Exactly 180° | The two rays form a straight line |
Complementary and Supplementary Angles
Complementary — two angles whose measures add to 90°. If one angle is 35°, its complement is 55°.
Supplementary — two angles whose measures add to 180°. If one angle is 35°, its supplement is 145°.
Polygons
A polygon is a closed figure formed by three or more straight line segments that meet only at their endpoints. Polygons are named by their number of sides. A polygon is regular if all its sides are equal in length and all its angles are equal in measure.
| Sides | Name | Example |
|---|
| 3 | Triangle | Traffic sign, roof truss |
| 4 | Quadrilateral | Room floor, window pane |
| 5 | Pentagon | Baseball home plate |
| 6 | Hexagon | Honeycomb cell, bolt head |
| 8 | Octagon | Stop sign |
Triangles
A triangle is a polygon with three sides and three angles. The sum of the three interior angles of any triangle is always 180°. Triangles are classified by their sides (equilateral: all three equal; isosceles: two equal; scalene: none equal) and by their angles (acute: all angles acute; right: one 90° angle; obtuse: one obtuse angle).
Tip — Triangle Angle Sum
A triangle can never have two right angles. Two right angles would add to 180°, leaving zero degrees for the third angle — which is impossible.
Quadrilaterals
A quadrilateral is a polygon with four sides. The sum of the interior angles of any quadrilateral is always 360°. Common quadrilaterals include rectangles (four right angles, opposite sides equal), squares (four right angles, all sides equal), parallelograms (opposite sides parallel and equal), trapezoids (exactly one pair of parallel sides), and rhombuses (all sides equal, opposite angles equal).
Draw small labeled diagrams of a rectangle, square, parallelogram, trapezoid, and rhombus, arranged in a visual chart.
Circles
A circle is the set of all points in a plane that are the same distance from a central point. That central point is the center. The distance from the center to any point on the circle is the radius. The distance across the circle through the center is the diameter.
diameter = 2 × radius and radius = diameter ÷ 2
Example 14.3.1
Radius and diameter
- Problem AA circle has a radius of 5 cm. Find the diameter.
- Solve
d = 2 × 5 = 10 cm
- Problem BA circle has a diameter of 14 inches. Find the radius.
- Solve
r = 14 ÷ 2 = 7 in
Answers: 10 cm diameter; 7 in radius.
- Name four basic geometric objects and state one way each is different from the others.
- Classify each angle: 47°, 90°, 130°, 180°, 23°, 91°.
- Two angles are complementary. One measures 64°. Find the other.
- Two angles are supplementary. One measures 115°. Find the other.
- A triangle has angles of 45° and 70°. Find the third angle.
- A circle has a diameter of 22 ft. Find its radius.
- Name the quadrilateral that has four right angles and four equal sides.
Sections 14.4 – 14.7
Perimeter, Area, and Composite Figures
Perimeter · Area of All Common Shapes · Formula Reference · Composite Figures
Section 14.4
Perimeter
The perimeter of a figure is the total length of its boundary — the distance around the outside of a flat shape. Perimeter is a one-dimensional measurement expressed in linear units such as centimeters, inches, feet, and meters. Think of perimeter as the length of fence needed to enclose a yard, or the length of frame needed to border a picture.
The perimeter of any polygon is found by adding the lengths of all its sides.
P = sum of all side lengths
Example 14.4.1
Perimeter of a triangle
- ProblemA triangle has sides of 7 cm, 10 cm, and 13 cm. Find its perimeter.
- ThinkAdd all three sides.
- Solve
P = 7 + 10 + 13 = 30 cm
- Check7 + 10 = 17; 17 + 13 = 30. ✓
Answer: The perimeter is 30 cm.
Example 14.4.2
Perimeter of a rectangle
- ProblemA rectangle is 8 feet long and 5 feet wide. Find its perimeter.
- ThinkA rectangle has two pairs of equal sides. Use P = 2l + 2w.
- Solve
P = 2(8) + 2(5) = 16 + 10 = 26 ft
- Check16 + 10 = 26. ✓
Answer: The perimeter is 26 feet.
Circumference of a Circle
The perimeter of a circle has its own name: circumference. Because a circle has no straight sides to add, its perimeter is calculated differently using a special number called pi (symbol: π). Pi is the ratio of the circumference of any circle to its diameter. No matter what size the circle, this ratio is always the same irrational number: approximately 3.14159. In most problems at this level, students use π ≈ 3.14 or leave answers in terms of π.
C = πd or equivalently C = 2πr
Example 14.4.3
Circumference of a circle
- ProblemA circle has a radius of 6 inches. Find its circumference. Use π ≈ 3.14.
- ThinkI know the radius. Use C = 2πr.
- Solve
C = 2 × 3.14 × 6 = 2 × 18.84 = 37.68 in
- Check2 × 6 = 12; 12 × 3.14 = 37.68. ✓
Answer: The circumference is approximately 37.68 inches.
About Pi
π is irrational — it cannot be written as an exact decimal. When you multiply by 3.14, your answer is an approximation. If a problem says "leave in terms of π," your answer will look like 12π rather than a decimal number.
Section 14.5
Area
The area of a figure is the amount of surface it covers. Area is a two-dimensional measurement expressed in square units, such as square centimeters (cm²), square inches (in²), and square feet (ft²). Think of area as the number of unit squares that fit inside a flat figure.
Area and perimeter are different measurements. Perimeter is the distance around the outside. Area is the space inside. They measure different things, use different units, and answer different questions.
Draw a 4 by 3 grid of unit squares inside a labeled rectangle. Show that 12 unit squares fill the interior, illustrating A = 12 cm² for a 4 cm × 3 cm rectangle.
Area of a Rectangle and Square
Rectangle: A = lw Square: A = s²
Example 14.5.1
Area of a rectangle
- ProblemA room is 12 feet long and 9 feet wide. Find its area.
- Solve
A = 12 × 9 = 108 ft²
- Check10 × 9 = 90; 2 × 9 = 18; 90 + 18 = 108. ✓
Answer: The area is 108 square feet.
Area of a Parallelogram
A parallelogram has a base and a height. The height is not the slanted side — it is the perpendicular distance between the two parallel bases, measured at a right angle.
A = bh
Draw a parallelogram with a horizontal base labeled b and a vertical dashed line showing the perpendicular height h from the top side down to the base. Mark the right angle. Label both b and h.
Area of a Triangle
A triangle is half of a parallelogram with the same base and height. If you take any parallelogram and cut it in half diagonally, you get two identical triangles. This is why the formula includes a factor of one-half.
A = ½bh
Draw a triangle with a horizontal base labeled b and a dashed vertical line from the apex to the base labeled h. Mark the right angle where h meets the base.
Example 14.5.2
Area of a triangle
- ProblemA triangle has a base of 8 inches and a height of 5 inches. Find its area.
- Solve
A = ½ × 8 × 5 = ½ × 40 = 20 in²
- CheckHalf of 40 is 20. ✓
Answer: The area is 20 square inches.
Area of a Trapezoid
A trapezoid has two parallel sides called bases, labeled b₁ and b₂. The height h is the perpendicular distance between the two bases.
A = ½(b₁ + b₂)h
Example 14.5.3
Area of a trapezoid
- ProblemA trapezoid has bases of 6 m and 10 m, and a height of 4 m. Find its area.
- Solve
A = ½(6 + 10)(4) = ½(16)(4) = ½(64) = 32 m²
Answer: The area is 32 square meters.
Area of a Circle
A = πr²
Example 14.5.4
Area of a circle
- ProblemA circle has a radius of 5 cm. Find its area. Use π ≈ 3.14.
- Solve
A = 3.14 × 5² = 3.14 × 25 = 78.5 cm²
Answer: The area is approximately 78.5 square centimeters.
Student Warning — Radius vs. Diameter
The area formula A = πr² requires the radius. If you are given the diameter, divide by 2 first. Using the diameter directly will give an answer four times too large.
Section 14.6
Perimeter and Area Formulas
| Shape | Perimeter / Circumference | Area |
|---|
| Rectangle | P = 2l + 2w | A = lw |
| Square | P = 4s | A = s² |
| Triangle | P = a + b + c | A = ½bh |
| Parallelogram | P = 2a + 2b | A = bh |
| Trapezoid | P = a + b₁ + c + b₂ | A = ½(b₁ + b₂)h |
| Circle | C = 2πr or C = πd | A = πr² |
Example 14.6.1
Choosing the right formula — tiling a floor
- ProblemA square tile has a side of 4 inches. How many tiles are needed to cover a floor that is 48 by 36 inches?
- ThinkFind the area of the floor, find the area of one tile, then divide.
- Solve
Area of floor: 48 × 36 = 1,728 in²
Area of tile: 4² = 16 in²
Number of tiles: 1,728 ÷ 16 = 108
- Check108 × 16 = 1,728. ✓
Answer: 108 tiles are needed.
Section 14.7
Composite Figures
A composite figure (also called a compound figure) is a shape made by combining two or more simpler shapes. Real buildings, rooms, and objects rarely come in the shape of a perfect rectangle or a single triangle. Most are combinations of standard shapes.
To find the area of a composite figure, identify the simpler shapes inside, decide whether to add or subtract areas, apply the correct formula for each part, and combine the results.
Example 14.7.1
L-shaped figure — addition
- ProblemAn L-shaped floor can be divided into two rectangles: 10 m × 6 m and 4 m × 3 m. Find the total area.
- Solve
A₁ = 10 × 6 = 60 m²
A₂ = 4 × 3 = 12 m²
Total A = 60 + 12 = 72 m²
Answer: The total area is 72 square meters.
Example 14.7.2
Rectangle with a circular hole — subtraction
- ProblemA rectangular metal plate is 20 cm × 15 cm. A circular hole with radius 3 cm is cut from the center. Find the remaining area. Use π ≈ 3.14.
- ThinkSubtract the area of the circle from the area of the rectangle.
- Solve
A_rect = 20 × 15 = 300 cm²
A_circle = 3.14 × 3² = 3.14 × 9 = 28.26 cm²
Remaining = 300 − 28.26 = 271.74 cm²
Answer: Approximately 271.74 square centimeters of metal remain.
Student Warning — Composite Perimeters
When two rectangles are joined to form an L-shape, the inner edges where they meet are inside the figure. Do not include them in the perimeter. Only add the outer boundary edges.
- Rectangle: L = 15 cm, W = 8 cm. Find P and A.
- Square: side = 9 in. Find P and A.
- Triangle: base = 10 m, height = 7 m, sides 10, 7, 8 m. Find P and A.
- Parallelogram: base = 14 ft, height = 6 ft. Find A.
- Trapezoid: b₁ = 7 m, b₂ = 13 m, h = 5 m. Find A.
- Circle: r = 10 cm. Find C and A. (π ≈ 3.14)
- Circle: d = 20 in. Find C and A. (π ≈ 3.14)
- L-shape: two rectangles, 10 m × 4 m and 6 m × 3 m. Find total A.
- Rectangle 20 ft × 12 ft with a 4 ft square removed from a corner. Find remaining A.
Sections 14.8 – 14.12
Volume, Units, Variables, and Coordinate Geometry
Volume · Volume Formulas · Units · Variables in Formulas · Coordinate Geometry
Section 14.8
Volume
Volume is the amount of three-dimensional space a solid figure occupies. Volume is expressed in cubic units, such as cubic centimeters (cm³), cubic inches (in³), and cubic feet (ft³). Think of volume as the number of unit cubes that fit inside a solid figure.
Length, Area, and Volume Compared
One dimension → length → measured in units (cm, ft, m)
Two dimensions → area → measured in square units (cm², ft², m²)
Three dimensions → volume → measured in cubic units (cm³, ft³, m³)
Draw a rectangular box labeled with length 4, width 3, and height 2. Show a small unit cube in one corner. Label the total as V = 24 cubic units.
| Solid Figure | Description |
|---|
| Rectangular prism | Box shape with six rectangular faces; all opposite faces equal and parallel |
| Cube | Rectangular prism where all six faces are squares and all edges are equal |
| Cylinder | Two circular bases connected by a curved surface |
| Cone | Circular base narrowing to a single point (the apex) |
| Sphere | Every point on the surface is the same distance from the center |
| Triangular prism | Two triangular faces and three rectangular faces |
Section 14.9
Volume Formulas
Rectangular prism: V = lwh Cube: V = s³ Prism (any base): V = Bh Cylinder: V = πr²h
Example 14.9.1
Volume of a rectangular prism
- ProblemA storage box is 5 ft long, 3 ft wide, and 4 ft tall. Find its volume.
- Solve
V = 5 × 3 × 4 = 60 ft³
- Check5 × 3 = 15; 15 × 4 = 60. ✓
Answer: The volume is 60 cubic feet.
Example 14.9.2
Volume using base area
- ProblemA triangular prism has a triangular base with area 15 cm² and a height of 8 cm. Find its volume.
- ThinkUse V = Bh where B is the area of the base.
- Solve
V = 15 × 8 = 120 cm³
Answer: The volume is 120 cubic centimeters.
Example 14.9.3
Volume of a cylinder
- ProblemA cylindrical tank has a radius of 4 m and a height of 10 m. Find its volume. Use π ≈ 3.14.
- Solve
V = 3.14 × 4² × 10 = 3.14 × 16 × 10 = 502.4 m³
Answer: Approximately 502.4 cubic meters.
Section 14.10
Units in Geometry
Every measurement in geometry must include a unit. A measurement without a unit is incomplete. Saying that a room's area is "48" communicates nothing useful. Saying that a room's area is 48 ft² communicates exactly what is meant.
Units also prevent serious errors. In construction, confusing inches with feet can waste thousands of dollars of materials. If a problem gives measurements in different units, they must be converted to the same unit before any formula is applied.
| Conversion | Value |
|---|
| 1 foot | 12 inches |
| 1 yard | 3 feet = 36 inches |
| 1 mile | 5,280 feet |
| 1 meter | 100 centimeters |
| 1 kilometer | 1,000 meters |
| 1 sq ft | 144 sq in |
| 1 sq yd | 9 sq ft |
| 1 cubic ft | 1,728 cubic in |
Example 14.10.1
Unit conversion before computing
- ProblemA rectangle is 2 feet long and 9 inches wide. Find its area in square inches.
- ThinkConvert feet to inches first. 2 feet = 24 inches.
- Solve
A = 24 × 9 = 216 in²
Answer: The area is 216 square inches.
Student Warning — Never Mix Units
If you multiply 2 ft by 9 in, the result of 18 has no valid unit. The units ft and in are different and cannot be combined directly. Always convert to a single unit first.
Section 14.11
Using Variables in Geometry Formulas
Every geometry formula introduced so far uses variables. The formula A = lw uses the variables A, l, and w. When students work with geometry formulas, they are already doing algebra. The skill of substitution from Chapter 9 is exactly what students need here: replace each variable with its known value, then evaluate the expression.
Example 14.11.1
Using substitution with a geometry formula
- ProblemUse A = ½bh to find the area of a triangle with b = 14 cm and h = 9 cm.
- Solve
A = ½(14)(9) = ½(126) = 63 cm²
Answer: The area is 63 square centimeters.
Example 14.11.2
Solving for a missing dimension
- ProblemA rectangle has an area of 84 square meters. Its width is 7 meters. Find its length.
- ThinkUse A = lw and solve for l.
- Solve
84 = l × 7
l = 84 ÷ 7 = 12 m
- Check12 × 7 = 84. ✓
Answer: The length is 12 meters.
Example 14.11.3
Trapezoid area with order of operations
- ProblemFind the area of a trapezoid with b₁ = 5 ft, b₂ = 11 ft, and h = 8 ft.
- ThinkAdd the bases first (inside the parentheses), then multiply.
- Solve
A = ½(5 + 11)(8) = ½(16)(8) = ½(128) = 64 ft²
Answer: The area is 64 square feet.
Section 14.12
Introduction to Coordinate Geometry
Chapter 12 introduced the coordinate plane for graphing patterns. In geometry, the coordinate plane is used to locate points, describe figures, and find distances. Recall that the plane is formed by a horizontal x-axis and a vertical y-axis meeting at the origin (0, 0). Every point has an ordered pair (x, y): x tells how far left or right from the origin, and y tells how far up or down.
Example 14.12.1
Finding side lengths from coordinates
- ProblemA rectangle has vertices at A(1,1), B(5,1), C(5,4), and D(1,4). Find its length, width, perimeter, and area.
- ThinkAB is horizontal (same y), so find the distance by subtracting x-coordinates. AD is vertical, so find the distance by subtracting y-coordinates.
- Solve
Length AB = |5 − 1| = 4 units
Width AD = |4 − 1| = 3 units
Perimeter = 2(4) + 2(3) = 14 units
Area = 4 × 3 = 12 square units
Answer: Length 4, width 3, perimeter 14 units, area 12 square units.
Example 14.12.2
A triangle in the coordinate plane
- ProblemA triangle has vertices at P(−3, 0), Q(3, 0), and R(0, 4). Find its base, height, and area.
- Solve
Base PQ = |3 − (−3)| = 6 units
Height (R above x-axis) = |4 − 0| = 4 units
Area = ½(6)(4) = 12 square units
Answer: Base 6, height 4, area 12 square units.
Distance and Negative Coordinates
When coordinates include negative numbers, use absolute value to find distances. Distance is always a positive quantity: |3 − (−3)| = |6| = 6, not −6.
- Rectangular prism: l = 8, w = 5, h = 3 cm. Find V.
- Cube: side = 4 in. Find V.
- Cylinder: r = 5 m, h = 12 m. Find V. (π ≈ 3.14)
- A rectangle has A = 96 m² and w = 8 m. Find l.
- A triangle has A = 30 in² and b = 10 in. Find h.
- Convert: rectangle 3 ft × 18 in. Find area in sq in.
- Plot A(2,3), B(2,−2), C(6,−2), D(6,3). Find P and A.
- Triangle with vertices (−4,0), (4,0), (0,3). Find A.
Sections 14.13 – 14.15
Word Problems, Common Mistakes, and the Road Ahead
Geometry Word Problems · Common Errors · Preparing for Geometry and Physics
Section 14.13
Geometry Word Problems
Word problems in geometry require students to translate a real situation into a geometric model, apply the correct formula, solve, check, and explain. This section connects the Chapter 13 problem-solving strategies to geometric contexts.
The Problem-Solving Method Applied to Geometry
Read: Identify the shape, the given measurements, and the question being asked.
Represent: Sketch the figure. Label the given measurements. Write the formula.
Solve: Substitute the values and compute, following the order of operations.
Check: Verify the calculation, confirm the unit is appropriate, and judge whether the answer is reasonable.
Explain: Write the answer in a complete sentence connected to the original context.
Example 14.13.1
Fencing a yard
- ReadA rectangular yard is 30 feet long and 20 feet wide. Fencing costs $4.50 per foot. How much will it cost to fence the entire yard?
- RepresentSketch a rectangle labeled 30 ft and 20 ft. I need the perimeter, then multiply by the cost rate.
- Solve
P = 2(30) + 2(20) = 60 + 40 = 100 ft
Cost = 100 × $4.50 = $450.00
- Check100 feet × $4.50/ft = $450. The perimeter of a 30 by 20 rectangle is 100 ft. Reasonable. ✓
- ExplainThe total cost to fence the yard is $450.00.
Answer: $450.00
Example 14.13.2
Painting a wall with a window
- ReadA wall is 12 ft wide and 8 ft tall. A window measuring 3 ft × 4 ft is cut in. One gallon covers 40 ft². How many gallons are needed to paint the wall (not the window)?
- RepresentComposite figure. Subtract window area from wall area, then divide by coverage.
- Solve
Wall area: 12 × 8 = 96 ft²
Window area: 3 × 4 = 12 ft²
Paintable: 96 − 12 = 84 ft²
Gallons: 84 ÷ 40 = 2.1 → round up to 3 gallons
- Check3 × 40 = 120 ft² covered. More than enough. ✓
- ExplainThe painter needs to purchase 3 gallons of paint.
Answer: 3 gallons
Example 14.13.3
Filling a pool
- ReadA rectangular pool is 25 m long, 10 m wide, and 2 m deep. How many cubic meters of water are needed to fill it?
- Solve
V = 25 × 10 × 2 = 500 m³
- ExplainThe pool holds 500 cubic meters of water.
Answer: 500 m³
Example 14.13.4
Comparing circular gardens
- ProblemTwo circular garden beds have radii of 3 feet and 6 feet. How many times greater is the area of the larger garden?
- Solve
A₁ = π(3)² = 9π
A₂ = π(6)² = 36π
Ratio = 36π ÷ 9π = 4
- ExplainThe larger garden has 4 times the area of the smaller one. When the radius doubles, the area quadruples because area uses the square of the radius.
Answer: The larger garden is 4 times the area of the smaller garden.
Section 14.14
Common Geometry Mistakes
Mistake 1 — Confusing Perimeter and Area
Area of a 6 × 4 rectangle: "P = 6 + 4 + 6 + 4 = 20 ft²"
Error: The student computed the perimeter but labeled it as area. Perimeter uses linear units; area uses square units. Correct area: 6 × 4 = 24 ft².
Mistake 2 — Using Diameter Instead of Radius in Circle Formulas
Circle with d = 10 cm: "A = π(10)² = 100π cm²"
Error: The formula requires the radius. The radius is 10 ÷ 2 = 5 cm. Correct area: A = π(5)² = 25π cm².
Mistake 3 — Forgetting the ½ in the Triangle Formula
Triangle with base 8, height 5: "A = 8 × 5 = 40 in²"
Error: A triangle is half of a parallelogram. Correct area: A = ½(8)(5) = 20 in².
Mistake 4 — Using the Slant Side as the Height
For a triangle, the height is labeled along the slanted side of the figure.
Error: The height must be the perpendicular distance from the base to the opposite vertex. It is a vertical dashed line, not the slanted side of the shape.
Mistake 5 — Mixing Linear, Square, and Cubic Units
Volume of a box: "V = 60 ft"
Error: Volume is measured in cubic units. The correct answer is 60 ft³. After multiplying three lengths, the unit is cubed.
Mistake 6 — Including Interior Segments in the Perimeter of a Composite Figure
Student adds the perimeters of both rectangles separately to find the perimeter of an L-shape.
Error: The edges where the two rectangles meet are interior to the composite figure and are not part of the outer boundary. Only add the edges on the outside of the combined shape.
Mistake 7 — Forgetting to Convert Units
Rectangle 2 ft × 18 in: "A = 2 × 18 = 36 ft²"
Error: The two measurements use different units. Convert first: 2 ft = 24 in. Correct area: 24 × 18 = 432 in².
Mistake 8 — Squaring π in the Circle Area Formula
"A = (π × 5)² = (5π)²"
Error: The exponent in A = πr² applies only to r, not to πr. Square the radius first, then multiply by π: A = π × 5² = π × 25 = 25π.
Section 14.15
Preparing for Geometry and Physics
Chapter 14 introduces the most fundamental tools of geometric reasoning. These tools open many doors.
In Algebra I, students will write and solve equations that include geometric formulas. They will use the coordinate plane to graph linear equations, find slopes, and describe relationships. Every formula substitution and coordinate-plane skill practiced here transfers directly.
In the MGU Geometry textbook, students will prove properties of triangles, quadrilaterals, and circles using logical reasoning. They will study congruence and similarity, learn the Pythagorean theorem, and apply coordinate geometry to distances and slopes. Every formula in this chapter is a foundation for more precise and powerful results.
In introductory physics, students will apply perimeter, area, and volume in the context of force, pressure, density, and energy. Geometric fluency built in this chapter will make that work significantly easier.
Four Key Ideas to Carry Forward
1. Geometry gives arithmetic a spatial dimension. Numbers describe quantities. Geometry puts those quantities into space. Together they give a much richer picture of measurement.
2. Formulas summarize relationships. A = lw is not just a rule to memorize. It captures why the area of a rectangle equals length times width: that is how many unit squares fit inside.
3. Units carry meaning. Square units describe surfaces. Cubic units describe space. Linear units describe lengths. These distinctions reflect genuine differences in what is being measured.
4. Geometry connects to algebra. Every formula is an algebraic equation. Solving for a missing dimension is solving an equation. Graphing a figure on the coordinate plane is combining geometry and algebra.
- A rectangular room is 14 ft × 11 ft. Carpet costs $3.25 per ft². How much will it cost to carpet the room?
- Using the same room, there is a 3 ft doorway where no baseboard trim is needed. How many feet of trim are needed for the rest of the perimeter?
- A circular garden has a diameter of 12 meters. How many meters of border material are needed around the edge? (π ≈ 3.14)
- An aquarium is 30 in × 12 in × 15 in. The owner fills it to 2 inches from the top. How many cubic inches of water are in the aquarium?
- An L-shaped office has a main section of 20 ft × 15 ft and an extension of 8 ft × 10 ft. Tile costs $2.10 per ft². Find the total tiling cost.
- A student finds the area of a 9 m × 4 m rectangle as "26 m²." Identify the error and give the correct answer.
- A student computes the volume of a 3 × 5 × 2 box as "10 ft³." What is wrong? What is correct?
Chapter 14 — Back Matter
Chapter Summary
- Sec 14.1Geometry studies shape, size, position, and space. Measurement geometry asks how big; coordinate geometry asks where. Both branches depend on the arithmetic skills built throughout this textbook.
- Sec 14.2Points, lines, segments, rays, and planes are the building blocks of geometry. A line extends infinitely in both directions. A segment has two endpoints. A ray has one endpoint and extends infinitely in one direction.
- Sec 14.3Angles are formed by two rays sharing a vertex and are measured in degrees. Acute angles are less than 90°, right angles are exactly 90°, obtuse angles are between 90° and 180°, and straight angles are exactly 180°. Complementary angles sum to 90°; supplementary angles sum to 180°.
- Sec 14.4Perimeter is the total distance around a flat figure, measured in linear units. It is found by adding all side lengths. The perimeter of a circle is called circumference and uses the formula C = 2πr or C = πd.
- Sec 14.5Area is the amount of surface inside a flat figure, measured in square units. Formulas: rectangle A = lw; square A = s²; parallelogram A = bh; triangle A = ½bh; trapezoid A = ½(b₁ + b₂)h; circle A = πr².
- Sec 14.6All perimeter and area formulas are collected in a reference table. Choosing the right formula requires identifying the shape, the known information, and the question being asked.
- Sec 14.7Composite figures are made by combining or removing simpler shapes. Areas are found by addition (joining shapes) or subtraction (removing a piece). Only outer boundary edges count in the perimeter of a composite figure.
- Sec 14.8Volume is the amount of three-dimensional space inside a solid figure, measured in cubic units. Length is one-dimensional, area is two-dimensional, and volume is three-dimensional.
- Sec 14.9Volume formulas: rectangular prism V = lwh; cube V = s³; prism V = Bh; cylinder V = πr²h. The base-times-height pattern connects area to volume for all prisms and cylinders.
- Sec 14.10Every measurement requires a unit. Length uses linear units, area uses square units, and volume uses cubic units. Measurements in different units must be converted to the same unit before any computation.
- Sec 14.11Geometry formulas are algebraic expressions. Substitution replaces known variables with values. Solving for an unknown dimension requires solving an equation. Order of operations governs all formula evaluations.
- Sec 14.12Geometric figures can be represented on the coordinate plane using ordered pairs at each vertex. Horizontal distances are found by subtracting x-coordinates. Vertical distances are found by subtracting y-coordinates. Absolute value ensures a positive distance.
- Sec 14.13Geometry word problems use the Read, Represent, Solve, Check, Explain method. Always sketch and label the figure before computing. Include units in every answer and write a sentence explaining the result in context.
- Sec 14.14Eight recurring errors: confusing perimeter with area; using diameter instead of radius; omitting ½ in triangle area; using slant sides as heights; mixing linear, square, and cubic units; including interior segments in perimeter; failing to convert units; and misapplying exponents in the circle area formula.
- Sec 14.15Geometry formulas are algebraic. Arithmetic is spatial. Algebra extends into coordinate geometry. These four ideas connect everything in this chapter to Algebra I, MGU Geometry, and introductory physics.
Key Terms Review
acute angle
An angle measuring less than 90°.
area
The amount of surface inside a flat figure, measured in square units.
base
A designated side of a polygon or solid from which height is measured perpendicularly.
circumference
The perimeter of a circle; C = 2πr or C = πd.
composite figure
A figure formed by combining or removing two or more simpler shapes.
coordinate plane
A flat surface defined by a horizontal x-axis and a vertical y-axis meeting at the origin.
cube
A rectangular prism with all edges equal; V = s³.
degree
The unit used to measure angles. A full rotation equals 360°.
diameter
The distance across a circle through its center; d = 2r.
geometry
The branch of mathematics that studies shape, space, size, and position.
height
The perpendicular distance from a base to the opposite side or vertex.
line
A straight path extending infinitely in both directions with no endpoints.
line segment
A portion of a line with two definite endpoints and a measurable length.
obtuse angle
An angle measuring more than 90° but less than 180°.
ordered pair
A pair of numbers (x, y) that locates a point on the coordinate plane.
origin
The point (0, 0) where the x-axis and y-axis intersect.
perimeter
The total length of the boundary of a flat figure, measured in linear units.
pi (π)
The ratio of a circle's circumference to its diameter; approximately 3.14159. An irrational number.
plane
A flat, two-dimensional surface extending infinitely in all directions with no thickness.
point
An exact location in space with no size, length, or width.
polygon
A closed figure made of three or more straight line segments meeting only at endpoints.
quadrant
One of the four regions of the coordinate plane divided by the x-axis and y-axis.
quadrilateral
A polygon with four sides. Interior angles sum to 360°.
radius
The distance from the center of a circle to any point on the circle; r = d ÷ 2.
ray
A portion of a line with one endpoint, extending infinitely in one direction.
rectangle
A quadrilateral with four right angles and opposite sides equal.
rectangular prism
A solid with six rectangular faces; V = lwh.
right angle
An angle measuring exactly 90°; marked with a small square at the vertex.
square
A rectangle with all four sides equal; a regular quadrilateral.
straight angle
An angle measuring exactly 180°; the two sides form a straight line.
triangle
A polygon with three sides. Interior angles always sum to 180°.
vertex
The point where two sides of an angle or polygon meet.
volume
The amount of three-dimensional space inside a solid figure, measured in cubic units.
Concept Review Questions
Write complete sentences where indicated. These questions test understanding, not just computation.
- What is the difference between a line and a line segment? Why does the distinction matter in geometry?
- Can a triangle have two obtuse angles? Explain why or why not.
- Describe in your own words the difference between perimeter and area. Give one real-world example of each.
- Why is area measured in square units and volume measured in cubic units?
- A rectangle and a parallelogram have the same base and height. Do they have the same area? Explain using the formulas.
- Why is the formula for the area of a triangle ½bh and not bh?
- What is pi, and why does it appear in circle formulas?
- What is a composite figure? Describe a strategy for finding its area by subtraction.
- If the radius of a circle is doubled, what happens to its area? Explain using the formula.
- A student says the height of a triangle is the slanted side. What is wrong with this, and how would you correct it?
- What does the coordinate pair (−3, 5) mean geometrically? Which direction does each number indicate?
- Explain why mixing feet and inches in the same formula produces an invalid answer.
Formula Review
State the formula and identify what each variable represents.
- Perimeter of a rectangle
- Area of a rectangle
- Area of a square
- Perimeter of a triangle
- Area of a triangle
- Area of a parallelogram
- Area of a trapezoid
- Circumference of a circle (two forms)
- Area of a circle
- Volume of a rectangular prism
- Volume of a cube
- Volume using base area and height
Skill Practice
- Triangle: sides 5, 12, 13 m
- Rectangle: l = 20 cm, w = 7 cm
- Square: side = 11 ft
- Circle: r = 8 m (π ≈ 3.14)
- Circle: d = 14 in (π ≈ 3.14)
- Pentagon: all sides = 6 cm
- Rectangle: l = 13 ft, w = 6 ft
- Square: side = 8 m
- Triangle: b = 16 m, h = 9 m
- Parallelogram: b = 12 cm, h = 7 cm
- Trapezoid: b₁ = 8 ft, b₂ = 14 ft, h = 6 ft
- Circle: r = 7 cm (π ≈ 3.14)
- Circle: d = 10 in (π ≈ 3.14)
- Triangle: b = 9 in, h = 4 in
- Rectangular prism: 8 × 5 × 3 cm
- Cube: side = 4 in
- Cube: side = 10 m
- Rectangular prism: 12 × 9 × 2 ft
- Triangular prism: B = 20 cm², h = 7 cm
- Cylinder: r = 5 m, h = 12 m (π ≈ 3.14)
- A = 96 ft², w = 8 ft. Find l.
- A = 45 m², s = ? (square). Find s.
- A = 30 in², b = 10 in (triangle). Find h.
- P = 50 cm, w = 9 cm (rectangle). Find l.
- V = 120 cm³, l = 5 cm, w = 4 cm (rectangular prism). Find h.
- V = 64 in³ (cube). Find s.
- An L-shaped floor has a larger part of 10 m × 8 m and a smaller part of 4 m × 3 m. Find the total area.
- A rectangle 18 ft × 10 ft has a 4 ft square removed from one corner. Find the remaining area.
- A large circle has r = 8 cm. A small circle with r = 3 cm is removed from its center. Find the remaining area. (π ≈ 3.14)
- A shape consists of a rectangle 12 cm × 5 cm with a triangle on top (same 12 cm base, height 4 cm). Find the total area.
- Plot these points: A(2,3), B(−1,4), C(−3,−2), D(0,−5), E(4,0).
- A rectangle has vertices at (0,0), (6,0), (6,4), (0,4). Find the perimeter and area.
- A rectangle has vertices at (−2,1), (5,1), (5,−3), (−2,−3). Find the length, width, perimeter, and area.
- A triangle has vertices at (0,0), (8,0), (0,6). Find the base, height, and area.
- Point A is at (3,7) and point B is at (3,−2). Find the distance between them.
- Point C is at (−5,4) and point D is at (3,4). Find the distance between them.
Answers may be placed in the answer key.
Real-World Applications
Use the Read, Represent, Solve, Check, Explain method for each problem.
- Carpeting: A rectangular room is 14 ft × 11 ft. Carpet costs $3.25 per square foot. What is the total cost to carpet the room?
- Baseboard trim: Using the same room, one doorway is 3 feet wide and needs no trim. How many feet of baseboard are needed for the rest of the perimeter?
- Garden border: A circular garden has a diameter of 12 meters. A stone border is placed around the entire edge. How many meters of border are needed? (π ≈ 3.14)
- Aquarium water: A rectangular aquarium is 30 in × 12 in × 15 in. The owner fills it to 2 inches from the top. How many cubic inches of water does it hold?
- Composite office: An L-shaped office has a main section of 20 ft × 15 ft and an extension of 8 ft × 10 ft. Tile costs $2.10 per square foot. What is the total tiling cost?
- Painting a cylinder: A cylindrical water tower has a radius of 5 m and a height of 12 m. Only the curved side surface needs painting. The curved surface area is 2πrh. Find the area to be painted. (π ≈ 3.14)
- Scale drawing: On a map, 1 inch represents 50 miles. A rectangular region on the map measures 3 in × 2 in. What is the actual area of the region in square miles?
- Juice box: A juice box is 8 cm × 5 cm × 4 cm. Find its volume in cubic centimeters. (Note: 1 cm³ = 1 mL. How many milliliters does it hold?)
Answers may be placed in the answer key.
Error Analysis
Each problem contains a student error. Identify the mistake, explain what went wrong, and show the correct solution.
Problem 1
Rectangle 9 m × 4 m. Area: "9 + 4 + 9 + 4 = 26 m²"
What is the error? What is the correct area?
Problem 2
Triangle with base 10 cm and slanted side 8 cm. Height used = 8 cm. A = ½(10)(8) = 40 cm²
What is the geometric error? What information would be needed to find the correct area?
Problem 3
Circle with diameter 8 in. Area: "A = π(8)² = 64π in²"
What is the error? What is the correct area?
Problem 4
Box 3 ft × 5 ft × 2 ft. Volume: "V = 3 + 5 + 2 = 10 ft³"
What is the error? What is the correct volume?
Problem 5
Rectangle 2 feet × 18 inches. Area: "A = 2 × 18 = 36 ft²"
What is the error? Find the correct area in square inches.
Problem 6
L-shaped figure made of rectangles 6 × 4 and 3 × 2. Perimeter: "P = 2(6)+2(4) + 2(3)+2(2) = 30 m"
What is the error in this approach to finding the perimeter of a composite figure?
Answers may be placed in the answer key.
Chapter 14 Checkpoint
Chapter 14 Checkpoint
Complete without assistance — assesses all key skills and concepts from Chapter 14
Section A — Vocabulary (1 point each)
- Name the type of angle that measures exactly 90°.
- What is the name for the perimeter of a circle?
- What do you call a closed figure with five straight sides?
- What type of unit is used to measure volume?
- What is the name for the point where two sides of an angle meet?
Section B — Formulas (2 points each)
Write the formula and identify the variables.
- Area of a triangle
- Volume of a rectangular prism
- Circumference of a circle using the radius
- Area of a trapezoid
- Area of a circle
Section C — Computation (3 points each)
- Find the perimeter of a triangle with sides 7, 9, and 11 cm.
- Find the area of a rectangle 13 ft long and 6 ft wide.
- Find the area of a triangle with base 16 m and height 9 m.
- Find the volume of a cube with side 5 in.
- Find the circumference and area of a circle with radius 7 cm. Use π ≈ 3.14.
- Find the area of a trapezoid with bases 8 ft and 14 ft and height 6 ft.
Section D — Problem Solving (5 points each)
- A rectangular park is 80 m long and 55 m wide. A walking path runs along the entire perimeter. How many meters long is the path?
- A square courtyard has a side of 20 feet. Sod costs $1.75 per square foot. What is the total cost to sod the courtyard?
- A composite figure is a rectangle 10 cm wide and 6 cm tall with a triangle on top. The triangle has the same 10 cm base and a height of 4 cm. Find the total area of the figure.
- A rectangular pool is 50 m long, 20 m wide, and 2 m deep. Water is pumped in at 5 m³ per minute. How many minutes to fill the pool?
Section E — Coordinate Geometry (4 points each)
- A rectangle has vertices at (1,2), (7,2), (7,5), and (1,5). Find the perimeter and area.
- A triangle has vertices at (−5,0), (5,0), and (0,6). Find its area.
Section F — Error Identification (3 points each)
- A student computes the area of a circle with radius 9 as
A = π × 9 = 9π. What error was made? What is the correct answer?
- A student finds the perimeter of a 5 × 8 rectangle as
5 × 8 = 40 ft. What error was made? What is the correct perimeter?
Answers may be placed in the answer key.
Bridge Note
Connecting to Algebra I, Geometry, and Physics
Chapter 14 completes the foundational work of Arithmetic and Pre-Algebra Foundations. In Algebra I, students will write and solve equations that include geometric formulas, use the coordinate plane to graph linear equations, and find slopes. The skills practiced here — substituting into formulas, solving for unknowns, and plotting points — transfer directly. In the MGU Geometry textbook, students will prove the properties of triangles, quadrilaterals, and circles using logical reasoning, study congruence and similarity, and learn the Pythagorean theorem. In introductory physics, perimeter, area, and volume appear in the context of force, pressure, density, and energy. The goal of this book was to build a strong, connected foundation. Arithmetic is not just a set of calculations. It is a language for describing quantities, relationships, patterns, and structures, and geometry gives that language its spatial dimension.
MGU Library Connection
- Appendix E: Formula Sheet — all geometry formulas from this chapter
- Appendix G: Geometry Formula Reference — expanded formula reference with notes
- Appendix D: Fraction, Decimal, and Percent Conversion Chart — unit conversion reference
- Appendix J: Algebra I Readiness Check — assess readiness for the next course
- MGU Geometry Textbook — proofs, congruence, similarity, transformations, Pythagorean theorem
- MGU Algebra I Textbook — geometric formula equations, graphing, coordinate geometry
- MGU Geometry Terms Dictionary — expanded definitions and diagrams for every term in this chapter