The Central Idea
Length measures one-dimensional extent. Distance measures separation between locations. Perimeter measures the total length around a boundary.
These ideas are closely connected because each ultimately compares a spatial quantity with a chosen unit of length.
A useful first summary is
\[ \boxed{\text{perimeter}=\text{sum of boundary lengths}}. \]Distance formulas and perimeter formulas are therefore built from the more basic idea of measuring length.
What Is Length?
Length describes how far an object or segment extends in one dimension.
A line segment with endpoints \(A\) and \(B\) has a length commonly written as
\[ AB \]or
\[ |AB|. \]The exact notation depends on context. The geometric segment itself is often written \(\overline{AB}\), while its numerical length is written \(AB\).
Objects and Their Measurements Are Different
It is important to distinguish a geometric object from the number used to measure it.
For example,
\[ \overline{AB} \]denotes the segment joining \(A\) and \(B\), while
\[ AB=7\text{ cm} \]states that the segment has a measured length of \(7\) centimeters.
The segment is an object. The value \(7\text{ cm}\) is a measurement of that object.
Length Is One-Dimensional
A length measures extent along one dimension.
Even when a segment appears inside a two-dimensional diagram or a three-dimensional object, the length itself is still a one-dimensional quantity.
This distinction will matter later when comparing length with area and volume.
\[ \text{length}\sim L,\qquad \text{area}\sim L^2,\qquad \text{volume}\sim L^3. \]Choosing a Unit of Length
Measurement requires a unit. Common units include millimeters, centimeters, meters, kilometers, inches, feet, yards, and miles.
Saying that a segment has length \(5\) is incomplete unless the unit is already established by context.
For example,
\[ 5\text{ cm},\qquad 5\text{ m},\qquad 5\text{ mi} \]represent very different physical lengths.
Length on a Number Line
On a number line, the length of the segment between coordinates \(a\) and \(b\) is the absolute difference of the coordinates:
\[ \boxed{|b-a|}. \]The absolute value is necessary because length is nonnegative.
Example: Length on a Number Line
Suppose point \(A\) is at \(-3\) and point \(B\) is at \(5\).
Their separation is
\[ |5-(-3)|=|8|=8. \]Therefore,
\[ \boxed{AB=8}. \]Why Absolute Value Appears
Reversing the order of the endpoints changes the sign of the subtraction:
\[ 5-(-3)=8 \]but
\[ -3-5=-8. \]The geometric separation is the same in either direction, so absolute value removes the sign:
\[ |5-(-3)|=|-3-5|=8. \]What Is Distance?
Distance measures how far apart two locations are according to a specified rule of measurement.
In ordinary Euclidean geometry, distance means the length of the shortest straight segment joining two points.
For points \(A\) and \(B\), this distance may be written
\[ d(A,B). \]Distance Is Never Negative
A distance cannot be negative.
\[ \boxed{d(A,B)\ge 0}. \]Furthermore, the distance between a point and itself is zero:
\[ d(A,A)=0. \]Distinct points in ordinary Euclidean geometry have positive distance.
Distance Is Symmetric
The distance from \(A\) to \(B\) is the same as the distance from \(B\) to \(A\):
\[ \boxed{d(A,B)=d(B,A)}. \]Direction may matter for other quantities, such as displacement vectors, but ordinary distance records magnitude rather than orientation.
Distance on the Coordinate Plane
Let
\[ A=(x_1,y_1) \]and
\[ B=(x_2,y_2). \]The horizontal change is
\[ x_2-x_1, \]and the vertical change is
\[ y_2-y_1. \]The Distance Formula
The horizontal and vertical changes form the legs of a right triangle. Applying the Pythagorean theorem gives the Euclidean distance formula:
\[ \boxed{ d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2} }. \]The squares remove sign differences and combine the perpendicular changes into a single straight-line distance.
Example: Distance Between Two Points
Find the distance between
\[ A=(1,2) \]and
\[ B=(7,10). \]Compute the coordinate changes:
\[ \Delta x=7-1=6, \qquad \Delta y=10-2=8. \]Then
\[ d=\sqrt{6^2+8^2} =\sqrt{36+64} =\sqrt{100} =10. \]Therefore,
\[ \boxed{d(A,B)=10}. \]The Distance Formula Comes from the Pythagorean Theorem
The formula is not an isolated rule to memorize. It is a direct application of
\[ a^2+b^2=c^2. \]In a coordinate diagram,
\[ a=|x_2-x_1|, \qquad b=|y_2-y_1|, \]and \(c\) is the straight-line distance between the points.
Horizontal and Vertical Distance
When two points share the same \(y\)-coordinate, their distance is simply the absolute horizontal difference:
\[ d=|x_2-x_1|. \]When two points share the same \(x\)-coordinate, their distance is the absolute vertical difference:
\[ d=|y_2-y_1|. \]These are special cases of the general distance formula.
Distance in Three Dimensions
The same Pythagorean reasoning extends naturally to three-dimensional coordinates.
For
\[ A=(x_1,y_1,z_1) \]and
\[ B=(x_2,y_2,z_2), \]the Euclidean distance is
\[ \boxed{ d=\sqrt{ (x_2-x_1)^2+ (y_2-y_1)^2+ (z_2-z_1)^2 } }. \]Example: Three-Dimensional Distance
Find the distance between
\[ A=(1,2,3) \]and
\[ B=(5,5,15). \]The coordinate changes are
\[ 4,\qquad3,\qquad12. \]Therefore,
\[ d=\sqrt{4^2+3^2+12^2} =\sqrt{16+9+144} =\sqrt{169} =13. \] \[ \boxed{d=13}. \]Length of a Path
The straight-line distance between two endpoints is not always the same as the length of a route traveled between them.
Suppose a path travels from \(A\) to \(B\) to \(C\). Its total path length is
\[ AB+BC. \]The direct distance from the starting point to the ending point is
\[ AC. \]Usually,
\[ AC\le AB+BC. \]The Triangle Inequality
The fact that a direct route cannot be longer than a detour through a third point is expressed by the triangle inequality:
\[ \boxed{AC\le AB+BC}. \]In an ordinary triangle with three distinct noncollinear vertices, the inequality is strict:
\[ AC<AB+BC. \]Equality occurs when the route does not create a genuine detour, such as when the relevant points lie on the same straight line in the appropriate order.
Broken-Line Distance
A route made from several connected segments is sometimes called a polygonal path or broken line.
If its consecutive segment lengths are
\[ \ell_1,\ell_2,\ldots,\ell_n, \]then the total path length is
\[ \boxed{ L=\ell_1+\ell_2+\cdots+\ell_n }. \]Perimeter is a special case in which the path closes around a figure.
What Is Perimeter?
The perimeter of a closed plane figure is the total length of its boundary.
For a polygon with side lengths
\[ s_1,s_2,\ldots,s_n, \]the perimeter is
\[ \boxed{ P=s_1+s_2+\cdots+s_n }. \]Perimeter Is a Length
Perimeter measures a boundary, so its units are units of length.
If a rectangle is measured in centimeters, its perimeter is reported in centimeters, not square centimeters:
\[ \boxed{\text{cm}} \]rather than
\[ \text{cm}^2. \]Square units belong to area, which will be developed in the next lesson.
Perimeter of a Rectangle
A rectangle has two sides of length \(l\) and two sides of width \(w\).
Adding all four sides gives
\[ P=l+w+l+w. \]Combining like terms,
\[ \boxed{P=2l+2w=2(l+w)}. \]Example: Rectangle Perimeter
A rectangle has length \(12\text{ m}\) and width \(5\text{ m}\).
\[ P=2(12)+2(5). \] \[ P=24+10=34. \]Therefore,
\[ \boxed{P=34\text{ m}}. \]Perimeter of a Square
A square has four equal sides. If each side has length \(s\), then
\[ P=s+s+s+s. \]Therefore,
\[ \boxed{P=4s}. \]Perimeter of a Triangle
If a triangle has side lengths \(a\), \(b\), and \(c\), then
\[ \boxed{P=a+b+c}. \]No special triangle type is required. Perimeter is always found by adding the lengths of all boundary sides.
Regular Polygons
A regular polygon has equal side lengths.
If a regular polygon has \(n\) sides and each side has length \(s\), then
\[ \boxed{P=ns}. \]For example, a regular hexagon with side length \(4\text{ cm}\) has
\[ P=6(4)=24\text{ cm}. \]Irregular Polygons
An irregular polygon may have sides of different lengths.
There is usually no shorter formula than adding the side lengths that make up the boundary:
\[ P=s_1+s_2+\cdots+s_n. \]The central task is to identify every boundary segment exactly once.
Perimeter from Coordinates
When the vertices of a polygon are given by coordinates, use coordinate differences or the distance formula to determine the side lengths.
Then add those side lengths.
For a triangle with vertices \(A\), \(B\), and \(C\),
\[ \boxed{ P=d(A,B)+d(B,C)+d(C,A) }. \]Example: Perimeter from Coordinates
Let
\[ A=(0,0),\qquad B=(4,0),\qquad C=(4,3). \]Then
\[ AB=4, \qquad BC=3. \]The third side is
\[ AC=\sqrt{(4-0)^2+(3-0)^2}=5. \]Therefore,
\[ P=4+3+5=12. \] \[ \boxed{P=12}. \]Closed Boundaries and Open Paths
A perimeter belongs to a closed boundary.
An open chain of segments has a total path length, but not a perimeter unless the path closes to form a boundary.
This distinction prevents a common language error: not every sum of segment lengths is a perimeter.
Shared Edges in Composite Figures
When figures are joined together, an edge inside the combined figure is not part of the exterior perimeter.
Only segments exposed on the outside boundary are counted.
A reliable strategy is to trace the entire exterior boundary once and record each exterior segment as it is encountered.
Example: Two Rectangles Joined Along an Edge
Suppose two rectangles share a full edge.
If their separate perimeters are \(P_1\) and \(P_2\), and the shared edge has length \(s\), then simply adding \(P_1+P_2\) counts the shared edge twice.
The perimeter of the combined figure is therefore
\[ \boxed{P=P_1+P_2-2s}. \]The subtraction removes both copies of the interior shared edge.
Missing Side Lengths
Composite figures often omit some boundary lengths because they can be inferred from other measurements.
Before adding the perimeter, identify any equal total horizontal or vertical spans.
For rectilinear figures, opposite overall horizontal travel must balance, and opposite overall vertical travel must balance because the boundary returns to its starting point.
A Useful Rectilinear Shortcut
In a closed shape made only of horizontal and vertical segments, the total rightward length equals the total leftward length, and the total upward length equals the total downward length.
\[ \text{right}=\text{left}, \qquad \text{up}=\text{down}. \]This balance can help recover missing side lengths without introducing unnecessary variables.
Perimeter and Circumference
A circle also has a boundary length. That boundary length is given the special name circumference.
Thus, circumference plays the role of perimeter for a circle.
The next lesson develops circumference together with area, surface area, and volume, so the circular formulas are deferred there.
Exact and Approximate Lengths
Some distances are exact whole numbers or fractions. Others naturally involve radicals.
For example, the distance between \((0,0)\) and \((1,1)\) is
\[ \sqrt{(1-0)^2+(1-0)^2} = \sqrt{2}. \]The exact distance is
\[ \boxed{\sqrt{2}}, \]while a decimal approximation is
\[ \sqrt{2}\approx1.414. \]Keep Exact Values Until the End
When a calculation involves square roots or other nonterminating values, retaining the exact form usually prevents unnecessary rounding error.
For example, prefer
\[ 3+\sqrt{5} \]during intermediate work rather than replacing \(\sqrt{5}\) immediately with a rounded decimal.
Approximate only when the context requires a decimal measurement.
Measured Lengths Are Not Always Exact
A geometric diagram may describe ideal exact lengths, but physical measurement has finite precision.
A ruler reading of
\[ 12.4\text{ cm} \]represents a measured value whose precision depends on the measuring instrument and procedure.
Later in this section, Bounds, Precision, and Measurement Error will treat these issues systematically.
Perimeter from Measured Data
If side lengths are measured rather than exact, the computed perimeter also inherits measurement uncertainty.
Suppose a triangle is measured as
\[ 4.2\text{ cm},\qquad 5.1\text{ cm},\qquad 6.4\text{ cm}. \]Using the reported values,
\[ P=4.2+5.1+6.4=15.7\text{ cm}. \]The arithmetic is exact relative to the reported measurements, but the physical measurements themselves may be approximate.
Unit Consistency
Lengths must be expressed in compatible units before they are added.
For example,
\[ 2\text{ m}+35\text{ cm} \]should not be reported as \(37\) of any single unit.
Convert first:
\[ 2\text{ m}=200\text{ cm}. \]Then
\[ 200\text{ cm}+35\text{ cm}=235\text{ cm}. \]Example: Mixed Units in a Perimeter
A rectangle is \(2\text{ m}\) long and \(75\text{ cm}\) wide.
Convert the length:
\[ 2\text{ m}=200\text{ cm}. \]Then
\[ P=2(200)+2(75) =400+150 =550\text{ cm}. \]Equivalently,
\[ \boxed{P=5.5\text{ m}}. \]Perimeter and Scale
If every length in a figure is multiplied by the same positive scale factor \(k\), then the perimeter is also multiplied by \(k\).
If the original perimeter is \(P\), the scaled perimeter is
\[ \boxed{P'=kP}. \]This is a first example of a scaling law. Later lessons will compare how length, area, and volume scale differently.
Example: Scaling a Perimeter
A polygon has perimeter
\[ 18\text{ cm}. \]Every side length is enlarged by a scale factor of \(3\).
The new perimeter is
\[ P'=3(18)=54\text{ cm}. \] \[ \boxed{P'=54\text{ cm}}. \]Length Is Additive Along a Straight Segment
If point \(B\) lies between \(A\) and \(C\), then the whole segment length equals the sum of its parts:
\[ \boxed{AB+BC=AC}. \]This is often called the segment addition principle.
It is one of the simplest examples of decomposing a quantity into pieces and then accumulating those pieces.
Example: Segment Addition
Suppose \(B\) lies between \(A\) and \(C\), with
\[ AB=7 \]and
\[ BC=11. \]Then
\[ AC=AB+BC=7+11=18. \] \[ \boxed{AC=18}. \]Finding a Missing Segment
Segment addition can also be rearranged.
If
\[ AB+BC=AC, \]then
\[ \boxed{BC=AC-AB}. \]For example, if \(AC=25\) and \(AB=9\), then
\[ BC=25-9=16. \]Length versus Displacement
In applications, the total distance traveled and the net change in position are different quantities.
A person who walks \(3\text{ km}\) east and then \(3\text{ km}\) west has traveled
\[ 6\text{ km} \]in total, even though the final position is the starting position.
Distance records accumulated path length. Displacement additionally records direction and is treated naturally with vectors.
Why Perimeter Matters
Perimeter appears whenever a problem concerns a boundary rather than an interior region.
Examples include:
- fencing a property;
- framing a picture;
- installing baseboard around a room;
- placing trim around an object;
- measuring the boundary of a map region;
- computing the length of a closed route.
Reading a Perimeter Problem Carefully
Before calculating, identify exactly which edges belong to the requested boundary.
Ask:
- Is the path open or closed?
- Which segments lie on the outside?
- Are any edges shared internally?
- Are any lengths missing but inferable?
- Are all measurements in compatible units?
Most perimeter errors occur before the arithmetic begins.
Common Error: Confusing Length with Area
A rectangle with length \(8\text{ cm}\) and width \(3\text{ cm}\) has perimeter
\[ 2(8)+2(3)=22\text{ cm}. \]Multiplying
\[ 8\cdot3=24 \]produces its area, not its perimeter.
Boundary measurements add lengths. Area measures the interior region.
Common Error: Omitting Absolute Value on a Number Line
The distance between \(7\) and \(-2\) is not
\[ -2-7=-9. \]Distance must be nonnegative:
\[ |-2-7|=|-9|=9. \] \[ \boxed{d=9}. \]Common Error: Forgetting to Close the Boundary
When computing the perimeter of a polygon, every side of the closed boundary must be included.
Adding only the visually prominent horizontal and vertical edges can omit a slanted or otherwise less obvious side.
Tracing the boundary from a starting vertex until returning to that same vertex is a reliable check.
Common Error: Counting an Interior Edge
If two shapes are joined, the shared edge lies inside the combined figure.
It should not be counted in the perimeter unless the problem explicitly asks for the total length of all drawn segments rather than the exterior boundary.
Common Error: Adding Unlike Units
An expression such as
\[ 3\text{ m}+40\text{ cm} \]must be converted before the numerical values are added.
For example,
\[ 3\text{ m}=300\text{ cm}, \]so
\[ 300\text{ cm}+40\text{ cm}=340\text{ cm}. \]Common Error: Rounding Too Early
Suppose a perimeter contains a side of exact length \(\sqrt{13}\).
Replacing it too early with a short decimal can introduce avoidable error, especially if several such values are added.
Keep exact values through the calculation and round once at the end when a decimal answer is required.
A Reliable Procedure for Finding Distance
- Identify the two locations or endpoints.
- Choose the coordinate differences appropriate to the dimension.
- Use absolute difference in one dimension.
- Use the Pythagorean distance formula in ordinary two- or three-dimensional Euclidean coordinates.
- Simplify exactly when possible.
- Approximate only if the context requires a decimal.
- Attach the correct unit when the coordinates represent measured lengths.
- Check that the result is nonnegative and reasonable.
A Reliable Procedure for Finding Perimeter
- Identify the closed exterior boundary.
- List every boundary segment exactly once.
- Determine any missing side lengths.
- Convert measurements into compatible units.
- Add the boundary lengths.
- Use a shortcut formula only when the figure's structure justifies it.
- Report the answer in linear units.
- Check that no interior edge was counted and no exterior edge was omitted.
Example: Full Coordinate-Perimeter Workflow
A quadrilateral has vertices
\[ A=(0,0),\quad B=(6,0),\quad C=(6,4),\quad D=(0,4). \]The side lengths are
\[ AB=6,\qquad BC=4,\qquad CD=6,\qquad DA=4. \]Therefore,
\[ P=6+4+6+4=20. \]Since the figure is a rectangle, this agrees with
\[ 2(6)+2(4)=20. \] \[ \boxed{P=20}. \]Example: A Perimeter with a Diagonal Side
Consider a triangle with vertices
\[ A=(0,0),\qquad B=(6,0),\qquad C=(6,8). \]Two side lengths are immediate:
\[ AB=6,\qquad BC=8. \]The diagonal side is
\[ AC=\sqrt{6^2+8^2}=10. \]Therefore,
\[ \boxed{P=6+8+10=24}. \]Example: Recovering a Missing Length
Point \(B\) lies between \(A\) and \(C\). Suppose
\[ AC=19.5\text{ m} \]and
\[ AB=7.2\text{ m}. \]Then
\[ BC=AC-AB. \] \[ BC=19.5-7.2=12.3. \] \[ \boxed{BC=12.3\text{ m}}. \]Practice 1: Number-Line Distance
Find the distance between \(-11\) and \(4\).
Practice 2: Coordinate Distance
Find the distance between
\[ A=(-2,1) \]and
\[ B=(4,9). \]Practice 3: Three-Dimensional Distance
Find the distance between
\[ A=(0,0,0) \]and
\[ B=(2,3,6). \]Practice 4: Segment Addition
Point \(B\) lies between \(A\) and \(C\). If
\[ AB=8.4\text{ cm} \]and
\[ BC=6.7\text{ cm}, \]find \(AC\).
Practice 5: Missing Segment
Point \(Q\) lies between \(P\) and \(R\). If
\[ PR=31 \]and
\[ PQ=12, \]find \(QR\).
Practice 6: Rectangle Perimeter
Find the perimeter of a rectangle with length \(14\text{ m}\) and width \(9\text{ m}\).
Practice 7: Regular Polygon
A regular octagon has side length \(5.5\text{ cm}\). Find its perimeter.
Practice 8: Perimeter from Coordinates
Find the perimeter of the triangle with vertices
\[ A=(0,0),\qquad B=(3,0),\qquad C=(3,4). \]Practice 9: Scaling
A polygon has perimeter \(28\text{ cm}\). Every side is enlarged by a scale factor of \(2.5\). Find the new perimeter.
Practice 10: Mixed Units
A rectangle is \(1.8\text{ m}\) long and \(60\text{ cm}\) wide. Find the perimeter in meters.
Practice Answers
Practice 1
\[ |4-(-11)|=15. \] \[ \boxed{15}. \]Practice 2
\[ d=\sqrt{(4-(-2))^2+(9-1)^2} =\sqrt{6^2+8^2} =10. \] \[ \boxed{10}. \]Practice 3
\[ d=\sqrt{2^2+3^2+6^2} =\sqrt{4+9+36} =\sqrt{49} =7. \] \[ \boxed{7}. \]Practice 4
\[ AC=AB+BC=8.4+6.7=15.1. \] \[ \boxed{AC=15.1\text{ cm}}. \]Practice 5
\[ QR=PR-PQ=31-12=19. \] \[ \boxed{QR=19}. \]Practice 6
\[ P=2(14)+2(9)=28+18=46. \] \[ \boxed{P=46\text{ m}}. \]Practice 7
\[ P=8(5.5)=44. \] \[ \boxed{P=44\text{ cm}}. \]Practice 8
The side lengths are
\[ 3,\qquad4,\qquad5. \]Therefore,
\[ \boxed{P=3+4+5=12}. \]Practice 9
\[ P'=2.5(28)=70. \] \[ \boxed{P'=70\text{ cm}}. \]Practice 10
Convert \(60\text{ cm}\) to meters:
\[ 60\text{ cm}=0.6\text{ m}. \]Then
\[ P=2(1.8)+2(0.6) =3.6+1.2 =4.8. \] \[ \boxed{P=4.8\text{ m}}. \]What to Remember
- Length measures one-dimensional extent.
- Distance measures separation between locations.
- Ordinary Euclidean distance is nonnegative and symmetric.
- On a number line, distance is an absolute difference.
- On the coordinate plane, the distance formula comes from the Pythagorean theorem.
- Perimeter is the total length of a closed boundary.
- Perimeter is measured in linear units, not square units.
- Interior shared edges are not part of the exterior perimeter.
- Compatible units are required before lengths are added.
- Scaling every length by \(k\) scales the perimeter by \(k\).
The unifying idea is that spatial boundaries and separations can be decomposed into lengths, measured consistently, and recombined.
Where We Go Next
Length describes one-dimensional extent and perimeter accumulates length around a boundary. The next lesson, Circumference, Area, Surface Area, and Volume, extends spatial measurement from one-dimensional boundaries to two-dimensional regions and three-dimensional objects.
That progression will make the role of dimension visible in both formulas and units.