Lesson Description
This lesson focuses on calculating the area of common polygons and circles. Students will apply formulas, substitute algebraic expressions, and solve multi-step problems involving composite reasoning.
Learning Objectives
- Find the area of triangles, parallelograms, trapezoids, and regular polygons.
- Use the formula for the area of a circle.
- Substitute algebraic expressions into area formulas.
- Solve real-world problems involving area.
Key Formulas
- Triangle: \( A = \frac{1}{2}bh \)
- Parallelogram: \( A = bh \)
- Trapezoid: \( A = \frac{1}{2}(b_1 + b_2)h \)
- Regular Polygon: \( A = \frac{1}{2}aP \)
- Circle: \( A = \pi r^2 \)
Lesson Examples
Example 1: Area of a Triangle
Find the area of a triangle with base 10 cm and height 6 cm.
\[ A = \frac{1}{2}(10)(6) = 30 \text{ cm}^2 \]
Example 2: Area of a Trapezoid
Bases measure 8 m and 12 m, height is 5 m.
\[ A = \frac{1}{2}(8 + 12)(5) = \frac{1}{2}(20)(5) = 50 \text{ m}^2 \]
Example 3: Area of a Circle
Find the area of a circle with radius 7 inches.
\[ A = \pi (7)^2 = 49\pi \approx 153.94 \text{ in}^2 \]
Practice Problems
- Find the area of a triangle with base 14 ft and height 9 ft.
- Find the area of a parallelogram with base 11 cm and height 4 cm.
- Find the area of a trapezoid with bases 6 and 10 and height 8.
- Find the area of a circle with radius 5.
- A regular hexagon has apothem 6 and perimeter 36. Find its area.
Challenge Problems
- The side of a square is \( x + 3 \). Write and simplify an expression for its area.
- A circle has diameter \( 2x \). Express its area in terms of \( x \).
- A triangular garden has base \( 3x \) and height \( 2x + 4 \). Write its area as a polynomial.
Summary
Area formulas allow us to quantify two-dimensional space. Mastery of substitution, algebraic manipulation, and correct formula selection is essential for solving geometry problems efficiently and accurately.