Finding Area of polygons 10.5

Geometry: Using Trigonometry to Find the Area of Polygons

Agenda

• Topic: How to use trigonometry to find the area of polygons.

• Homework: HW #7: Area of Polygons Using Trigonometry.

• Next Quiz: Friday, May 1.

• Do Now: Sit in your assigned seat.

• Materials Required: Pencil, Calculator.

Weekly Schedule

• Tuesday: Quiz on Area of Polygon Using Trig.

• Wednesday: Practice Area Using Trig.

• Thursday: Circles and Arcs.

• Friday: Areas of Sectors and Circles.

Opener

• Task: Find the measures of the numbered angles in the polygons:   1. m∠1 = 45°   2. m∠2 = 22.5°   3. m∠3 = 67.5°   4. m∠1 = 72°   5. m∠2 = 36°   6. m∠3 = 54°

• Instructions: Use trigonometry to solve for the missing sides of the polygons mentioned above.

Using Trig for Area

• Example Problem: What is the area of a regular nonagon with 10-cm sides?   - Step 1: Calculate the central angle: 360°/9 = 40°.   - Step 2: Split the central angle in half to create a right triangle: 40°/2 = 20°.   - Step 3: Use SOHCAHTOA to solve for the apothem:     - $a = 5 an(20°)$.   - Step 4: Area Calculation:     - $A = rac{1}{2} imes ext{Perimeter} imes ext{Apothem}$.     - Perimeter of the nonagon: $p = (9)(10) = 90 ext{ cm}$.     - Area: $A ext{ (final)} = 618.2 ext{ cm}^2$.     - Conclusion: Area of nonagon is approximately 618.2 cm².

You Try

• Problem: Find the area of a pentagon with 4-inch sides:   - Step 1: Calculate central angle: 360°/5 = 72°.   - Step 2: Split the angle: 72°/2 = 36°.   - Step 3: Apothem calculation:     - $a = 2 an(36°)$.   - Step 4: Area Calculation:     - $A = rac{1}{2} imes ext{Perimeter} imes ext{Apothem}$.     - Perimeter: $p = (5)(4) = 20 ext{ in}$.     - Area: $A ext{ (final)} = 27.5276… ext{ in}^2 ightarrow 28 ext{ in}^2$ (rounded).

Another Example

• Scenario: Area of a standard stop sign (regular octagon) with a radius of 16.2 inches.   - Step 1: Calculate central angle: 360°/8 = 45°.   - Step 2: Split the angle: 45°/2 = 22.5°.   - Step 3: Apothem calculations:     - $a = 16.2 imes ext{cos}(22.5°)$.   - Step 4: Area Calculation:     - Area of octagon: $A = rac{1}{2} imes R imes ext{Perimeter}$ where Perimeter = 8(16.2sin(22.5°)).     - $A ext{ (final)} = 742.2924… ext{ in}^2 ightarrow 742 ext{ in}^2$ (rounded).

You Try (Tabletop Shape)

• Task: What is the area of a regular decagon with a radius of 9.5 inches?   - Step 1: Calculate central angle: 360°/10 = 36°.   - Step 2: Split the angle: 36°/2 = 18°.   - Step 3: Apothem calculation:     - $a = 9.5 imes ext{cos}(18°)$.   - Step 4: Area Calculation:     - $A = rac{1}{2} imes ext{Perimeter} imes ext{Apothem}$.     - $A ext{ (final)} = 265.23809… ext{ in}^2 ightarrow 265 ext{ in}^2$ (rounded).

Theorem 10-8: Area of a Triangle Given SAS

• Definition: The area of a triangle is half the product of the lengths of two sides and the sine of the included angle.   - Formula: $ext{Area of } riangle ABC = rac{1}{2} b c imes ext{sin}(A)$, where $b$ and $c$ are the lengths of two sides, and $A$ is the included angle.   - Illustration:     - $A = rac{1}{2} imes ext{side length} imes ext{side length} imes ext{sine of included angle}$.

How This Works

• Scenario: Finding the area of triangle △ABC where only m∠A and lengths of sides b and c are known.   - Step 1: To apply the formula $A = rac{1}{2}bh$, height (h) is needed.   - Step 2: Height is found using the sine ratio:     - $ext{sin}(A) = rac{h}{c} ightarrow h = c imes ext{sin}(A)$.   - Step 3: Substitute height into the area formula:     - $A = rac{1}{2} b c imes ext{sin}(A)$.

Finding Area of a Triangle

• Example Problem: What is the area of the triangle given:   - Formula: $A = rac{1}{2} imes ext{side length} imes ext{side length} imes ext{sine of included angle}$.   - Values Used:     - Side lengths: 12 and 21.     - Angle: 48°.   - Calculation: $A = rac{1}{2} imes 12 imes 21 imes ext{sin}(48°) = 93.6362… ext{ cm}^2 ightarrow 94 ext{ cm}^2$ (rounded).

You Try (Triangle Example)

• Problem: Find the area of the triangle:   - Formula: $A = rac{1}{2} imes ext{side length} imes ext{side length} imes ext{sine of included angle}$.   - Values Used:     - Side lengths: 16 and 10.     - Angle: 34°.   - Calculation: $A = rac{1}{2} imes 16 imes 10 imes ext{sin}(34°) = 44.73543… ext{ in}^2 ightarrow 45 ext{ in}^2$ (rounded).