1. 11-2 Areas of Regular Polygons Warm Up Lesson Presentation Lesson Quiz
Holt Geometry

2. Warm Up Find the unknown side lengths in each special right triangle.
11.2 Areas of Regular Polygons
Warm Up
Find the unknown side lengths in each special right triangle.
1. a 30°-60°-90° triangle with hypotenuse 2 ft
2. a 45°-45°-90° triangle with leg length 4 in.
3. a 30°-60°-90° triangle with longer leg length 3m

3. 11.2 Areas of Regular Polygons
Objectives
Develop and apply the formula for the area of a regular polygon.

4. Vocabulary Apothem Center of a Polygon Radius of a Polygon
11.2 Areas of Regular Polygons
Vocabulary
Apothem
Center of a Polygon
Radius of a Polygon
central angle of a regular polygon

5. 11.2 Areas of Regular Polygons
The center of a regular polygon is equidistant from the vertices. The apothem is the distance from the center to a side. A central angle of a regular polygon has its vertex at the center, and its sides pass through consecutive vertices. Each central angle measure of a regular n-gon is

6. Finding the Area of an Equilateral Triangle
11.2 Areas of Regular Polygons
Finding the Area of an Equilateral Triangle

7. 11.2 Areas of Regular Polygons

8. Finding the Area of Regular Polygons
11.2 Areas of Regular Polygons
Finding the Area of Regular Polygons
Regular pentagon DEFGH has a center C, apothem BC, and central angle DCE.

9. total area of the polygon:
11.2 Areas of Regular Polygons
To find the area of a regular n-gon with side length s and apothem a, divide it into n congruent isosceles triangles.
area of each triangle:
total area of the polygon:
The perimeter is P = ns.

10. 11.2 Areas of Regular Polygons

11. Example 3A: Finding the Area of a Regular Polygon
11.2 Areas of Regular Polygons
Example 3A: Finding the Area of a Regular Polygon
Find the area of regular heptagon with side length 2 ft to the nearest tenth.
Step 1 Draw the heptagon. Draw an isosceles triangle with its vertex at the center of the heptagon. The central angle is .
Draw a segment that bisects the central angle and the side of the polygon to form a right triangle.

12. Step 2 Use the tangent ratio to find the apothem.
11.2 Areas of Regular Polygons
Example 3A Continued
Step 2 Use the tangent ratio to find the apothem.
The tangent of an angle is
opp. leg
adj. leg
Solve for a.

13. Step 3 Use the apothem and the given side length to find the area.
11.2 Areas of Regular Polygons
Example 3A Continued
Step 3 Use the apothem and the given side length to find the area.
Area of a regular polygon
The perimeter is 2(7) = 14ft.
Simplify. Round to the nearest tenth.
A  14.5 ft2

14. 11.2 Areas of Regular Polygons
The tangent of an angle in a right triangle is the ratio of the opposite leg length to the adjacent leg length. See page 525.
Remember!

15. Example 3B: Finding the Area of a Regular Polygon
11.2 Areas of Regular Polygons
Example 3B: Finding the Area of a Regular Polygon
Find the area of a regular dodecagon with side length 5 cm to the nearest tenth.
Step 1 Draw the dodecagon. Draw an isosceles triangle with its vertex at the center of the
dodecagon. The central angle is
Draw a segment that bisects the central angle and the side of the polygon to form a right triangle.

16. Step 2 Use the tangent ratio to find the apothem.
11.2 Areas of Regular Polygons
Example 3B Continued
Step 2 Use the tangent ratio to find the apothem.
The tangent of an angle is
opp. leg
adj. leg
Solve for a.

17. Step 3 Use the apothem and the given side length to find the area.
11.2 Areas of Regular Polygons
Example 3B Continued
Step 3 Use the apothem and the given side length to find the area.
Area of a regular polygon
The perimeter is 5(12) = 60 ft.
Simplify. Round to the nearest tenth.
A  cm2

18. Find the area of a regular octagon with a side length of 4 cm.
11.2 Areas of Regular Polygons
Check It Out! Example 3
Find the area of a regular octagon with a side length of 4 cm.
Step 1 Draw the octagon. Draw an isosceles triangle with its vertex at the center of the octagon. The
central angle is
Draw a segment that bisects the central angle and the side of the polygon to form a right triangle.

19. Check It Out! Example 3 Continued
11.2 Areas of Regular Polygons
Check It Out! Example 3 Continued
Step 2 Use the tangent ratio to find the apothem
The tangent of an angle is
opp. leg
adj. leg
Solve for a.

20. Check It Out! Example 3 Continued
11.2 Areas of Regular Polygons
Check It Out! Example 3 Continued
Step 3 Use the apothem and the given side length to find the area.
Area of a regular polygon
The perimeter is 4(8) = 32cm.
Simplify. Round to the nearest tenth.
A ≈ 77.3 cm2

21. Find the area of each regular polygon to the nearest tenth.
11.2 Areas of Regular Polygons
Lesson Quiz:
Find the area of each regular polygon to the nearest tenth.
1. a regular nonagon with side length 8 cm
A ≈ cm2
2. a regular octagon with side length 9 ft
A ≈ ft2