1. Areas of Regular Polygons and Composite Figures

2. Regular Polygons

3. Concept

4. Identify Segments and Angles in Regular Polygons
In the figure, pentagon PQRST is inscribed in Identify the center, a radius, an apothem, and a central angle of the polygon. Then find the measure of a central angle.
center: point X
radius: XR or XQ
apothem: XN
central angle: RXQ

5. Identify Segments and Angles in Regular Polygons
A pentagon is a regular polygon with 5 sides. Thus, the measure of each central angle of pentagon PQRST is or 72.
Answer: mRXQ = 72°

6. Example 1
In the figure, hexagon ABCDEF is inscribed in Find the measure of a central angle.
A. mDGH = 45°
B. mDGC = 60°
C. mCGD = 72°
D. mGHD = 90°

7. Area of a Regular Polygon
FURNITURE The top of the table shown is a regular hexagon with a side length of 3 feet and an apothem of 1.7 feet. What is the area of the tabletop to the nearest tenth?
Step 1 Since the polygon has 6 sides, the polygon can be divided into 6 congruent isosceles triangles, each with a base of 3 ft and a height of 1.7 ft.

8. Area of a Regular Polygon
Step 2 Find the area of one triangle.
Area of a triangle
b = 3 and h = 1.7
Simplify.
= 2.55 ft2
Step 3 Multiply the area of one triangle by the total number of triangles.

9. Area of a Regular Polygon
Since there are 6 triangles, the area of the table is ● 6 or 15.3 ft2.
Answer: 15.3 ft2

10. Example 2
UMBRELLA The top of an umbrella shown is a regular hexagon with a side length of 2 feet and an apothem of 1.5 feet. What is the area of the entire umbrella to the nearest tenth?
A. 6 ft2
B. 7 ft2
C. 8 ft2
D. 9 ft2

11. Example 3A
A. Find the area of the regular hexagon. Round to the nearest tenth.
Step 1 Find the measure of a central angle.
A regular hexagon has 6 congruent central angles, so

12. Example 3A Step 2 Find the apothem.
Apothem PS is the height of isosceles ΔQPR. It bisects QPR, so mSPR = It also bisects QR, so SR = 2.5 meters.
ΔPSR is a 30°-60°-90° triangle with a shorter leg that measures 2.5 meters, so

13. Example 3A Step 3 Use the apothem and side length to find the area.
Area of a regular polygon
≈ 65.0 m Use a calculator.
Answer: about 65.0 m2

14. Example 3B
B. Find the area of the regular pentagon. Round to the nearest tenth.

15. Example 3B
Step 2 Apothem CD is the height of isosceles ΔBCA. It bisects BCA, so mBCD = 36. Use trigonometric ratios to find the side length and apothem of the polygon.
AB = 2DB or 2(9 sin 36°). So, the pentagon’s perimeter is 5 ● 2(9 sin 36°). The length of the apothem CD is 9 cos 36°.

16. Example 3B Step 3 Area of a regular polygon
a = 9 cos 36° and P = 10(9 sin 36°)
Use a calculator.
Answer: cm2

17. Example 3
A. Find the area of the regular hexagon. Round to the nearest tenth.
A m2
B m2
C m2
D m2

18. Example 3
B. Find the area of the regular pentagon. Round to the nearest tenth.
A m2
B m2
C m2
D m2

19. Composite Figures

20. Example 4
POOL The dimensions of an irregularly shaped pool are shown. What is the area of the surface of the pool?
The figure can be separated into a rectangle with dimensions 16 feet by 32 feet, a triangle with a base of 32 feet and a height of 15 feet, and two semicircles with radii of 8 feet.

21. Example 4 Area of composite figure  953.1
Answer: The area of the composite figure is square feet to the nearest tenth.

22. Example 4
Find the area of the figure in square feet. Round to the nearest tenth if necessary.
A ft2
B ft2
C ft2
D ft2

23. Example 5 Find the area of the shaded figure.
To find the area of the figure, subtract the area of the smaller rectangle from the area of the larger rectangle. The length of the larger rectangle is or 150 feet. The width of the larger rectangle is or 70 feet.

24. Example 5 area of shaded figure
= area of larger rectangle – area of smaller rectangle
Area formulas
Substitution
Simplify.
Simplify.
Answer: The area of the shaded figure is square feet.

25. Example 5
INTERIOR DESIGN Cara wants to wallpaper one wall of her family room. She has a fireplace in the center of the wall. Find the area of the wall around the fireplace.
A. 168 ft2
B. 156 ft2
C. 204 ft2
D. 180 ft2