1. Areas of Regular Polygons and Composite Figures
LESSON 11–4
Areas of Regular Polygons and Composite Figures

2. Five-Minute Check (over Lesson 11–3) TEKS Then/Now New Vocabulary
Example 1: Identify Segments and Angles in Regular Polygons
Example 2: Real-World Example: Area of a Regular Polygon
Key Concept: Area of a Regular Polygon
Example 3: Use the Formula for the Area of a Regular Polygon
Example 4: Find the Area of a Composite Figure by Adding
Example 5: Find the Area of a Composite Figure by Subtracting
Lesson Menu

3. Find the area of the circle. Round to the nearest tenth.
A ft2
B ft2
C ft2
D ft2
5-Minute Check 1

4. Find the area of the sector. Round to the nearest tenth.
A m2
B m2
C m2
D m2
5-Minute Check 2

5. Find the area of the sector. Round to the nearest tenth.
A in2
B in2
C in3
D in2
5-Minute Check 3

6. Find the area of the shaded region. Assume that the polygon is regular
Find the area of the shaded region. Assume that the polygon is regular. Round to the nearest tenth.
A units2
B units2
C units2
D units2
5-Minute Check 4

7. Find the area of the shaded region. Assume that the polygon is regular
Find the area of the shaded region. Assume that the polygon is regular. Round to the nearest tenth.
A units2
B units2
C units2
D units2
5-Minute Check 5

8. The area of a circle is 804. 2 square centimeters
The area of a circle is square centimeters. The area of a sector of the circle is square centimeters. What is the measure of the central angle that defines the sector?
A. 110°
B. 120°
C. 135°
D. 150°
5-Minute Check 6

9. Mathematical Processes G.1(E), G.1(B)
Targeted TEKS
G.11(A) Apply the formula for the area of regular polygons to solve problems using appropriate units of measure.
G.11(B) Determine the area of composite two-dimensional
figures comprised of a combination of triangles, parallelograms,
trapezoids, kites, regular polygons, or sectors of circles to solve problems using appropriate units of measure.
Also addresses G.10(B).
Mathematical Processes
G.1(E), G.1(B)
TEKS

10. Find areas of regular polygons.
You used inscribed and circumscribed figures and found the areas of circles.
Find areas of regular polygons.
Find areas of composite figures.
Then/Now

11. center of a regular polygon radius of a regular polygon apothem
central angle of a regular polygon
composite figure
Vocabulary

12. 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
Example 1

13. 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°
Example 1

14. 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°
Example 1

15. 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.
Example 2

16. Step 2 Find the area of one triangle.
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.
Example 2

17. 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
Example 2

18. 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
Example 2

19. Concept

20. A. Find the area of the regular hexagon. Round to the nearest tenth.
Use the Formula for the Area of a Regular Polygon
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
Example 3A

21. Use the Formula for the Area of a Regular Polygon
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
Example 3A

22. Step 3 Use the apothem and side length to find the area.
Use the Formula for the Area of a Regular Polygon
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
Example 3A

23. B. Find the area of the regular pentagon. Round to the nearest tenth.
Use the Formula for the Area of a Regular Polygon
B. Find the area of the regular pentagon. Round to the nearest tenth.
Step 1 A regular pentagon has 5 congruent central angles, so
Example 3B

24. Use the Formula for the Area of a Regular Polygon
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°.
Example 3B

25. Step 3 Area of a regular polygon
Use the Formula for the Area of a Regular Polygon
Step 3 Area of a regular polygon
a = 9 cos 36° and P = 10(9 sin 36°)
Use a calculator.
Answer: cm2
Example 3B

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

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

28. Find the Area of a Composite Figure by Adding
POOL The dimensions of an irregularly shaped pool are shown. What is the area of the surface of the pool?
ft2
ft2
953.1 ft2
852.5 ft2
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.
Example 4

29. Area of composite figure
Find the Area of a Composite Figure by Adding
Area of composite figure 
Answer: The area of the composite figure is square feet to the nearest tenth. The answer is C.
Example 4

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

31. Find the area of the shaded figure.
Find the Area of a Composite Figure by Subtracting
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.
Example 5

32. = area of larger rectangle – area of smaller rectangle
Find the Area of a Composite Figure by Subtracting
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.
Example 5

33. INTERIOR DESIGN Cara wants to wallpaper one wall of her family room
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
Example 5

34. Areas of Regular Polygons and Composite Figures
LESSON 11–4
Areas of Regular Polygons and Composite Figures