1. Areas of Regular Polygons
Warm Up
Lesson Presentation
Lesson Quiz

2. Warm-Up
1. An isosceles triangle has side lengths 20 meters,
26 meters, and 26 meters. Find the length of the
altitude to the base.
ANSWER
24 m 1 2
2. Solve 18 = x.
ANSWER 36

3. Warm-Up
3. Evaluate (4 cos 10º)(10 sin 20º).
ANSWER
13.47
4. Evaluate 10
tan 88º
0.3492
ANSWER

4. Example 1
In the diagram, ABCDE is a regular pentagon inscribed in F. Find each angle measure.
a. m AFB
SOLUTION
a. AFB is a central angle, so m AFB = , or 72°.
360° 5

5. Example 1
In the diagram, ABCDE is a regular pentagon inscribed in F. Find each angle measure.
b. m AFG
SOLUTION
b. FG is an apothem, which makes it an altitude of isosceles ∆AFB. So, FG bisects AFB and m AFG = m AFB = 36°. 1 2

6. Example 1
In the diagram, ABCDE is a regular pentagon inscribed in F. Find each angle measure.
c. m GAF
SOLUTION
c. The sum of the measures of right ∆GAF is 180°.
So, 90° + 36° + m GAF = 180°, and m GAF = 54°.

7. Guided Practice
In the diagram, WXYZ is a square inscribed in P.
1. Identify the center, a radius, an apothem, and a central angle of the polygon.
P, PY or XP, PQ, XPY
ANSWER
2. Find m XPY, m XPQ, and m PXQ.
90°, 45°, 45°
ANSWER

8. Example 2
You are decorating the top of a table by covering it with small ceramic tiles. The table top is a regular octagon with 15 inch sides and a radius of about 19.6 inches. What is the area you are covering?
DECORATING
SOLUTION
STEP 1
Find the perimeter P of the table top. An
octagon has 8 sides, so P = 8(15) = 120 inches.

9. Example 2
STEP 2
Find the apothem a. The apothem is height RS of ∆PQR. Because ∆PQR is isosceles, altitude RS bisects QP .
So, QS = (QP) = (15) = 7.5 inches. 1 2
To find RS, use the Pythagorean
Theorem for ∆ RQS.
a = RS ≈ √19.62 – 7.52 = ≈ √

10. Find the area A of the table top. 1 2 A = aP
Example 2
STEP 3
Find the area A of the table top. 1 2
A = aP
Formula for area of regular polygon
≈ (18.108)(120) 1 2
Substitute. ≈
Simplify.
So, the area you are covering with tiles is about
square inches.

11. Example 3
A regular nonagon is inscribed in a circle with radius 4 units. Find the perimeter and area of the nonagon.
SOLUTION
360°
The measure of central JLK is , or 40°. Apothem LM bisects the central angle, so m KLM is 20°. To find the lengths of the legs, use trigonometric ratios for right ∆ KLM. 9

12. Example 3
sin 20° = MK LK
cos 20° = LM LK
sin 20° = MK 4
cos 20° = LM 4
4 sin 20° = MK
4 cos 20° = LM
The regular nonagon has side length
s = 2MK = 2(4 sin 20°) = 8  sin 20° and apothem
a = LM = 4  cos 20°.
So, the perimeter is P = 9s = 9(8 sin 20°) = 72 sin 20° ≈ 24.6 units,
and the area is A = aP = (4 cos 20°)(72 sin 20°) ≈ 46.3 square units. 1 2

13. Guided Practice
Find the perimeter and the area of the regular polygon. 3.
about 46.6 units, about units2
ANSWER

14. Guided Practice
Find the perimeter and the area of the regular polygon. 4.
70 units, about units2
ANSWER

15. Guided Practice
Find the perimeter and the area of the regular polygon. 5.
30 3  52.0 units, about units2
ANSWER

16. Guided Practice
6. Which of Exercises 3–5 above can be solved
using special right triangles?
Exercise 5
ANSWER

17. Lesson Quiz
1. Find the measure of the central angle of a regular polygon with 24 sides.
ANSWER
15°

18. Lesson Quiz
Find the area of each regular polygon. 2.
ANSWER
110 cm2 3.
ANSWER
374.1 cm2

19. Lesson Quiz
Find the perimeter and area of each regular polygon. 4.
ANSWER
99.4 in. ; in.2 5.
ANSWER
22.6 m; 32 m2