1. EXAMPLE 2
Find the area of a regular polygon
DECORATING
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?
SOLUTION
STEP 1
Find the perimeter P of the table top. An
octagon has 8 sides, so P = 8(15) = 120 inches.

2. √ EXAMPLE 2 Find the area of a regular polygon 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 = ≈ √

3. Find the area of a regular polygon
EXAMPLE 2
Find the area of a regular polygon
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.
ANSWER

4. EXAMPLE 3
Find the perimeter and area of a regular polygon
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

5. EXAMPLE 3
Find the perimeter and area of a regular polygon
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°).

6. EXAMPLE 3
Find the perimeter and area of a regular polygon
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
ANSWER

7. GUIDED PRACTICE
for Examples 2 and 3
Find the perimeter and the area of the regular polygon. 3.
SOLUTION
The measure of the central angle is = or 72°. Apothem a bisects the central angle, so angle is 36°. To find the lengths of the legs, use trigonometric ratios for right angle.
360 5

8. GUIDED PRACTICE
for Examples 2 and 3
sin 36° = b
hyp
sin 36° = b 8
8 sin 36° = b
The regular pentagon has side length = 2b = 2
(8 sin 36°) = sin 36° 20°
So, the perimeter is P = 5s = 5(16 sin 36°)
= 80 sin 36°
≈ 46.6 units,
and the area is A = aP = 1 2
≈ units2.

9. GUIDED PRACTICE
for Examples 2 and 3
Find the perimeter and the area of the regular polygon. 4.
SOLUTION
The regular nonagon has side length = 7.
So, the perimeter is P = 10 · s = 10 · 7 = 70 units

10. GUIDED PRACTICE
for Examples 2 and 3
The measure of central is = or 36°. Apothem a bisects the central angle, so angle is 18°. To find the lengths of the legs, use trigonometric ratios for right angle.
360 10
tan 18° =
opp
adj
tan 18° =
3.5 a
a =
3.5
tan 18°
≈10.8
and the area is A = aP = 1 2
≈ 377 units2.

11. GUIDED PRACTICE
for Examples 2 and 3 5.
SOLUTION
The measure of central angle is = 120°. Apothem a bisects the central angle, so is 60°. To find the lengths of the legs, use the trigonometric ratios.
360° 3

12. GUIDED PRACTICE
for Examples 2 and 3
cos 60° = a x
sin 60° = b 10
x cos 60° = 5 b
10 sin 60° =
x = 5
x = 10
The regular polygon has side length
s = 2 = 2 (10 sin 60°) = 20 sin 60° and apothem a = 5.

13. GUIDED PRACTICE
for Examples 2 and 3
So, the perimeter is P = 3 s = 3(20 sin 60°)
= 60 sin 60°
= units
and the area is A = aP 1 2
= × 1 2
= units2

14. GUIDED PRACTICE
for Examples 2 and 3
6. Which of Exercises 3–5 above can be solved
using special right triangles?
Exercise 5 can be solved using special right triangles.
The triangle is a Right Triangle
ANSWER