1. 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

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

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

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

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

6. GUIDED PRACTICE
for Example 1
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

7. GUIDED PRACTICE
for Example 1
2. Find m XPY, m XPQ, and m PXQ.
90°, 45°, 45°
ANSWER

8. 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.

9. √ 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 = ≈ √

10. 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

11. 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

12. 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°.

13. 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

14. GUIDED PRACTICE
for Examples 2 and 3
Find the perimeter and the area of the regular polygon. 3.
about 46.6 units, about units2
ANSWER

15. GUIDED PRACTICE
for Examples 2 and 3
Find the perimeter and the area of the regular polygon. 4.
70 units, about units2
ANSWER

16. GUIDED PRACTICE
for Examples 2 and 3
Find the perimeter and the area of the regular polygon. 5.
30 3  52.0 units, about units2
ANSWER

17. GUIDED PRACTICE
for Examples 2 and 3
6. Which of Exercises 3–5 above can be solved
using special right triangles?
Exercise 5
ANSWER

18. Daily Homework Quiz
1. Find the measure of the central angle of a regular polygon with 24 sides.
ANSWER
15°
Find the area of each regular polygon. 2.
ANSWER
110 cm2

19. Daily Homework Quiz 3.
ANSWER
374.1 cm2

20. Daily Homework Quiz
Find the perimeter and area of each regular polygon. 4.
ANSWER
99.4 in. ; in.2

21. Daily Homework Quiz 5.
ANSWER
22.6 m; 32 m2