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Warm-Up Exercises
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
2. Solve 18 = 12
ANSWER
36
x.
Warm-Up Exercises
3. Evaluate (4 cos 10º)(10 sin 20º).
ANSWER
4. Evaluate
ANSWER
13.47
10 .
tan 88º
0.3492
Find angle measures in a regular polygon
Warm-Up1Exercises
EXAMPLE
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
72°.
°
360
AFB =
, or
5
Find angle measures in a regular polygon
Warm-Up1Exercises
EXAMPLE
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 = 1 m AFB = 36°.
2
Find angle measures in a regular polygon
Warm-Up1Exercises
EXAMPLE
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°.
Warm-Up
Exercises
GUIDED
PRACTICE
for Example 1
In the diagram, WXYZ is a square inscribed in
1.
Identify the center, a radius,
an apothem, and a central
angle of the polygon.
ANSWER
P, PY or XP, PQ, XPY.
P.
Warm-Up
Exercises
GUIDED
PRACTICE
2.
Find m
ANSWER
90°, 45°, 45°
XPY, m
for Example 1
XPQ, and m
PXQ.
Find the area of a regular polygon
Warm-Up2Exercises
EXAMPLE
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.
Find the area of a regular polygon
Warm-Up2Exercises
EXAMPLE
STEP 2 Find the apothem a. The apothem is height
RS of ∆PQR. Because ∆PQR is isosceles,
altitude RS bisects QP .
So, QS = 1 (QP) = 1 (15) = 7.5 inches.
2
2
To find RS, use the Pythagorean
Theorem for ∆ RQS.
a = RS ≈ √19.62 – 7.52 = √327.91 ≈ 18.108
Find the area of a regular polygon
Warm-Up2Exercises
EXAMPLE
STEP 3 Find the area A of the table top.
1
Formula for area of regular polygon
A = 2 aP
1
≈ 2 (18.108)(120)
Substitute.
≈ 1086.5
Simplify.
ANSWER
So, the area you are covering with tiles is about
1086.5 square inches.
Find the perimeter and area of a regular polygon
Warm-Up3Exercises
EXAMPLE
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 9 , 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.
Find the perimeter and area of a regular polygon
Warm-Up3Exercises
EXAMPLE
4
sin 20° = MK
LK
LM
cos 20° =
LK
sin 20° = MK
4
cos 20° = LM
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°.
Find the perimeter and area of a regular polygon
Warm-Up3Exercises
EXAMPLE
ANSWER
So, the perimeter is P = 9s = 9(8 sin 20°) = 72 sin 20°
≈ 24.6 units,
1
1
and the area is A =
aP =
(4 cos 20°)(72 sin 20°)
2
2
≈ 46.3 square units.
Warm-Up
Exercises
GUIDED
PRACTICE
for Examples 2 and 3
Find the perimeter and the area of the regular polygon.
3.
ANSWER
about 46.6 units, about 151.5 units2
Warm-Up
Exercises
GUIDED
PRACTICE
for Examples 2 and 3
Find the perimeter and the area of the regular polygon.
4.
ANSWER
70 units, about 377.0 units2
Warm-Up
Exercises
GUIDED
PRACTICE
for Examples 2 and 3
Find the perimeter and the area of the regular polygon.
5.
ANSWER
30 3  52.0 units, about 129.9 units2
Warm-Up
Exercises
GUIDED
PRACTICE
6.
for Examples 2 and 3
Which of Exercises 3–5 above can be solved
using special right triangles?
ANSWER
Exercise 5
Warm-Up
Exercises
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
Warm-Up
Exercises
Daily
Homework
Quiz
3.
ANSWER 374.1 cm2
Warm-Up
Exercises
Daily
Homework
Quiz
Find the perimeter and area of each regular polygon.
4.
ANSWER 99.4 in. ; 745.6 in.2
Warm-Up
Exercises
Daily
Homework
Quiz
5.
ANSWER 22.6 m; 32 m2