11.6 Areas of Regular Polygons
Hubarth
Geometry
Apothem of a polygon is the height of an isosceles triangle that has two radii as legs
P
N
M
Q
𝑎𝑝𝑜𝑡ℎ𝑒𝑚
𝑃𝑄
Theorem 11.11
Area of a Regular Polygon
The area of a regular n-gon with side length s is one half the product of the apothem a and
the perimeter p
1
1
𝐴 = 𝑎𝑃, 𝑜𝑟 𝐴 = 𝑎 ∙ 𝑛𝑠
2
2
a
s
Ex 1 Find Angle Measures in a Regular Polygons
In the diagram, ABCDE is a regular pentagon inscribed
in
F. Find each angle measure.
a. m∠AFB
b. m∠AFG
c. m∠GAF
a.
∠AFB is a central angle, so m∠AFB =
360
5
or 72°.
b. FG is an apothem, which makes it an altitude of isosceles
1
∆AFB. So, FG bisects ∠AFB and m∠AFG = 2m∠AFB = 36°.
c. The sum of the measures of right ∆GAF is 180°.
So, 90° + 36° + m∠GAF = 180°, and m∠GAF = 54°.
Ex 2 Find the Area of a Regular Polygon
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?
Find the perimeter P of the table top. An octagon has 8 sides,
so P = 8(15) = 120 inches.
Find the apothem a. The apothem is height RS of ∆PQR. Because ∆PQR is isosceles,
altitude 𝑅𝑆 bisects 𝑄𝑃.
1
2
1
2
So, QS = (QP) = (15) = 7.5 inches.
To find RS, use the Pythagorean Theorem for ∆ RQS.
a = RS =
19.62 − 7.52 = 327.91 ≈ 18.11
Ex 2 continued
Find the area A of the table top.
1
A = 2aP
≈
1
(18.108)(120)
2
≈ 1086.5
So, the area you are covering with tiles is about
1086.5 square inches.
Ex 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.
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.
LM
MK
sin 20° =
cos 20° =
LK
LK
sin 20° =
MK
4
4 sin 20° = MK
cos 20° =
4
LM
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,
1
1
and the area is A = 2aP = 2(4 ∙ cos 20°)(72 ∙ sin 20°) ≈ 46.3 square units.
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.
center= P, radius = 𝑃𝑌or𝑋𝑃,apothem = 𝑃𝑄, central angle = XPY.
2. Find m∠XPY, m∠XPQ, and m∠PXQ.
𝑚∠𝑋𝑃𝑌 = 90, 𝑚∠𝑋𝑃𝑄 = 45, 𝑚∠𝑃𝑋𝑄 = 45
Find the perimeter and the area of the regular polygon.
3.
4.
P= 46.6 units,
A = 151.5 units2
5.
P = 70 units,
A = 377.0 units2
𝑃 = 30 3 = 52,
𝐴 = 129.9 𝑢𝑛𝑖𝑡𝑠 2