CK-12 Geometry - Second Edition

10.6. Area and Perimeter of Regular Polygons http://www.ck12.org

Solution:In this problem we need to find the apothem and the length of the side before we can find the area of the
entire polygon. Each central angle for a regular pentagon is^360
◦
5 =^72
◦. So, half of that, to make a right triangle with

the apothem, is 36◦. We need to use sine and cosine.

sin 36◦=

. 5 n
4
cos 36◦=

a

4

4 sin 36◦=

1

2

n 4 cos 36◦=a

8 sin 36◦=n a≈ 3. 24

n≈ 4. 7

Using these two pieces of information, we can now find the area.A=^12 ( 3. 24 )( 5 )( 4. 7 )≈ 38. 07 units^2.

Example 6:The area of a regular hexagon is 54

√

3 and the perimeter is 36. Find the length of the sides and the
apothem.

Solution:Plug in what you know into both the area and the perimeter formulas to solve for the length of a side and
the apothem.

P=sn A=

1

2

aP

36 = 6 s 54

√

3 =

1

2

a( 36 )

s= 6 54

√

3 = 18 a

3

√

3 =a

Know What? RevisitedFrom the picture to the right, we can see that the total distance across the Pentagon is the
length of the apothem plus the length of the radius. If the total area of the Pentagon is 34 acres, that is 2,720,000
square feet. Therefore, the area equation is 2720000=^12 a( 921 )( 5 )and the apothem is 590.66 ft. To find the radius,
we can either use the Pythagorean Theorem, with the apothem and half the length of a side or the sine ratio. Recall
from Example 5, that each central angle in a pentagon is 72◦, so we would use half of that for the right triangle.

sin 36◦=

460. 5

r

→r=

460. 5

sin 36◦

≈ 783. 45 f t.

Therefore, the total distance across is 590. 66 + 783. 45 = 1374. 11 f t.