CK-12 Geometry-Concepts

10.12. Area of Regular Polygons http://www.ck12.org

1. The octagon can be split into 8 congruent triangles. So, if we find the area of one triangle and multiply it by 8, we
   will have the area of the entire octagon.

Aoctagon= 8

(

1

2

· 12 · 14. 49

)

= 695. 52 units^2

2. 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.

3. 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

Explore More

Use the regular hexagon below to answer the following questions. Each side is 10 cm long.