CK-12 Geometry - Second Edition

http://www.ck12.org Chapter 10. Perimeter and Area

tan 67. 5 ◦=

AB

6

AB= 6 ·tan 67. 5 ◦≈ 14. 49

The apothem is used to find the area of a regular polygon. Let’s continue with Example 3.

Example 4:Find the area of the regular octagon in Example 3.

Solution: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

From Examples 3 and 4, we can derive a formula for the area of a regular polygon.

Theareaofeachtriangleis:A 4 =^12 bh=^12 sa, wheresis the length of a side andais the apothem.

If there arensides in the regular polygon, then it is made up ofncongruent triangles.

A=nA 4 =n

(

1

2

sa

)

=

1

2

nsa

In this formula we can also substitute the perimeter formula,P=ns, fornands.

A=

1

2

nsa=

1

2

Pa

Area of a Regular Polygon: If there arensides with lengthsin a regular polygon andais the apothem, then
A=^12 asnorA=^12 aP, wherePis the perimeter.

Example 5:Find the area of the regular polygon with radius 4.