9.4 Areas of Regular 9.4 Areas of Regular PolygonsPolygons

February 6, 2008February 6, 2008

DefinitionsDefinitions (associated with regular (associated with regular polygons only)polygons only)

Center of a polygonCenter of a polygon – the center of – the center of its circumscribed circle.its circumscribed circle.

Radius of a polygonRadius of a polygon – the radius of its – the radius of its circumscribed circle, or the circumscribed circle, or the distance from the center to a distance from the center to a vertex.vertex.

Apothem of a polygonApothem of a polygon – distance – distance from the center to any side of the from the center to any side of the polygon.polygon.

The center of circle A is:The center of circle A is:

AA

The center of pentagon The center of pentagon BCDEF is:BCDEF is:

AA

A radius of circle A is:A radius of circle A is:

AFAF

A radius of pentagon A radius of pentagon BCDEF is:BCDEF is:

AFAF

An apothem of pentagon An apothem of pentagon BCDEF is:BCDEF is:

AGAG

BB

CC

DDEE

FF

AA

GG

Area of a regular polygonArea of a regular polygon

The area of a regular polygon is: The area of a regular polygon is:

A = ½ PaA = ½ PaArea Area

Perimeter Perimeter apothem apothem

Where does the equation come from?

The apothem is the height of a triangle between the center and two consecutive vertices of the polygon.

you can find the area o any regular n-gon by dividing the polygon into congruent triangles.

a

G

F

E

D C

B

A

H

Hexagon ABCDEF with center G, radius GA, and apothem GH

Ex 1: A regular octagon has an apothem Ex 1: A regular octagon has an apothem of 4 in. Side length is 3. Find its area.of 4 in. Side length is 3. Find its area.

aa44

A = ½ PaA = ½ Pa

= ½ (24)(4)= ½ (24)(4)

Step 1: Find Perimeter

3 x 8 = 24

Step 2: Use Equation

Ex. 2: Finding the area of a regular polygon

A regular pentagon is inscribed in a circle with a side length of 7.5 and an apothem of 5.2. Find the area of the pentagon.

B

C

A

D