1. Sec. 10 – 3 Area of Regular Polygons
Objective:
1) To find the area of a regular polygon.

2. A = Area of 1 triangle • # of triangles
= ( ½ • apothem • side length s) • # of sides
= ½ • apothem • # of sides • side length s
= ½ • apothem • perimeter of a polygon
This approach can be used to find the area of any regular polygon. F A H a E G B D C
Hexagon ABCDEF with center G, radius GA, and apothem GH

3. ** Regular Polygon – All sides  All s 
Vertex s
Regular Pentagon inscribed inside of a circle.
Center of circle and the reg. pentagon.
Radius of circle and the reg. pentagon.
** 5  isosceles Δs
Apothem -  distance from the center to a side.

4. Find the missing  measures.
# of sides
Vertex  =
360 n
m1 = 360/n
= 360/5
= 72° 1 2
Apothem bisects vertex  of the Is. Δ formed by the radii. 3
m2 + m = 180
36 + m3 +90 = 180
m3 = 54
m2 = ½ m1
m2 = ½ (72)
m2 = 36

5. Finding the area of a reg. polygon?
Suppose reg. n-agon w/ side s
The radii divides the figure into n  isos Δs each w/ area = ½ as s
Apothem Side
p = ns
Area of n-gon = n • ½ as
Perimeter of n-gon = ns
Perimeter of reg. Polygon
# of sides
Length of sides

6. The area formula for any reg. Polygon
A = ½ ap
Apothem
Area of any reg. Polygon
Perimeter of reg. polygon

7. Example 2: find the area of a regular decagon with a 12
Example 2: find the area of a regular decagon with a 12.3in apothem and an 8in side.
p = ns
= (10)(8cm)
= 80cm
A = ½ ap
= ½ (12.3in)(80in)
= 492in2

8. Example 3: Find the apothem of a reg. hexagon with sides of 10mm.60 2 3
a = √3 • short side
a = √3 • (5mm)
a = 5√3 = 8.66 30
Δ shortcut a 60
Could you find the area of it now??
5mm

9. Example 4: Find the area of a regular pentagon with 7
Example 4: Find the area of a regular pentagon with 7.2ft sides and a 6.1ft radius.
p = ns
= (5)(7.2ft)
= 36ft
A = ½ ap
= ½ (4.92ft)(36ft)
= 88.6ft2
6.1ft
6.1ft a
a2 + b2 = c2
a = 6.12
a = 37.21
a2 = 24.25
a = 4.92
3.6 ft

10. What is the difference between the radius and the apothem?
The radius is longer because it is from the center all the way to
the edge of the circle but the apothem is just from the center to
the middle of the side
What is the formula for the area of a regular polygon?
A = ½asn
A = ½aP