Areas of Regular Polygons and Circles
In the picture to the right, polygon ABCDEF is inscribed in Circle G. AG and FG are radii for the cirle and GH is the apothem of the polygon. An apothem is the segment from the center of the polygon perpendicular to one of the sides.
= possum
1
Aregular polygon = Pa
2
P = perimeter of polygon
a = apothem
1
Cubs lose . . . Cubs lose
Robbie Schultz goes into hiding!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
2
Ex: Find the area of a regular pentagon with a perimeter
of 40 centimeters.
First find angle 1. 360 ÷ 5 = 72.
This is the exterior angle of a regular pentagon. So, the interior angle is 108. Divide 108 in half to find
angle 1. So, angle 1 is 54.
The perimeter of the pentagon is 40,
so each side is 8. Divide that by 2
and x = 4. To find the apothem (a),
use the trig function tangent.
a
1
x
a
Tan 54 = . This makes a = 5.5.
4
To find the area:
1
A = a P
2
1
A = (5.5)(40)
2
A = 110 3
Acircle = π r2
In Class ­ Page 613 (4,5).
4
Page 613
Ashaded region = Acircle ­ Asquare
Acircle = π 102 = 314.2
1
Asquare = 20 20 = 200 Remember, a square is also a rhombus.
2
Ashaded region = 314.2 ­ 200 = 114.2
5
Page 613
x
1
A = Atriangle ­ Acircle
1
Atriangle = a P
2
1
Atriangle = (3.6)(37.4) = 67.3
2
In an equilateral triangle, each angle is 60 degrees.
Therefore, angle 1 is 30 degrees, and you have a
30­60­90 triangle. Take 3.6 (the short leg) and multiply by √3 to get x. Multiply x by 2 to get the
length of one side, then multiply that by 3 to get the
perimeter.
Acircle = π (3.6)2 = 40.7
A = 67.3 ­ 40.7 = 26.6
6
Page 613
1
x
a
In a regular hexagon, 360 ÷ 6 = 60 which is the measure of an exterior angle. Therefore, 2
each interior angle is 120. Divide that by 2 to Acircle = π (10) = 314.2
get angle 1. So we have a 30­60­90 triangle. In that triangle the hypotenuse is 10, so x is 5 1
Ahexagon = a P
and a is 5√3. This makes the perimeter 60.
2
1
Ahexagon = (5
√3) (60) = 150√3
2
A = Acircle ­ Ahexagon
A = 314.2 ­ 259.8 = 54.4 in2
7
Page 615
A = Acircle ­ Atriangle
2
Acircle = π r
To get the height of the triangle, multiple 1.5 by √3.
Acircle = π (3.5)2 = 38.5
1
Atriangle = b h
2
1
Atrangle = (3) (2.6) = 3.9
2
A = 38.5 ­ 3.9 = 34.6
h
60
1.5
8
Page 615
A = Asquare ­ Afour circles
Asquare = 6 6 = 36
Aone circle = π (1.5)2 = 7.07
Afour circles = (7.07) (4) = 28.28
A = 36 ­ 28.28 = 7.7
9
Page 616
Did you notice that this is a rhombus?
1 d
A = d
1 2
2
1
A = (20) (26)
2
A = 260 cm2
10