Download Regular Polygon Area

Regular Polygon Area
GEOMETRY
NAME_________________________
DATE __________ Per.___________
1
Here is a method for finding the area of a regular polygon using the formula:
A = ap
Find the area of this regular trigon (n = 3),
2
also known as “triangle”.
a) To find the interior angle sum use the formula:
Area & Perimeter
180(n − 2) = 180(3 − 2) = 180
interior
angle
radius
s
apothem
a
r
b) To find the interior angle measure divide by n:
180(n − 2) 180(3 − 2) 180
=
=
= 60
n
3
3
c) To find the base angle measure divide by 2:
60 ÷ 2 = 30

opp 
d) To find the apothem a, use trigonometry:
sin =


a

hyp 
sin30° = ⇒ a = r sin30°
r
a
r
For r = 10, a = 10 ⋅sin30° = 10( 0.5) = 5
30°

adj 
x
 cos =

cos30° = ⇒ x = r cos30°

hyp 
r
For r = 10, x = 10 ⋅ cos30° = 10(0.866 ) = 8.66
e) To find the adjacent side x:
x
base angle
r
x 30°
f) To find the polygon side s, multiply the ∆ adjacent side by 2.
s = 2x = 2r cos30° = 2(10cos30°) = 2(8.66 ) = 17.321
g) To find the perimeter p, multiply the polygon side by n:
p = ns = 3(17.321) = 51.962
h) To find the area, substitute the apothem a and the perimeter p, into the formula:
1
1
ap = (5)(51.962) = 129.905
2
2
1. Pentagon
radius = 10
n= 5
Interior ∠ Sum = 180(n–2) = 180(5–2) =
∠ Measure = 180(5–2)/5 = 540 ÷ 5 =
Base ∠ = 180(n–2)/n ÷ 2 = 108 ÷ 2 =
Apothem = r sin 54°=10 sin 54° =
Right ∆ Base = r cos 54° = 10 cos 54° =
PolySide = 2rcos 54° = 20 cos 54° =
Perimeter = n 2rcos 54° = 5(20) cos 54° =
Polygonal Area = ap÷2 =
n=
2. Hexagon
radius = 10
Interior ∠ Sum = 180(n–2) =
∠ Measure = 180(n–2)/ n =
Base ∠ = 180(n–2)/ n ÷ 2 =
Apothem = rsin base∠ =
Right ∆ Base = r cos base∠ =
PolySide = 2rcos base∠ =
Perimeter = 2rn sin base∠ =
Polygonal Area = ap÷2 =