Finding the Area of Regular Polygons
1) Divide the polygon into n triangles
2) Calculate the value of each central angle
360
n
a. x° =
3) Calculate the base and height of the triangle
a. Draw the apothem. Using one of the right triangles this creates, use SOHCAHTOA and the
provided dimension to determine the length of the apothem and side.
b. The radius may be used in the SOHCAHTOA calculations, but if it isn’t given it’s not necessary to
find as it has no bearing on the area of the triangle
c. The apothem will be the height and the base will be the side length
4) Calculate the area of each triangle
a. Areatriangle =
b. Areatriangle =
1
2
1
2
∙ base ∙ height
∙ side ∙ apothem
5) Multiply by n
Areapolygon =
1
∙ side ∙ apothem ∙ n
2
Apothem – perpendicular distance from the center to the side (bisects the side)
Radius – distance from the center to a vertex. If a circle is inscribed around the
polygon, this line would be the radius of that circle
Central angle – formed by two consecutive radii
SOH CAH TOA
sin θ =
opposite
hypotenuse
cos θ =
adjacent
hypotenuse
tan θ =
opposite
adjacent
Hypotenuse
Opposite
θ
Adjacent
Overly Detailed Example with a Regular Hexagon
1) Divide the hexagon into 6 triangles
2) Calculate the value of the central angle
a. φ =
φ
360
n
3) Calculate the base and height of one triangle
a. Draw the apothem and consider one of the right
triangles this creates
θ
The apothem bisects the vertex, so θ =
-
Depending on which dimension is labeled (apothem, radius, or side) use the
relevant trig function relationships to find the missing peices
If it helps to use another angle, you can find the third angle (the base angles of the
isosceles triangle) by
δ
apothem
δ
φ
2
-
-
θ
θ
180−φ
2
= δ or 90 − θ = δ
-
sin θ =
side⁄
2
radius
side = 2 ∙ radius ∙ sin θ
radius
-
cos θ =
apothem
radius
apothem = radius ∙ cos θ
δ
-
tan θ =
side⁄
2
apothem
side = 2 ∙ apothem ∙ tan θ
side
2
-
tan δ =
apothem
side⁄
2
apothem =
For a Regular Hexagon:
φ=
360
= 60°
6
θ=
60
= 30°
2
δ=
180 − 60
= 90 − 30 = 60°
2
If we’re given the apothem, we need the side
- apothem = 8
- side = 2 ∙ apothem ∙ tan θ
-
side = 2 ∙ 8 ∙
-
side ≈ 9.24
√3
3
If we’re given the side, we need the apothem
- side = 10
side
2
10
∙
2
-
apothem =
-
apothem =
-
apothem ≈ 8.66
∙ tan δ
√3
Areahexagon =
side
2
∙ tan δ
sin
cos
tan
30°
1
2
√3
2
√3
3
60°
√3
2
1
2
√3
If we’re given the radius, we need both the
apothem and the side
- radius = 12
- side = 2 ∙ radius ∙ sin θ
1
-
side = 2 ∙ 12 ∙ 2
-
side = 12
apothem = radius ∙ cos θ
-
apothem = 12 ∙
-
apothem = 10.39
1
∙ side ∙ apothem ∙ n
2
√3
2