Math 212
AREA OF REGULAR POLYGONS
Notes 39
C. Brownell
Make sure you understand the difference between perimeter and area. This is a common mistake.
Perimeter is the total distance around a shape. Think of a fence. Area involves covering a surface
like a carpet.
You already know how to calculate the area of many different shapes; rectangles, triangles,
trapezoids, rhomboids(diamonds), and circles.
rectangle area
trapezoid
area
= bxh
triangle
area
= h(b + B)
2
= bxh
2
rhombus
area
parallelogram
area
= Dxd
2
circle =
area
= bxh
πr2
To calculate the area of a regular polygon, you
need to know understand the word “apothem”.
Apothem – An apothem is a line segment that drops from the center of a regular polygon and is
perpendicular to any one side. (It is like the height of a triangle). It DOES NOT go to a vertex
(corner).
Here is the formula to find the area of a regular polygon:
(𝑝𝑒𝑟𝑖𝑚𝑒𝑡𝑒𝑟 )(𝑎𝑝𝑜𝑡ℎ𝑒𝑚)
2
= 𝑟𝑒𝑔𝑢𝑙𝑎𝑟 𝑝𝑜𝑙𝑦𝑔𝑜𝑛 𝑎𝑟𝑒𝑎
p(ap) = A
2
*Remember that the perimeter of a regular polygon is calculated by multiplying the length of
one side times the number of sides.
Example 1: A regular pentagon has side lengths of 7 cm each. The apothem has a length of 8.5 cm.
Find the area of this shape.
perimeter x apothem = area
2
I need the perimeter.
5 sides X 7 cm = 35 cm perimeter.
35 x 8.5 = area
2
297.5 = area
2
area = 148. 75 cm2
Math 212
AREA OF REGULAR POLYGONS
Notes 39
Here are some examples of how to solve questions when you are given the area of the regular
polygon.
Example 2: The area of a regular polygon is 70 cm2. The perimeter is 28 cm. Find the apothem
length.
perimeter x apothem = area
2
28(ap) = 70
2
28(ap) = 70
2
1
Now cross-multiply.
28ap = 140
Divide both sides by 28
apothem = 5 cm
Example 3: The area of a regular polygon is 267.75 cm2. The apothem is 8.5 cm long. Each side
is 7 cm long. What is the name of this figure?
perimeter x apothem = area
2
perimeter x 8.5 = 267.75
2
I have the area. I need to work backwards.
perimeter x 8.5 = 267.75
2
1
Now crosss-multiply.
8.5(perimeter) = 535.5
perimeter = 63 cm
Now divide both sides by 8.5
Now go back to the question. You are asked
to find the name of the figure.
63 cm perimeter ÷ 7 cm per side = 9 sides
This is a NONAGON!