Name_________________________________________________ Date______________ Period_____________
Developing Formulas for Regular Polygons
Area of Regular Polygons
The area of a regular
polygon with apothem
a and perimeter P
1
is A  aP.
2
The apothem is
the distance from
the center to a side.
The center is
equidistant from
the vertices.
Find the area of a regular hexagon with side length 10 cm.
Step 1
Draw a figure and find the measure of a central angle. Each central
360
angle measure of a regular n-gon is
.
n
A central angle has its
vertex at the center. This
central angle measure is
360
 60.
n
Step 2
Use the tangent ratio to find the apothem. You could also use the
30°-60°-90°  Thm. in this case.
leg opposite 30 angle
Write a tangent ratio.
tan 30 
leg adjacent to 30 angle
5 cm
a
5 cm
a
tan 30
tan 30 
Step 3
Substitute the known values.
Solve for a.
Use the formula to find the area.
1
A  aP
2
1 5 
A 
60
2  tan30 
a
A  259.8 cm2
5
, P  6  10 or 60 cm
tan30
Simplify.
Find the area of each regular polygon. Round to the nearest tenth when necessary.
1.
2.
Use the formula for the area of a regular polygon to find each measurement. Round to the nearest tenth when
necessary.
3.
A regular hexagon with an apothem of 3 m
5.
8.
A regular decagon with a perimeter of 70 ft
6.
the area of the regular triangle
7.
4.
the area of the regular octagon
The side length of a regular nonagon in which A  99 in2 and a  5.5 in.
9.
Answer Key
1) A = 695.3 cm^2
2) A = 58.1 m^2
3) A = 32.2 m^2
4) A = 377.0 ft^2
5) A = 27.6 ft^2
6) A = 1086 mm^2
7) s = 4in
8) A = 1122.4 in^2
9) A = 85.6 m^2