7.4 Notes
Area/Perimeter of Regular Polygons
Name: _________________________
Zac is helping his family to design and build a gazebo. The gazebo will be the shape of a regular polygon. As
part of the design, Zac will need to calculate several things so that his parents can purchase the right amount
of lumber for the construction. Meaning, he will need to calculate the perimeter of gazebo to order enough
railing, the area of the floor to buy enough lumber for the base, and the surface area of the pyramidal roof to
cover the gazebo. The problem is that his parents keep changing the design of the gazebo - from a hexagon
(____ sides), octagon (____ sides), decagon (____ sides), dodecagon (____ sides), or even some other type of
n-gon (____ sides).
But, since Zac knows that all regular polygons are cyclic, meaning they can be inscribed in a circle, he is
wondering how he can use this idea to help him find the perimeter and area of all regular polygons.
For his first attempt at creating a drawing of the gazebo, Zac has inscribed a regular hexagon inside a circle
and given it a radius of 2 inches. He is wondering if this is enough information to find the perimeter and area
of his regular hexagon.
1. To get started with the task of finding both the area and perimeter of the gazebo, Zac decides to
write down what he already knows about the figure. Decide if you agree or disagree with each
statement. Justify your response.
What Zac thinks he knows:
Agree/Disagree? Justify.
Two radii drawn to two consecutive vertices of the
regular hexagon form a central angle whose measure
can be found based on the rotational symmetry of the
figure.
The hexagon can be decomposed into 6 congruent
isosceles triangles.
The length of the altitudes of each of these 6
congruent triangles (the segment drawn from the
vertex of the circle, perpendicular to a side of the
polygon) can be found using trig.
The length of the sides of the triangle that form the
chords of the circle can be found, also, using trig.
2. Now, based on what you and Zac know to be true,
find the perimeter of the hexagon with radius of 2 ft.
3. Find the area of the hexagon that Zac has inscribed in
the circle with a radius of 2 ft.
4. What if Zac had inscribed an octagon inside a circle and kept
the same radius of 2 inches? Modify your strategy to find the
perimeter and area of the octagon.
P=
A=
5. Modify your strategy to find the perimeter and area of any regular n-gon inscribed in a circle of any given
radius.
n=9
P = _________________________________________
n = 22
A = _________________________________________