http://www.ck12.org Chapter 10. Perimeter and Area
Practice
Use the regular hexagon below to answer the following questions. Each side is 10 cm long.
- Each dashed line segment isa(n)____.
- The red line segment isa(n)__.
- There are _____ congruent triangles in a regular hexagon.
- In a regular hexagon, all the triangles are _____.
- Find the radius of this hexagon.
- Find the apothem.
- Find the perimeter.
- Find the area.
Find the area and perimeter of each of the following regular polygons. Round your answer to the nearest hundredth.
9.
10.
11.
12.- If the perimeter of a regular decagon is 65, what is the length of each side?
- A regular polygon has a perimeter of 132 and the sides are 11 units long. How many sides does the polygon
have? - The area of a regular pentagon is 440. 44 in^2 and the perimeter is 80 in. Find the length of the apothem of the
pentagon. - The area of a regular octagon is 695. 3 cm^2 and the sides are 12 cm. What is the length of the apothem?
A regular 20-gon and a regular 40-gon are inscribed in a circle with a radius of 15 units.
17.ChallengeDerive a formula for the area of a regularhexagon with sides of lengths. Your only variable will
bes. HINT: Use 30-60-90 triangle ratios.
18.Challengein the following steps you will derive an alternate formula for finding the area of a regular polygon
withnsides. We are going to start by thinking of a polygon withnsides asncongruent isosceles triangles.
We will find the sum of the areas of these triangles using trigonometry. First, the area of a triangle is^12 bh. In
the diagram to the right, this area formula would be^12 sa, wheresis the length of a side andais the length of
the apothem. In the diagram,xrepresents the measure of the vertex angle of each isosceles triangle.
(a) The apothem,a, divides the triangle into two congruent right triangles. The top angle in each isx◦
2. Find
sin(x◦
2)
and cos(x◦
2)
.
(b) Solve your sin equation to find an expression forsin terms ofrandx.
(c) Solve your cos equation to find an expression forain terms ofrandx.
(d) Substitute these expressions into the equation for the area of one of the triangles,^12 sa.
(e) Since there will bentriangles in an n-gon, you need to multiply your expression from part d bynto get
the total area.
(f) How would you tell someone to find the value ofxfor a regular n-gon?Use the formula you derived in problem 18 to find the area of the regular polygons described in problems 19-22.
Round your answers to the nearest hundredth.
- Decagon with radius 12 cm.
- 20-gon with radius 5 in.
- 15-gon with radius length 8 cm.
- 45-gon with radius length 7 in.