10.12. Area of Regular Polygons http://www.ck12.org
- 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.
Summary
This chapter covers perimeter and area of all the basic geometric figures. Perimeter and area are compared and
calculated for rectangles, parallelograms, triangles, and then for composite shapes of those figures. The chapter then
branches into perimeter and area for other special geometric figures, namely trapezoids, rhombuses, and kites, as
well as similar polygons. The chapter wraps up with the circumference of circles and arc length followed by the area
of a circle and the area of sectors and segments.