A regular octagon has a radius of 12. Find each of the following to the nearest tenth.

Question

a) Its perimeter.

 

 

b) Its area.

Answer 1

regular octagon with a radius of 12

 

draw the diagram

there are 8 congruent triangles and each triangle has a central angle of 360/8=45º

use the law of cosines to find the length of one of the sides

let "c" be the length of a side of the regular octagon and "r" be the length of a radius

c²=r²+r²-2*r*r*cos(360/8)

c²=r²+r²-2*r*r*cos45º

c²=12²+12²-2*12*12*(0.707)

c²=144+144-2*144*(0.707)

c²=288-288*(0.707)

c²=288-203.616

C²=84.384

c=√84.384

c=9.186

now multiply this number by 8 to get the perimeter

P=8*9.186

P=73.488

P=73.5 units

 

there are 8 congruent triangles; find the area of one of them and multiply the answer by 8

use the following formula for finding the area of one of the triangles:

Area=(1/2)(r²)(sin[360/8])

A=(1/2)(12²)(sin45º)

A=(1/2)(144)(0.707)

A=72*0.707

A=50.904

now multiply this answer by 8 to get the area

A=8*50.904

A=407.232

A=407.2 square units

Answer 2

 

  The number of degrees inside the regular polygon is 180*(n-2)=

 

 180*(12-2) = 1800, So each angle is 1800/12 = 150 degrees.

 

The radii form 12 congruent isoceles triangles by SSS.

 

The base angles are 75 degrees each, which leaves 30 degrees for

the angle at the center.

 

The height of the triangle bisects the 30 degree angle at the center.

So two congruent right triangles are formed with angles 90, 75 and 15.

 

We can now find the missing half of the side measure using trig.

sin 15 = ?/12 

 

SO HALF the side measure is ? = 12 * sin 15 = 3.10582854123

 

The side measure is then  6.211657082460

 

 

The perimeter is then   74.54

 

 

The HEIGHT of each right triangle is found by using pythagorean theorem in reverse: 

 

sqrt(144 - 3.10582854123^2 ) = 11.59111

 

The area of each triangle is 1/2*base*height = 18

 

The total area is 18*36 = 648