Question

A Target consists of two concentric similar octagons. The outside octagon has a side length of 2 feet and an area of 19.28 square feet. If the inside octagon has a side length of 1.5 feet, what is the area of the inside octagon?

Answer 1

Answer: The apothem of the roof is 18.5 feet.

The perimeter of the hexagon is 122.3 feet.

So, each side is 15.3 feet.

Explanation:

If a regular octagon is divided into 8 congruent isosceles triangles, then each one would have an angle at the center that measures 45 degree. If an altitude is drawn from that angle, it would divide the angle in half.

Diagram shows a regular octagon divided into 8 equals sized triangles. A triangle has a leg of 20 feet and altitude drawn from vertex to the base makes an angle of 22.5 degrees and a right angle is marked at the base.

We can find the length of the apothem, a, with the cosine of 22.5 degree.

cos(22.5degree)=a/20

a=18.5 ft

The area of a polygon is A=ap/2. Use this to solve for the perimeter:

1,131.4=(18.5)p/2

p=122.3

Since a regular octagon has 8 congruent sides, each side is one-eighth of the perimeter. The side length is approximately 15.3 feet.

Answer 2

General formula for regular polygon is:

`A= (1)/(4) *n* a^(2) *cot( (180)/(n) )`

For the inside octagon we have:

`A= (1)/(4) *8* 1.5^(2) *cot( (180)/(8) )`

`A=10.86 ft^(2)`