Question

Directions: Develop a strategy and create a complete solution to each problem. Identify the measurements you are solving for, and the approach you will take to solving the problem. Include a sketch or diagram (you may use Geogebra or other software) and show all your steps.

Answer 1

The diagram for each is shown below. I used GeoGebra to make it.

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Problem 1

The regular octagon has a perimeter of 80 feet. Assuming this is a regular octagon (i.e. all 8 sides are the same length), then we know that each side is 80/8 = 10 feet long.

We'll use n = 8 and s = 10 in the formula below to find the area.

`A = 0.25n*s^2*\cot(180/n)\\\\A = 0.25*8*10^2*\cot(180/8)\\\\A = 0.25*8*10^2*\cot(22.5)\\\\A = 0.25*8*10^2*(1)/(\tan(22.5))\\\\A \approx 482.8427\\\\A \approx 483\\\\`

The octagon has an area of roughly 483 square feet.

The square also has a perimeter of 80 feet. All four sides are the same length, so each side is 80/4 = 20 feet long. The area of this square is s^2 = 20^2 = 400 square feet.

Answers:

Area of the octagon = 483 square feet (approximate)

Area of the square = 400 square feet (exact)

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Problem 2

Divide 360 over the central angle to find out how many sides this regular polygon has

360/18 = 20

This is a 20 sided polygon (aka icosagon).

Since this polygon has 20 sides of the same length, and the perimeter is 144 ft, this means each side must be 144/20 = 7.2 feet long exactly.

The inputs we'll use are

n = 20, s = 7.2

And we'll use that formula from problem 1 to find the area of this regular polygon.

`A = 0.25n*s^2*\cot(180/n)\\\\A = 0.25*20*(7.2)^2*\cot(180/20)\\\\A = 0.25*20*(7.2)^2*\cot(9)\\\\A = 0.25*20*(7.2)^2*(1)/(\tan(9))\\\\A \approx 1636.5244\\\\A \approx 1636.5\\\\`

Answer: The area is roughly 1636.5 square feet

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Problem 3

There are two types of radius when it comes to describing a regular polygon.

• inradius
• circumradius

The inradius stretches from the center of the polygon to the midpoint of any given side. This produces the inscribed circle. It is the largest circle that fits perfectly inside the polygon without spilling over.

The circumradius goes from the center of the polygon to the vertex or corner point. This helps form the circumscribed circle. It is the smallest circle where it completely encloses the polygon without going inside the boundaries.

I'll assume your teacher is talking about the circumradius here.

If we know the circumradius is r = 5.5 feet and we have n = 6 sides, then the area of the regular hexagon is...

`A = 0.5n*r^2*\sin(360/n)\\\\A = 0.5*6*(5.5)^2*\sin(360/6)\\\\A = 0.5*6*(5.5)^2*\sin(60)\\\\A \approx 78.5918\\\\`

The hexagon has an area of roughly 78.5918 square feet.

The cost to refinish the floor is $2.50 per square foot.

The total cost is 2.50*78.5918 = 196.4795 = 196.48 dollars.

Answer: $196.48