SOLUTION: You are laying concrete around a gazebo that is a regular octagon with a radius of 8 feet. The concrete will form a circle that extends 15 feet from the vertices of the octagon.

Algebra ->  Customizable Word Problem Solvers  -> Geometry -> SOLUTION: You are laying concrete around a gazebo that is a regular octagon with a radius of 8 feet. The concrete will form a circle that extends 15 feet from the vertices of the octagon.       Log On

Ad: Over 600 Algebra Word Problems at edhelper.com


    

Question 908060: You are laying concrete around a gazebo that is a regular octagon with a radius of 8 feet. The concrete will form a circle that extends 15 feet from the vertices of the octagon. What is the area of the concrete to the nearest square foot?
Answer by KMST(5328) About Me  (Show Source):
You can put this solution on YOUR website!
The whole thing looks like this:
The area of the concrete is the area of the big red circle minus the area of the green octagon. That is one way to calculate it.

The radius of the red circle, in feet, is
8%2B15=23 ,
so the area of the big red circle is
pi%2A23%5E2square feet = approximately1661.9025square feet.

Now we have to figure out the area of the octagon.

Maybe you have been given a convenient formula to calculate the area of a regular octagon,
such as 2sqrt%282%29%2Aradius%5E2, and that would make it easy.
If so, you do not need to read further.

Otherwise you would have to visualize, reason and calculate.
The octagon looks like this:
It is made of 8 wedges>
Each wedge is an isosceles triangle with legs measuring 8 feet and a vertex angle measuring 45%5Eo .
Here is a close-up of one of those wedges:

Depending on what you have learned in math, it may be easier or harder to figure out how to calculate the area of that wedge.
In the USA, they would teach you in pre-calculus class, that the area of
triangle ABC is %281%2F2%29%2Ab%2Ac%2AsinA, so you would calculate it as
%281%2F2%29%2A8%2A8%2Asin%2845%5Eo%29=%281%2F2%29%2A8%2A8%2A%28sqrt%282%29%2F2%29 .
Before pre-calculus, applying what you learned from trigonometry to right triangle ABD, you would calculate that
h=8%2Asin%2845%5Eo%29=8%2A%28sqrt%282%29%2F2%29 ,
and would calculate the area of the wedge as a triangle:
%281%2F2%29%2Abase%2Aheight=%281%2F2%29%2A8%2Ah=%281%2F2%29%2A8%2A8%2A%28sqrt%282%29%2F2%29 .
Before learning any trigonometry, you would still know that ABD is a right isosceles triangle, with a hypotenuse measuring 8 feet, and two legs measuring h . Then using the Pythagorean theorem, you would figure out that
h%5E2%2Bh%5E2=8%5E2--->2%2Ah%5E2=8%5E2--->h%5E2=8%5E2%2F2--->h=8%2Fsqrt%282%29=8%2Asqrt%282%29%2F2
Either way, you should be able to calculate that the area of that wedge is %281%2F2%29%2A8%2A8%2A%28sqrt%282%29%2F2%29= approximately22.62742square feet.
Then the area of the octagon (made of 8 such wedges) is 8 times the area of one wedge, or approximately
8%2A22.62742square feet = 181.0194square feet.

The area of the concrete, in square feet is approximately
1661.9025-181.0194=1480.8831 , so
the area of the concrete to the nearest square foot is highlight%281481%29 square feet.