What is the area of the stop sign to the nearest square inch?

Question

A stop sign is a regular octagon with a 16.2 inches radius. The area should be rounded to the nearest square inch. 

Answer 1

So, I assume by it's radius, your are referring to the mid point to each of the corners as this would form a circumscribed circle with this radius.

 

From this center point, draw out line segments to each of the eight vertices of the octagon. Resulting will be 8 congruent triangles. So, if we can find the area of one triangle, we can multiply this by 8 and get the area of the octagon.

 

Since there are 8 congruent triangles from one point, and all angles are used around that point, the sum of these 8 angles is 360 degrees. Therefore each angle is 45 degrees.

 

So, now let's take one triangle and look at it.

We have an isosceles triangle with two legs equal to the radius, and a 45 degree angle between them. To find the area of this triangle, we must find the base and the height, neither of which is given by the radius.

 

Draw a perpendicular bisector from your 45 degree angle through the base of the triangle, to make two congruent right triangles.

Using trigonometry, we can take sine and cosine functions of 22.5 degrees to get the base (x) and height (y) of the right triangle.

Sin (22.5) = x/16.2  -> x=6.199

Cos (22.5) = y/16.2  -> y=14.967

 

The base of the isosceles triangle is 2x, and the height of this triangle is y.

The formula for the area of this triangle = 1/2*b*h = 1/2*2x*y = x*y

 

So the area for an individual triangle is 92.787 sq inches. To get the area of the octagon, multiply this value by the number of isosceles triangles in the octagon, 8. For a grand total of 742 sq inches (after rounding).

 

Hope this helps,

Chris