Find a polynomial for the shaded area of the figure

Question

The picture is a circle shaded inside with a radius of 5

 

I can't figure out how to answer the question.

Answer 1

If the problem asks for a polynomial for an area I gather that this may not answer your question in the spirit intended.

You described a single circle with its interior completely shaded. If this is question about the area of a circle, read on.

My favorite way of approaching this is to imagine that he circle is the side of a toilet paper roll. Except instead of terminating at a cardboard cylinder, it winds all the way to a point in the center so that there is no hole. Now suppose you used a razor blade to make perfectly straight cut from the outside of the roll to the exact center. This cut is the length of your radius (r). If you carefully unwrap the roll and lay it down flat with the outside down and the center up, the face that used to be a circle will look like a long right triangle. Since the triangle is an unwrapped version of the circle, it will have the same area. The long leg on the bottom started as the perimeter of the circle right? So that side must be 2π times the radius (2πr). And the vertical leg is the cut you made from the outside to the center which is the length of the roll's radius (r). The way you find the area of a right triangle is to multiply the two legs to get the area of a rectangle.

 

(2πr) x (r) = 2πr²

 

Then you chop the rectangle in half from corner to corner to get the area of the right triangle.

 

(2πr) ÷ 2 = πr²

 

You'll find in your text that the formula for the area of a circle is πr². Does this answer your question or were you looking for something else?