# Finding a Shaded Region Using the Arcs of Circles

• Apr 24th 2008, 04:53 PM
happy_go_lucky
Finding a Shaded Region Using the Arcs of Circles
I have a geometry problem, which I believe I have worked out but seems way too easy.

The question describes a scenario where arcs of circles are contained within a square of length A. Therefore, its area is A^2. The requirement is to work out the area of a shaded region created by these arcs in terms of A.

A perfect circle is formed within the square. Because the length of the square is A, the radius of the circle within it can be defined as 1/2 of A.

Therefore the area of the circle is simply Pi x (1/2 A)^2 = Pi(1/4 x A^2).

The area of the square that does not include the circle is the area of the square minus the area of the circle:

A^2 - Pi(1/4 x A^2)

But from looking at the diagram and considering its symmetry, I can tell that each corner forms an arc with a radius of 1/2 of A, just like the central circle.

Consequently, these arcs forms 4 quarter-circles, which all combine to create an area equal to the main circle.

Because these quarter-circles include all the shaded regions, I can use their combined area and subtract from it the area of the square not contained in the circle:

Pi(1/4 x A^2) - (A^2 - Pi(1/4 x A^2)) = 2Pi(1/4 x A^2) - A^2

Therefore, 2Pi(1/4 x A^2) - A^2 expresses the area of the shaded region/s in terms of A.

I have included a link to a rough MS Paint diagram I drew to illustrate the problem. The diagram is meant to be symmetrical - please disregard the slight asymmetry of the arcs.

I am wondering - have I gone about this problem correctly? Once again, it seems far too simple. Is there another way to do it that I have not used?

• Apr 24th 2008, 05:15 PM
icemanfan
Quote:

Originally Posted by happy_go_lucky
I have a geometry problem, which I believe I have worked out but seems way too easy.

The question describes a scenario where arcs of circles are contained within a square of length A. Therefore, its area is A^2. The requirement is to work out the area of a shaded region created by these arcs in terms of A.

A perfect circle is formed within the square. Because the length of the square is A, the radius of the circle within it can be defined as 1/2 of A.

Therefore the area of the circle is simply Pi x (1/2 A)^2 = Pi(1/4 x A^2).

The area of the square that does not include the circle is the area of the square minus the area of the circle:

A^2 - Pi(1/4 x A^2)

But from looking at the diagram and considering its symmetry, I can tell that each corner forms an arc with a radius of 1/2 of A, just like the central circle.

Consequently, these arcs forms 4 quarter-circles, which all combine to create an area equal to the main circle.

Because these quarter-circles include all the shaded regions, I can use their combined area and subtract from it the area of the square not contained in the circle:

Pi(1/4 x A^2) - (A^2 - Pi(1/4 x A^2)) = 2Pi(1/4 x A^2) - A^2

Therefore, 2Pi(1/4 x A^2) - A^2 expresses the area of the shaded region/s in terms of A.

I have included a link to a rough MS Paint diagram I drew to illustrate the problem. The diagram is meant to be symmetrical - please disregard the slight asymmetry of the arcs.

I am wondering - have I gone about this problem correctly? Once again, it seems far too simple. Is there another way to do it that I have not used?

Your solution is correct. To summarize: The white area inside the big circle is the same as the white area outside the circle but inside the square, which is easily computed as $B = A^2 - \pi\frac{A^2}{4}$. Then the shaded area is equal to the area of the circle - the white area inside the circle, or $\pi\frac{A^2}{4} - B$, which is the same as your expression.