Find the area of the shaded region in terms of pi

Question

The picture shows a square with a side length of 14 inches. In the square is a quarter circle (whose radius is the Side of the square) and two half circles ( with diameters that are sides of the square. 

Answer 1

I know it's very difficult to 'draw' a problem using words, but you need to know where the shaded area is before just subtracting things from the square.  John is correct in pointing out that there will be overlaps.  You must tell us where is the shading first.  Also, Mark's quarter circle is wrong, as it would have a radius of 14 making its area 196π square inches thus the quarter circle is 49π square inches.  Of course, the whole answer given by Mark could not work since these circles overlap, you'll just be subtracting some things more than once. 

 

But where is the shading anyway? I understand from your note that the half circles are are on the bottom and left. But what about the quarter circle? 

 

Let's say the square has it's left bottom corner at (0,0) on the coordinate plane.  Is the center of the quarter circle at (0,0)? is it at (14, 14)? is it at (0,14) ?  or (14,0) ?  I'm assuming the half circles, with this scenario have centers at (0,7) and (7,0). 

 

Depending on which way the quarter circle goes and where is exactly shaded, the answer could be very different.  Is it outside the circle overlaps or within it, for example?

 

For example, if we leave out the quarter circle for a moment, and consider the two half circles in the position you mention with the centers I've given, the overlapping region is twice the segment of a circle with 90 degree arc.  A segment is the area bounded by the circle and the chord to an arc - or simply the difference between a sector and the central triangle contained by the sector.

 

Since the sector is 1/4 of the circle, it has area (1/4)(49π). The triangle there is a right triangle with area (1/2)(49).  Their difference is (49/4)(π - 2). Doubling this, we have (49/2)(π - 2).  This is the overlap of the two half circles on the bottom and left. 

 

But once the quarter circle is added, things change quite a bit. I think I would need to use integral calculus to find the area, but I don't want to go that far unless I know where the shaded area really is.

 

Nice problem!

Answer 2

The two half circles make a circle with area = 49π

The quarter circle is one-fourth of a circle with area = 49π, or 12.25π

The area of the square is 14², or 196

The area of the shaded region is 196 - 49π - 12.25π, or

196 - 61.25π