+49 A square 10cm on each side has four quarter circles drawn with centers at the four corners. How many square centimeters are in the area of the shaded region? Express your answer in terms of pi.
I understand all of the question, but how should I calculate the area of the shaded region? Please don't tell me the answer, I would like it if I understand how to solve it.
We first notice that the area of the shaded region is the area of the square minus the areas of the four quarter circles. Each quarter circle has a radius half the side length, so if we sum the areas of the four quarter circles, we have the area of one full circle with radius 5 cm. Now, we know the area of a square is the square of its side length, so the square has an area of 100 cm sq. A circle has an area of pi times its radius squared, so the four quarter circles combined have an area of pi(5)sq=25pi cm sq. From this, we know that the area of the shaded region is 100-25pi.
We first notice that the area of the shaded region is the area of the square minus the areas of the four quarter circles. Each quarter circle has a radius half the side length, so if we sum the areas of the four quarter circles, we have the area of one full circle with radius 5 cm. Now, we know the area of a square is the square of its side length, so the square has an area of 100 cm sq. A circle has an area of pi times its radius squared, so the four quarter circles combined have an area of pi(5)sq=25pi cm sq. From this, we know that the area of the shaded region is 100-25pi.
