Problem of the Week 8 – Shaded Part

This is the eighth problem in our Problem of the Week.

Problem

Two semi-circles are placed inside a square as shown below. What is the area of the shaded part?

circle square shaded part

 

Solution 1

To get the area of the shaded part, we have to get the area of the square and then subtract the area of the two semicircles. 

The area of the square with side s is equal to s \times s, so the area of the square above is

A = 10 \times 10 = 100 sq. cm.

Now, the area of the semicircle is half the area of a circle. The diameter of the circle above is the same as the side of the square, so its radius is half of it. Therefore, if we let the radius r, then

r = \displaystyle \frac{10}{2} = 5 cm.

Now, the area of a circle with radius r is \displaystyle \frac{\pi r^2}{2}. So,

Area of a semicircle = \displaystyle \frac{\pi (5^2)}{2} = \frac{25 \pi}{2} sq. cm.

There are two semicircles, so we multiply the area by 2. That is

Area of 2 semicircles = \displaystyle \frac{25 \pi}{2} \times 2 = 25 \pi sq. cm.

Now,

Area of shaded part = area of square – area of 2 semicircles.

Area of shaded part = 100 - 25 \pi sq. cm.

Solution 2

To get the area of the shaded part, we subtract the area of the semicircles from the area of the square. But the two semicircles has area equal to one circle.

From above, the area of a square is the square of its side. So, if we let the area of a square be equal to A, then

A = 10 \times 10 = 100 square centimeters.

Now, the area of the circle above is equal to

A = \pi r^2

A = \pi 25^2

A = 25 \pi.

Now,

Area of shaded part = area of square – area of 2 semicircles.

Area of shaded part = 100 - 25 \pi.

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