Find the area of the shaded part of the diagram in terms of 𝜋.
To find the shaded area of this diagram, we’ll need to find two areas. It’s the area of the larger quarter circle minus the area of the smaller semicircle. And we know that the largest sector is indeed a quarter circle because we can see that this line is a tangent to the semicircle. And the angle between a tangent and a radius is 90 degrees.
To calculate these areas, we recall the formula for area of a sector with radius 𝑟 and angle 𝜃 radians. It’s a half 𝑟 squared 𝜃. Remember, a full turn is two 𝜋 radians. So a quarter of a turn, 90 degrees, must be two 𝜋 divided by four or 𝜋 over two radians. The radius of our larger quarter circle is 30 centimeters. So the area is a half multiplied by 30 squared multiplied by 𝜋 over two.
To multiply these three numbers, we add the denominator of one to 30 squared. And since 30 squared is 900, multiplying the numerators and we get 900𝜋. Then, for the denominators, we get four. And so the area of a quarter circle is 900𝜋 over four square units. Then, for the semicircle, 180 degrees is a half of 360 degrees. And a half of two 𝜋 is 𝜋 radians. So 180 degrees is equal to 𝜋 radians.
This time, the area of this sector is one-half multiplied by 15 squared multiplied by 𝜋. 15 squared is 225. So the area of our semicircle is 225𝜋 over two square units.
To subtract these two areas, we could make the denominators the same. However, 900 over four and 225 over two are fairly easy to evaluate. 900 divided by four is 225. And 225 divided by two is 112.5. And so the shaded area is the difference between these two numbers. It’s 225𝜋 minus 112.5𝜋. The area is therefore 112.5𝜋 square units. Now, in fact all units are centimeters.
So we found that the area of the shaded region is 112.5𝜋 centimeters squared.