A rectangle which measures 9.6 centimeters by 7.2 centimeters is drawn inside a circle with center 𝑀 and radius six centimeters. Using 3.14 as an estimate for 𝜋, find the area of the shaded part.
It’s always a good idea to start a question like this by filling any information about the lengths onto the diagram. We have this rectangle which is 9.6 centimeters by 7.2 centimeters. And we have a circle, and we’re told the radius is six centimeters. In order to find the area of this shaded part, we’ll need to do two area calculations.
First, we’ll need to find the area of the whole circle. And secondly, we’ll need to find the area of the rectangle. If we subtract the area of the rectangle from the area of the circle, we’ll be left with the area that’s shaded. First, let’s recall that to find the area of a circle, we multiply 𝜋 by the radius squared, remembering that it’s just the radius that’s squared and it doesn’t include the 𝜋. Let’s take a look at our circle.
We’re given that the radius is six centimeters so we can plug this into the formula. Be careful, because sometimes we’re given the diameter instead, which we would need to halve. We can go ahead and plug in six squared for our radius squared and we’re told to use 3.14 for 𝜋. Six squared is 36, so we’ll need to multiply 3.14 by 36. We can do this using long multiplication. We multiply 314 by 36 using whatever multiplication method you choose, which gives an answer of 11304. Because the sum of the decimal digits in our two values which were multiplied was two, our answer will have two decimal places as well. Our units here will be the square units of square centimeters.
Now we’ve found the area of the circle; let’s find the area of a rectangle. We’ll need to remember that we do this by multiplying the length by the width. Plugging in our values of 9.6 and 7.2, we multiply 9.6 and 7.2. This gives us 69.12 square centimeters.
To find the area of the shaded part then, we have the area of our circle, 113.04, subtract the area of the rectangle, 69.12. The final answer is 43.92 square centimeters.