Question Video: Finding the Side Length of a Square inside Which a Quarter Circle Is Drawn Using Circular Sectors and Areas of Circles Mathematics • 7th Grade

Video Transcript

A quarter circle has been drawn inside a square such that the circle’s radius equals the square’s length. The area of the remaining part of the square is 47.18 square centimeters. Find, to the nearest centimeter, the side length of the square.

Let’s see if we can draw a diagram to represent this information. We’ll break it down into stages. We’re told, firstly, that there’s a quarter circle. So, we’ve drawn a circle and split it into four pieces. We only need one of those pieces for a quarter circle. We’re told that this quarter circle has been drawn inside a square, but not a square of any size surrounding it. This square has the same length as the radius of the circle, so we could draw our square like this. The sides here we can denote as the letter 𝑟, which will be the same as the radius of this quarter circle.

We’re told that the area of the remaining part of the square is 47.18 square centimeters, which is this bit. It’s the area of the square, with the area of the quarter circle removed from it. We’re asked in the question to find the side length of the square. So, we need to work out the value of 𝑟. So, let’s write down what we’ve established.

The area of the remaining part is equal to the area of the square subtract the area of the quarter circle. We were given that the area of the remaining part is 47.18. And to find the area of a square, we multiply the length by the length. For this square then, we’ll have 𝑟 squared. We can use the fact that the area of a circle is equal to 𝜋𝑟 squared to help us find the area of a quarter circle. We simply take 𝜋𝑟 squared and divide it by four. So now, we have an equation set up and we can solve this to find the value of 𝑟.

If we look at the right-hand side of this equation, we can see that we have two instances of 𝑟 squared. To help us with rearranging, we want to have one instance of 𝑟 squared, and we can do this by taking 𝑟 squared as a common factor. On the right-hand side then, we’ll have 𝑟 squared and parentheses. Our first term will be one since 𝑟 squared times one gives us 𝑟 squared. We’ll then have subtract 𝜋 over four. This second term occurs because 𝜋 over four multiplied by 𝑟 squared gives us 𝜋𝑟 squared over four.

In order to find 𝑟 squared by itself, our next step will be to divide by one minus 𝜋 over four. To find 𝑟 then, we need to take the square root of both sides of this equation. So we have that 𝑟 is equal to the square root of all of 47.18 over one minus 𝜋 over four. We can then use our calculator to find the value of 𝑟, 14.827306 and so on. Notice that the units here are length units. And as we’re told that the area was square centimeters, the length units will be centimeters.

We were asked to find the answer to the nearest centimeter. So, we check our first decimal digit to see if it’s five or more. And as it is, then this rounds up to 15 centimeters. So, we found that the radius of this quarter circle is 15 centimeters to the nearest centimeter. But we know that the length of the square is the same as this radius. And so, this is our final answer, 15 centimeters.