A square is inscribed in a circle. The area of the square is what percent of the area of the circle?
Let’s firstly consider what we mean when a square is inscribed in a circle. This means that all four vertices, or corners, of the square will be on the circumference of the circle. If we let 𝑥 be the length of the square, then the area of the square will be equal to 𝑥 squared, as the area of any square is equal to the side length squared. If length 𝑟 is equal to the radius of the circle, then the area of the circle will be equal to 𝜋𝑟 squared. We need to calculate what percent the area of the square is of the area of the circle. This will be equal to 𝑥 squared out of, or divided by, 𝜋𝑟 squared.
In order to calculate this as a number, we will need to find an expression for 𝑥 squared in terms of 𝑟. If we draw the diagonal on the square, we create a right-angled triangle. This diagonal is equal to the diameter of the circle and is therefore equal to two 𝑟, twice the radius. As we are dealing with a right-angled triangle, we can use Pythagoras’ theorem. This states that 𝑎 squared plus 𝑏 squared is equal to 𝑐 squared, where 𝑐 is the length of the longest side of the right-angled triangle known as the hypotenuse.
Substituting in our lengths gives us 𝑥 squared plus 𝑥 squared is equal to two 𝑟 squared. 𝑥 squared plus 𝑥 squared is equal to two 𝑥 squared. Squaring two 𝑟 means multiplying it by itself, two 𝑟 multiplied by two 𝑟. This is equal to four 𝑟 squared. Both sides of this equation are divisible by two. On the left-hand side, the twos cancel, and we are left with 𝑥 squared. On the right-hand side, four divided by two is equal to two. This gives us the equation 𝑥 squared is equal to two 𝑟 squared.
We can now substitute this value into the calculation to work out the percent. As 𝑥 squared is equal to two 𝑟 squared, the percent will be equal to two 𝑟 squared divided by 𝜋𝑟 squared. The 𝑟 squareds here cancel, so we are left with two over 𝜋, or two divided by 𝜋. Typing this into the calculator gives us 0.6366 and so on. As the word percent means out of 100, we need to multiply this answer by 100.
Multiplying by 100, moves all the digits two places to the left. So, 0.6366 multiplied by 100 is equal to 63.66. Rounding this to one decimal place gives us an answer of 63.7 percent. The area of the square is 63.7 percent of the area of the circle. This will be true for any square that is inscribed in a circle no matter what the radius is.