A circle is inscribed in the square 𝐴𝐵𝐶𝐷 as shown. The center of both shapes is 𝑂. The length of the diagonal 𝐴𝐶 is eight centimeters. Work out the area of the circle. Give an exact answer in terms of 𝜋.
Now, try not to worry too much about the wording. The fact that the circle is inscribed inside a square means that each side of the square is a tangent to the circle. They touch at four distinct points on the circle circumference. So how are we going to calculate the area of the circle?
We know the formula for area of a circle with a radius of 𝑟 is 𝜋 𝑟 squared. But we don’t currently know the length of the radius. We can, however, work out the side length of the square. Remember the radius and a tangent meet at 90 degrees. So this means that the side length of the square has to be equal to the diameter of the circle. And then, once, we know the diameter of the circle, we can halve that to find the value of the radius.
We can use this right-angled triangle to help us work out the length of one side of the square. Remember a square has four equal sides. So we know that two sides in this triangle are of equal length. Let’s call them 𝑥 centimeters. Then, we can use Pythagoras’s theorem to form an equation in terms of 𝑥.
Remember Pythagoras’s theorem says that the sum of the squares of the smaller two sides is equal to the square of the longest side, the hypotenuse. This is often written as 𝑎 squared plus 𝑏 squared equals 𝑐 squared, where 𝑐 is the length of the hypotenuse of the triangle.
We can find the hypotenuse because it’s the longest side. It’s always the one directly opposite the right angle. In this case, the longest side of our triangle, the hypotenuse, is eight centimeters.
Since the length of the smaller two sides is 𝑥 centimeters, we can say that 𝑥 squared plus 𝑥 squared is equal to eight squared. And we can simplify the left-hand side of this equation: 𝑥 squared plus 𝑥 squared is two 𝑥 squared and eight squared is 64. So that means that two 𝑥 squared is equal to 64. And now, we can solve this equation for 𝑥.
The 𝑥 has two things happening to it: firstly, it’s being squared; then, it’s being multiplied by two. To solve, we need to reverse that process. The opposite of multiplying by two is dividing by two. So we’re going to divide both sides of this equation by two. 64 divided by two is 32. So we can see that 𝑥 squared is equal to 32.
The opposite of squaring is square rooting. So next, we’re going to square root both sides of our equation. And when we square root a number, we get two distinct numbers: we get a positive solution and a negative solution.
Since 𝑥 represents a length, so we can disregard the negative value; there’s no such thing as a negative length. And we can say that 𝑥 is equal to positive root 32. Where possible, we should simplify any surds. To simplify the square root of 32, we need to find the largest factor of 32, which is also a square number.
The largest factor of 32, which is a square number, is 16. So we write 32 as 16 multiplied by two. And we can say that the square root of 32 must be equal to the square root of 16 multiplied by two.
Now, remember one of the rules of surds we know is that the square root of 𝑎 multiplied by the square root of 𝑏 is the same as the square root of 𝑎 multiplied by 𝑏. We can reverse this for our problem and say well the square root of 16 multiplied by two must be the same as the square root of 16 multiplied by the square root of two.
But the square root of 16 is of course four. So this means that the square root of 16 multiplied by the square root of two is four multiplied by the square root of two. The square root of 32 simplifies to four root two.
So the length of the side of the square is four root two centimeters, which means our diameter is also four root two centimeters. The radius is half the length of the diameter. So we can divide four root two by two. And that tells us that the radius of our circle is two root two centimeters.
Let’s take this information and clear some space. Now that we know the length of the radius, we can work out the area using the formula we mentioned earlier. The formula is 𝜋 multiplied by the radius squared. Our radius is two root two. So this becomes 𝜋 multiplied by two root two all squared.
We can rewrite this a little and say that two root two all squared is the same as two squared multiplied by the square root of two squared. And of course, two squared is four and the square root of two squared is just two.
Four multiplied by two is eight. And just like with algebra, we write the letter after the number. And we found that the area of the circle in terms of 𝜋 is eight 𝜋 centimeters squared.