A circle has the equation 𝑥 squared plus 𝑦 squared equals 64. Which of the following is the length of its diameter? Circle your answer. The options are eight, 128, 64, or 16.
To answer this question, we can first recall the general equation of a circle. If a circle has its centre at the point 𝑎, 𝑏 and a radius of 𝑟 units, then the equation of the circle is 𝑥 minus 𝑎 all squared plus 𝑦 minus 𝑏 all squared equals 𝑟 squared. If in fact the centre of the circle is at the origin, the point zero, zero, then the values of 𝑎 and 𝑏 are both zero. So the equation simplifies to 𝑥 squared plus 𝑦 squared equals 𝑟 squared.
Looking at the equation we’ve been given, we can see that this is indeed a circle with its centre at the origin. And by comparing the right-hand side of the two equations, we can work out the radius of the circle. We have that 𝑟 squared is equal to 64. To solve this equation for 𝑟, we need to take the square root of each side of this equation.
Now, usually when we solve an equation by square rooting, we have to remember to take plus or minus the square root because there are two possible values. However, here 𝑟 is the radius of a circle; it’s a length. So it must take a positive value which means we’re only taking plus the square root. 64 is a square number and its square root is just eight. So we have that 𝑟 equals eight.
But we need to make sure we’re paying attention because the question hasn’t actually asked us for the length of the radius; it’s asked for the length of the diameter. And remember in any circle the diameter is always twice the length of the radius. So if the radius is eight, then the diameter is twice this; it’s 16. So we can circle our answer.
Watch out for the other possible values. There’s a value of eight to see if this will catch us out. But remember eight was the radius of the circle not the diameter. The value of 64 is the value of 𝑟 squared not the value of 𝑟. So this would be another common mistake: to forget that we need to square root to find the radius.
The value of 128 is actually two 𝑟 squared, is two times 64. So again, if we’d thought that the radius was 64, then by doubling it, we might see that the diameter was 128. The correct answer though found by solving 𝑟 squared equals 64 and then doubling to give the diameter is 16.