A sphere has a diameter of 18 centimetres. Work out the volume of the sphere, giving your answer in terms of 𝜋.
So in this problem, we’re told the diameter of a sphere and asked to calculate its volume. Let’s recall first the formula for calculating the volume of a sphere. The formula is that the volume is four-thirds 𝜋𝑟 cubed, where 𝑟 is the radius of the sphere.
Now, we haven’t been given the radius of the sphere; we’ve been given the diameter, but that’s not a problem because the two are very closely related. The radius of a sphere is half of the diameter. So if the diameter of the sphere is 18 centimetres, then the radius must be nine centimetres. Let’s substitute this value of 𝑟 into our formula for calculating the volume.
We have that the volume of the sphere is equal to four-thirds multiplied by 𝜋 multiplied by nine cubed. Now the question has asked us to give our answer in terms of 𝜋. We suggest we may not have access to our calculator for this problem. We need to evaluate the constant in the volume.
Let’s write that nine cubed as nine multiplied by nine multiplied by nine. Now, there is a factor of three in the denominator and I can cancel that with one of the nines in the numerator. If I divide them both by three, then I now have four multiplied by 𝜋 multiplied by three multiplied by nine multiplied by nine. Let’s think about how to perform this multiplication easily without a calculator.
The four and the three multiplied together to give 12 and the nine and the nine multiplied together to give 81. So I now have that the volume of the sphere is equal to 12 multiplied by 81 multiplied by 𝜋. So we just need to work out what 12 multiplied by 81 is. Well, we can do this by dividing the 12 up into 10 plus two. 10 times 81 is 810; two times 81 is 162. So we can add these together in order to find 12 times 81 is 972. Therefore, our answer for the volume of the sphere in terms of 𝜋 is 972𝜋 and the units for this volume are centimetres cubed.