Video: Calculating the Volume of a Hemisphere Giving Your Answer in Terms of πœ‹

A hemisphere has a radius of 15 inches. Work out its volume, giving your answer in terms of πœ‹.

02:13

Video Transcript

A hemisphere has a radius of 15 inches. Work out its volume, giving your answer in terms of πœ‹.

We recall firstly that a hemisphere is simply half a sphere. So the formula for finding the volume of a hemisphere is just half the formula for finding the volume of a sphere. It’s two-thirds πœ‹π‘Ÿ cubed. We’re told that the radius of this hemisphere is 15 inches. So we can substitute this value directly into our formula, giving two-thirds πœ‹ multiplied by 15 cubed.

Now we’re asked to give our answer in terms of πœ‹, which suggests that we don’t have access to a calculator for this problem. So we need to consider how to simplify the calculation without using a calculator. We can first write 15 cubed as 15 multiplied by 15 multiplied by 15. And then as three is a factor of 15, we can cancel the three in the denominator with a factor of three from one of the 15s in the numerator, giving two over one πœ‹ multiplied by five multiplied by 15 multiplied by 15.

We can perform this multiplication in any order. So perhaps the easiest is to think of it as two times five multiplied by πœ‹ multiplied by 15 multiplied by 15. Two times five is of course 10, and 15 multiplied by 15 we should know β€” it’s one of our square numbers β€” is 225. So we have 10 multiplied by πœ‹ multiplied by 225. 10 multiplied by 225 is 2,250. So our value in terms of πœ‹ is 2,250πœ‹.

The units given for the radius were inches. And so the units given for the volume will be cubic inches. And so we have our answer to the problem. It’s 2,250πœ‹ cubic inches.

Now we could’ve answered this problem a slightly different way. We could’ve simply calculated the volume of the full sphere using the formula four-thirds πœ‹π‘Ÿ cubed and then divided our answer by two at the end to give the volume of a hemisphere. But of course, it would give the same result of 2,250πœ‹ cubic inches.

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