The diagram shows a cone with a radius of 20 centimeters and a perpendicular height of 40 centimeters. We’re told that the volume of a cone is equal to a third times the area of its base times its perpendicular height. And we’ve got to work out the volume of the cone, giving our answer in terms of 𝜋.
Now we can see that our cone has a circular base with a radius of 20 centimeters. And we can see that the perpendicular height is 40 centimeters. And the fact that our diagram had this little right angle symbol here helped us to identify the fact that it was in fact a perpendicular height. Now, it’s important to remember when you’re calculating the volume of a cone that you do take the perpendicular height of that cone and not the slant height, the length of the slopy side of the cone.
Now, we should also notice that we’ve been asked to give our answer in terms of 𝜋. And this just means as a multiple of 𝜋. Now in this particular question, we can’t express the answer as an exact decimal. We’d need to round it to a few decimal places. And then, it wouldn’t be fully accurate. But by giving it as an exact number multiplied by 𝜋, we are representing the exact answer.
So let’s start our working out by just copying out that formula that we were given earlier in the question. “Volume equals a third times the area of the base times the perpendicular height.” Now, all of the dimensions that we were given in the questions were in terms of centimeters. So our volume is going to be in cubic centimeters. And we know that the perpendicular height was 40 centimeters. So we need to work out the area of the base of our cone.
Well, a cone has a circular base. And the formula for the area of a circle is 𝜋 times the radius squared. Now, it’s important to remember that it’s only the radius that’s squared. We don’t end up squaring 𝜋 as well. And we know that the radius is 20. So the formula for our volume becomes a third times 𝜋 times 20 squared times 40. And then, the units are gonna be cubic centimeters. Now, 20 squared means 20 times 20. And that’s 400.
Now, we can think of 400 as being four times 10 times 10. And 40 is just four times 10. Now it doesn’t matter what order we multiply those numbers together in. We’re gonna get the same answer. So I’m gonna say four times four is 16. 16 times 10 is 160, times 10 again is 1600, times 10 again is 16000. So that gives us a third times 𝜋 times 16000 cubic centimeters.
Again, it doesn’t matter what order we multiply these things together in. We’re still gonna get the same answer. So I’m gonna swap the 𝜋 and the 16000 around. And 16000 is the same as 16000 over one. It’s just a fraction form of the same number. So now, I’ve got a third times 16000 over one. This is a fraction calculation. I can multiply the one and the 16000 together and the three and the one together. So that’s 16000 over three times 𝜋, but more simply, 16000 over three 𝜋 cubic centimeters.
So we’ve given our answer as a multiple of 𝜋. In other words, we’ve given our answer in terms of 𝜋. The fact that the number we’ve got is a top heavy fraction doesn’t really matter. It’s a perfectly acceptable way to present your answer.
Now, if you’ve got your calculator with you, you may be tempted to turn that top heavy fraction into a mixed number like this, 5333 and a third 𝜋 cubic centimeters. And that’s absolutely fine. There’s absolutely no need to do it. But you can do it if you want to. If you do stop messing about with your calculator, it may give you an answer like this 5333.3 recurring 𝜋 cubic centimeters. And that’s not quite exact. 0.3 recurring isn’t quite the same as a third. So that’s not an exact answer. I wouldn’t advise doing that.
And if you go the extra step and actually multiply by 𝜋, you’ll get an answer like this, 16755.16082 and so on cubic centimeters. That’s wrong because the question asked us to specifically give our answer in terms of 𝜋. So don’t be tempted to do more work than you need to. Just give your answer in the simple form I’ve asked for, as a multiple of 𝜋.