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Volume of Cones - Concept

Teacher/Instructor Brian McCall
Brian McCall

Univ. of Wisconsin
J.D. Univ. of Wisconsin Law school

Brian was a geometry teacher through the Teach for America program and started the geometry program at his school

The formula for the volume of cylinders can also be applied to the volume of cones, which occupy only one third of the space of a corresponding cylinder with the same base and height. To find volume of cone dimensions and measurements, we simply use the formula for the area of a circle for one of the bases and then multiply by one third of the height of the cone. This formula is very similar to volume of cylinder and prism volume formulas.

When we're talking about the concept of volume, we're talking about the amount of space that's defined by a 3 dimensional figure. The volume of a cylinder can be calculated by finding the base area and since the base area is a circle, you can say that's pi r squared times its height. Now some text books will use a capital H some will use a lower case h as long as you remember that you're going to find your base area and then multiply by the height of your solid. But there's a special relationship between the volume of the cylinder and volume of a cone. Well if I drew 2 figures, that had the same radius and the same height then what I could do is I could say that the volume of this cone will be one third the volume of that cylinder. So we can say that the volume of any cone is equal to one third times the base area which is going to be pi r squared since the base of a cone is a circle times the height. So I could re-write this in terms of a radius we could say that this is one third pi r squared times h. So the only 2 things you need to know to calculate the volume of a cone is the radius and the height.
Let's look at a quick example, in this problem we're being asked to find the volume I see that I have a radius of 5 centimeters and the height is 10 centimeters and I know it's the height because we have a right angle there. So we're going to start by writing our formula just like every problem in Geometry. We're going to say the volume is equal to one third base area times our height. Now we need to identify our known variables, we know that our base area is equal to pi r squared, we know that our height is equal to 10 centimeters and we know that our radius is equal to 5 centimeters.
Now one thing to be careful of when you're taking a test or quiz they might give you a diameter in which case you have to divide by 2 to find your radius. So now we can just substitute in to our volume formula, so we're going to say that volume is equal to 1 third because it's a third of the volume of an identical cylinder that has, well it can't be identical because they're different shapes but a cylinder with the same radius and the same height. So we're going to say one third times pi times our radius now instead of writing radius I'm going to substitute in 5 centimeters so I'm going to erase that and I'm going to write 5 centimeters and we're going to square that and we need to multiply by our height and our height is 10 centimeters. So we're going to say times 10 centimeters, so if we do 5 squared that's going to be 25, so we're going to have one third 25 square centimeters times 10 centimeters.
Now we're going to check our dimensions here we should have something to the third dimension, because it's asking us to find volume. And since we have centimeters squared times centimeters that's going to be centimeters to the third or cubic centimeters. So this is going to be one third, 25 times 10 is 250, so we're going to say times 250 cubic centimeters. Now some teachers will say leave it as a fraction, some teachers will say put it in a decimal if they want a decimal just use your calculator so I'm going to say that the volume here is 250 cubic centimeters divided by 3. The key thing to these is remembering our volume formula and that the only 2 things that you need to know to calculate the volume of a cone is the radius and the height.
So hopefully caught Brian's mistake there, let's walk through it really quickly. When Brian set up the formula he set it up correctly with a pi in the formula for the volume of a cone. But then he made a common mistake which is he forgot to bring tot bring the pi down into his equation. So let's go ahead and put those there pi, pi and pi in the answer. So the actual answer should be 250 pi over 3 cubic centimeters. Remember to check your work.

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