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Video:How to Find the Volume and Surface Area of a Cone

with Eric Stone

Our expert shows us how to determine the volume and surface area of a cone. See these instructions for how to easily find the volume and surface area of a cone.See Transcript

Transcript:How to Find the Volume and Surface Area of a Cone

Hi, I am Eric Stone from South Burlington High School in South Burlington, Vermont, here for About.com. Today, we are going to talk about the surface area and volume of a cone.

Parts of a Cone

The parts of the cone that you need to understand are first, the radius -- the radius of a cone is the part that goes from the center out to the edge. The height of the cone is the distance from the bottom all the way to up the top. That is pretty much all you need to figure out the volume and surface area of a cone.

Formula for the Volume of a Cone

The formula is that the volume is equal to one third the area of the base times the height. Now the area of the base, as you recall, is that area that is carried through your third dimension. You have this circle that is moving through the third dimension of space, which is why you have three units of length for your volume -- area of the base times height -- but that circle is constantly changing. But it is always a circle, that is the point.

So, the volume can be rewritten to be one third pi-r-squared -- where r is the radius of the circular base -- times the height.

Formula for the Surface Area of a Cone

To find the surface area of a cone, it is a little more complicated. There are, essentially, two parts to a cone that you need to find the area of. First, you need to find the area of the base. Luckily, we already know how to do that, because we did that for the volume calculation. This part, though, is like one piece -- if you can imagine if you cut it out it would look something like a circle.

Figuring out how to calculate the area of that requires that you know this piece right here -- and we will call that s. Some people call that the slant length. And in order to find the length from here to here, the most convenient thing for you to know is the Pythagorean Theorem.

Pythagorean Theorem

The Pythagorean Theorem is a theorem that relates three sides of a right triangle. The right triangle we are going to be interested in today is the right triangle that goes from the tip of the cone down to the center of the circle and out, like so. See it? There is your triangle right there -- where this length is s, this length is h and this length is r.

So by Pythagorean Theorem we can find s, by using a-squared plus b-squared equals c-squared -- or in this case, r-squared plus h-squared equals s-squared.

Once you know that, you can find the area of this figure -- the area of the side -- by simply saying pi times r times s.

So the total area of a cone will be equal to the area of the circle at the bottom plus the area of the side. This will be, in total, pi-r-squared for the area of the circle plus pi-r-s.

And that is how you find the surface area and volume of a cone.

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