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Surface Area 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

Surface area is a two-dimensional property of a three-dimensional figure. Cones are similar to pyramids, except they have a circular base instead of a polygonal base. Therefore, the surface area of a cone is equal to the sum of the circular base area and the lateral surface area, calculated by multiplying half of the circumference by the slant height. Related topics include pyramid and cylinder surface area.

If you want to calculate the surface area of a cone, you only need to know 2 dimensions. The first is the slant height l and the second is the radius. So what we're going to do, we're going to separate this into two pieces the first is the base which is a circle with radius r and the second is this slant height l. So if I cut, if I took a scissors and cut the cone part and I fended out it would look like a sector. Well what I could do here is I could rearrange this sector into a parallelogram. So again if I cut this into really tiny pieces then I'll be able to organize it into a parallelogram where I would be able to calculate its area. And the way that we'll calculate its area, is first by saying well what are these lines that are going out?
Well those lines are going to be your l, your slant height and this side right here is going to be half of your circumference and half of a circumference is pi times r because the whole circumference is 2 pi r. So this down here is pi times r, so if our height l and our base is pi times r then the area of this is equal to pi times r times l. So the surface area of a cone which I'm going to write over here is equal to the base pi r squared plus this lateral area which is found using your slant height. So that's going be pi times r times l, so you only need to know 2 dimensions the radius and the slant height and you can calculate the surface area of any cone.

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