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Surface Area of Cylinders - 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. Cylinders are similar to prisms in that they have congruent, parallel bases, except cylinders have circles as their bases. To conceptualize surface area of cylinders, we can imagine that the lateral area of a cylinder can be "unrolled" into a rectangle with one side equaling the circumference of the circle and the other side equal to the height of the cylinder (unless it is oblique).

Surface area is the amount of area on the outside of a polyhedron so you can almost think of it as if you had a bucket of paint how much paint would you need to cover the outside of the cylinder? Well to do that let's kind of separate the difference pieces of the cylinder, we can take this top which is a circle and I'm going to draw that circle right there we have some radius r. So we have the top and the bottom is also going to be a circle, so I can draw the bottom right there. So these 2 are going to be congruent, now this middle piece in between the 2 congruent circles what is that going to look like? Well if you took pair of scissors and made a cut right here and unraveled it, it would be a rectangle. So the net for our cylinder is going to look something like this, where we're going to have a circle on top which you could fold over, you're going to wrap this part around and then you're going to fold the other base. So in order to calculate the surface area you're going to need to add up the area of this circle, the area of this rectangle and the area of that other circle.
Well let's start with the easy part, the area of the 2 circles, so I'm going to say the surface area is equal to 2 times pi r squared. So that piece right there will calculate the surface area of the top circle and the surface area of the bottom circle now you want to be careful on those homework problems, where they leave out the top. What you're going to do then is you're going to omit these 2 and you're just going to say pi r squared. And now what about this middle piece? Well the middle piece has a height of h so I'm going to say that this right here is h now what is this other distance? Well that other distance is the distance that you would walk around that circle which is also known as the circumference. So this dimension right here is your circumference which is equal to 2 times pi times r. So to find the area of this base you're going to need to multiply 2 times pi times r times h, so the surface area of the whole cylinder is going to be the area of the 2 circles plus the area of that middle piece the lateral area. So when you're calculating surface area you're going to need to know a couple of things, the radius of your circle and the height of your cylinder.

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