Geometry for Beginners - How to Find the Surface Area and Volume of Cones

Welcome to Geometry for Beginners. This article deals with the surface area and volume of cones. Probably the most common visual image people have of a cone is the ice cream cone; but my personal favorite comes from carnivals and state fairs--cinnamon sugar roasted pecans or cashews in their red and white striped cone-shaped package.
Both ice cream cones and cinnamon roasted nuts give us excellent examples of the applications of surface area and volume.

As with all the 3-dimensional figures, the application of surface area is the package or container.
For our visual images, surface area would be represented by the ice cream cone itself which holds the ice cream and the red and white cone-shaped paper that holds the nuts.
The application of volume would be the ice cream itself and the nuts that go in the paper cone.

For vendors at craft shows, fairs, and carnivals, both of these concepts are extremely important. Vendors cannot afford to run out of either the containers or the product that goes inside. Poor planning can be costly in terms of lost sales. These examples, of course, are not the only applications of cones, but they are some of the better tasting ones.

If you have already read the articles about prisms and pyramids, you know that they are similar to each other and have similar formulas.

The same is true for cylinders and cones. The difference is the issue of one base (cone) versus 2 bases (cylinder).

Formula for the Surface Area of Cones: SA = B + LA, where SA refers to surface area, B refers to the AREA of the base, and LA refers to lateral area.

This formula is exactly the same as the initial formula for the pyramid. CAUTION! This formula will get very different and may get difficult to memorize.

In other situations, I have advised only memorizing this initial formula and then substituting in the appropriate previously learned polygon formula.
This time, however, things are different. The shape we get when we open our cone is NOT any of the polygons when have learned before and we will need some new terminology. For a cone, the base is a circle, so the first change to the initial formula looks like SA = pi r^2 + LA.

It is this lateral area that will give us trouble.
Picture making a vertical slice in a cone like an Indian Tee Pee and then opening it and placing the opened shape out flat. The shape will look like a large wedge of pizza, but it will not be the entire pizza.

Now, using the same "limiting" process we used when calculating the area of circles, we will mentally slice this wedge into many pieces and fit them together alternating point up and point down.

We will, again, use the "taking the limit" of this process. The end result of this process is a rectangle whose length is half the circumference of the base circle--1/2( 2 pi r) or pi r and whose height is the slant height s. Slant height is the new terminology we must learn.

While the height of a cone is the perpendicular distance straight to the ground, the slant height is the height of the SIDE of the cone. It is the height of the material (leather) of the tee pee measured from top to bottom. It is the length or height of the slanted side of the cone. Making the final substitution, SA = B + LA becomes SA = (pi r^2) + (pi r) s, where r is the radius of the bottom circle and s is the slant height of the side of the cone.
Whew! Now you understand why I said you need to memorize this final formula. Fortunately, the volume formula is not that complicated.
The Formula for the Volume of Cones: V = (1/3) B h, where B is the AREA of the base and h is the perpendicular height of the cone.
The 1/3 comes from the fact that, just as with pyramids and prisms, it would take 3 cones to fill the cylinder having the same base and height.
Thus, V = (1/3) B h becomes V = (1/3) (pi r^2) h.

To summarize: (1) The formula for surface area of a cone is SA = B + SA or SA = pi r^2 + pi r s; and area is always measured in square units. (2) The formula for volume of a cone is V = (1/3) B h or V = (1/3) pi r^2 h; and volume is always measured with cubic units.