Cones

The Structure of a Cone

Cones are solids with a flat base, and one curved side. The flat base may be shaped like a circle or an ellipse. Most of the time, it's a circle. Cones are not polyhedrons as they have a curved surface, and no straight edges.

The flat bottom of a cone is called its base, and the pointy bit at the top of the cone is called its apex, as shown in the picture below.

How Do Mathematicians Build a (Circular) Cone?

You may have made a cone yourself at some stage, by taking a bit of cardboard shaped like the one shown below and bending it around until the edges meet.

I've made them as witches and wizards hats, and as lolly baskets for the Christmas tree.

Mathematicians have a slightly more abstract view of how to create a cone. They take a right-angled triangle and rotate it around one of its two shorter sides:

The side that the triangle is rotated around is called the axis of the cone.

Right and Oblique Cones

Just as with pyramids, there are two types of cones: right cones are cones in which the apex lies directly about the centre of the base, and oblique cones are cones in which it doesn't:

In the above picture, the cone on the left is a right cone, while the cone on the right is an oblique cone.

Finding the Surface Area of a Cone

For example, if we were asked to find the surface area of a cone with height \(h = 3\) cm and radius \(r = 4\) cm, then we would calculate \(s = \sqrt{3^2 + 4^2} = 5\) cm, and

Finding the Volume of a Cone

The formula for the volume of a right circular cone is

where \(h\) is the height of the cone, and \(r\) is the radius of its base.

For example, the volume of a cone with height \(h = 3\) cm and base radius \(r = 4\) cm is given by

Volumes of Cones and Volumes of Cylinders

Did you notice something about the formula for the volume of a cone? It looks a lot like the formula for the volume of a cylinder.

In fact, we can draw a cylinder of the same base radius and height around a cone, as shown in the picture below:

The formula for the volume of the cylinder is \(V = \pi r^2 h\), so the volume of the cone is equal to \(\dfrac{1}{3}\) of the volume of the cylinder of the same base radius and height.

If \(\dfrac{1}{3}\) is sounding awfully familiar to you, that's because the volume of a right pyramid is also equal to \(\dfrac{1}{3}\) of the volume of the prism of the same base and height. This isn't a coincidence: you can think of a cone as a "pyramid with a circular base".