Building blocks

# Cones Add to your resource collection Remove from your resource collection Add notes to this resource View your notes for this resource

## Solution

You may have used sports “cones” like these at school:

The mathematical name for a truncated cone such as this is a frustum of a cone.

What volume of plastic is needed to make such a “cone”?

Our chosen approach is to begin by finding the surface area of the frustum.

Once we have done this, if we know the thickness of the plastic, then a good estimate of the volume would be the surface area multiplied by the thickness. (This does not give the exact volume, but the error will be small in comparison to the error in measuring the thickness of the plastic.)

Therefore, to get a decent estimate of the volume of the “cone”, get a sports “cone”, measure various lengths, and use these to work out the surface area. Then estimate or measure the thickness and use these to work out the volume.

So let us return to the question of working out the surface area.

The images in the suggestions proposed thinking about a whole cone or about cutting the frustum open.

Throughout this solution, we’ve chosen to use algebra rather than actual measurements from a real “cone”. We will then put in our measurements at the end to arrive at a final numerical answer.

Why might we have done this?

We begin by drawing a sketch of the frustum within a complete cone, as in the suggestion, and indicating a variety of lengths and angles on it.

With an actual sports “cone”, we can easily find $r$ and $R$ (by measuring the diameters and halving them) and $s$ (by direct measurement). With a bit of effort, we can measure $h$ (by holding a straight edge vertically and seeing where the top of the “cone” lies on it). None of the other lengths or angles are easy to measure directly, so we will have to work them out from $r$, $R$ and $s$ (or possibly $h$) as needed.

Here are two different routes to finding the surface area, based on the two ideas in the suggestions.

The answer we obtained was in terms of $R$, $r$ and $s$. It might also be useful also to have an answer in terms of $\theta$ but not $s$. How can we rewrite it?

How can we check that our answer is sensible?

What volume of plastic is needed to make such a “cone”?