The Frustum of a Cone: Quick Revision

A brief note on The Frustum of a Cone. Read it before a test. Try each question at the end before you open its answer.

The shape and its measures

• A frustum is a cone with its top cut off by a plane parallel to the base. It has a bottom radius $R$, a top radius $r$ and a height $h$.
• Slant height: $l = \sqrt{h^2 + (R - r)^2}$.
• The small cone cut off is similar to the complete cone: $\dfrac{r}{R} = \dfrac{H - h}{H}$. So $H = \dfrac{hR}{R - r}$ and $L = \dfrac{lR}{R - r}$.
• The net of the curved surface is a ring with outer radius $L$, inner radius $L - l$ and angle $\dfrac{R}{L} \times 360^\circ$.

Areas and volume

• Curved surface: $\pi(R + r)\,l$.
• Total surface of a closed frustum: $\pi(R + r)\,l + \pi R^2 + \pi r^2$. An open bucket has the curved surface and the base only.
• Volume: $\dfrac{\pi h}{3}(R^2 + Rr + r^2)$.
• Each comes from the big cone minus the small cone. With $r = R$ the formulas give the cylinder.
• A cone cut parallel to its base: the small cone with scale $f$ has $f^2$ of the curved surface and $f^3$ of the volume.

A worked example

A frustum has bottom radius $6\ \text{cm}$, top radius $2\ \text{cm}$ and height $9\ \text{cm}$. Find its volume in terms of $\pi$.

1. The bracket is $R^2 + Rr + r^2 = 36 + 12 + 4 = 52\ \text{cm}^2$.
2. The volume is $\dfrac{\pi \times 9\ \text{cm}}{3} \times 52\ \text{cm}^2 = 156\pi\ \text{cm}^3$.

Practice with new numbers

A program to try

The program finds the volume of a frustum by the formula and as the difference of two cones.

Mistakes to avoid

• Using the height in place of the slant height in $\pi(R + r)l$.
• Writing $(R + r)^2$ in the bracket of the volume. It is $R^2 + Rr + r^2$.
• Counting the open end of a bucket as metal.
• Using the square for volumes of similar cones. Volumes go with the cube.

Check yourself

1. A frustum has radii $5\ \text{cm}$ and $2\ \text{cm}$ and height $4\ \text{cm}$. What is its slant height?

2. For that frustum, find the curved surface area in terms of $\pi$.

3. A small cone is cut from the top of a cone and has scale $\frac{1}{3}$. What fraction of the volume does it have?

4. A bucket is open at the top and its top is the wide end. Which circle counts as part of its surface?