What a frustum is

A cone with its top cut off

Look at a bucket, a drinking glass, a flower pot or a lampshade. Each one is wider at one end than at the other, and each has two circular ends. They are not cones, because they have no tip. They are not cylinders, because the ends are different. Each one is a cone with its pointed top cut off.

The solid that is left, when a plane parallel to the base cuts the top off a cone, is a frustum of a cone. The plural is frustums or frusta. Here are the names of its parts.

• $R$ is the radius of the larger circle, the bottom of the frustum.
• $r$ is the radius of the smaller circle, the top.
• $h$ is the height: the perpendicular distance between the two circles.
• $l$ is the slant height: the length along the side, from a point on one rim to the nearest point on the other rim.

NCERT and some state boards write the radii as $r_1$ and $r_2$ with $r_1 > r_2$. Cambridge IGCSE and GCSE usually write $R$ and $r$, as we do here. The frustum is in the older NCERT books, in several state boards and in the IGCSE and GCSE syllabuses for the higher tier. It is not in the current NCERT text. If your course leaves it out, you still meet cones and similar triangles, which is the reason it is worth learning.

The slant height

Draw the side view of the frustum. It is a trapezium with two parallel sides, the diameters of the two circles. Drop a perpendicular from the top-left corner to the base. It cuts off a right triangle on the side. The legs are the height $h$ and the difference of the radii $R - r$. The hypotenuse is the slant height. By Pythagoras,

The complete cone

Continue the two slanting sides upwards until they meet at the tip. You get the complete cone. It has height $H$ and slant height $L$. The part you cut off is a small cone, with height $H - h$, radius $r$ and slant height $L - l$.

The small cone has the same shape as the complete cone, only smaller. It is similar to it. In similar figures all lengths are in the same ratio. So

Take the first equality and solve for $H$. Cross-multiply: $rH = R(H - h) = RH - Rh$. Then $Rh = RH - rH = H(R - r)$, so

The second formula is the same step done for the slant heights. We want $H$ here, so we use the rearranged form.

Example 1. A frustum has bottom radius $9\ \text{cm}$, top radius $3\ \text{cm}$ and height $8\ \text{cm}$. Find its slant height and the height and slant height of the complete cone.

1. The slant height. $l = \sqrt{8^2 + (9 - 3)^2}\ \text{cm} = \sqrt{100}\ \text{cm} = 10\ \text{cm}$.
2. The complete cone. $H = \dfrac{8\ \text{cm} \times 9}{9 - 3} = 12\ \text{cm}$ and $L = \dfrac{10\ \text{cm} \times 9}{9 - 3} = 15\ \text{cm}$.
3. The small cone. Its height is $12\ \text{cm} - 8\ \text{cm} = 4\ \text{cm}$ and its slant height is $15\ \text{cm} - 10\ \text{cm} = 5\ \text{cm}$.
4. Check. The small cone has radius $3\ \text{cm}$ and height $4\ \text{cm}$, so its slant height is $\sqrt{3^2 + 4^2}\ \text{cm} = 5\ \text{cm}$. This agrees.

Example 2. A cone of height $20\ \text{cm}$ and base radius $12\ \text{cm}$ is cut by a plane parallel to the base, $8\ \text{cm}$ above the base. What is the radius of the circle made by the cut?

1. The small cone. Its height is $20\ \text{cm} - 8\ \text{cm} = 12\ \text{cm}$.
2. Similar cones. $\dfrac{r}{12\ \text{cm}} = \dfrac{12\ \text{cm}}{20\ \text{cm}}$, so $r = 7.2\ \text{cm}$.

The net of the curved surface

Cut a lampshade along its slant side and flatten it. You get a ring-shaped piece of a circle. The outer edge is an arc of radius $L$, and the inner edge is an arc of radius $L - l$. They are the two arcs of the complete cone's net and of the small cone's net, with the same centre.

The outer arc is as long as the bottom rim, $2\pi R$. A full circle of radius $L$ has circumference $2\pi L$. So the ring is the fraction $\dfrac{R}{L}$ of a full circle, and its angle is

For Example 1 the angle is $\dfrac{9}{15} \times 360^\circ = 216^\circ$.

Practise with new numbers

Try it in Python

This program takes a frustum and works out its slant height, and the height and slant height of the complete cone. It checks the ratio of the small cone to the complete cone in two ways.

Common mistakes

• Using the height as the slant height. The slant is longer, because it is the hypotenuse.
• Using $R + r$ in the right triangle. The base of the triangle is $R - r$, the difference.
• Forgetting the part that was cut off. The complete cone is taller than the frustum.
• Using the wrong ratio in similar triangles. The small cone is to the complete cone as $r$ is to $R$, and its height is $H - h$, not $h$.

Remember these

• A frustum is a cone with its top cut off by a plane parallel to the base.
• The slant height is $l = \sqrt{h^2 + (R - r)^2}$.
• The small cone cut off is similar to the complete cone: $\dfrac{r}{R} = \dfrac{H - h}{H}$.
• $H = \dfrac{hR}{R - r}$ and $L = \dfrac{lR}{R - r}$.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. A frustum has radii $10\ \text{cm}$ and $4\ \text{cm}$ and height $8\ \text{cm}$. Find its slant height.

2. For the same frustum, find the height of the complete cone.

3. A cone of height $30\ \text{cm}$ is cut by a plane parallel to its base, $10\ \text{cm}$ below the tip. What is the ratio of the radius of the cut circle to the base radius?

4. Is a frustum with $R = r$ still a frustum? What solid is it?

5. The flat net of the curved surface of a lampshade is a ring with outer radius $25\ \text{cm}$. The bottom radius of the shade is $10\ \text{cm}$. What is the angle of the ring?