# Frustum - Definition, Volume, and Surface Area

## TL;DR

A frustum is the 3D solid that remains when you slice the top off a cone or pyramid with a cut parallel to the base. For a conical frustum with radii R and r and height h, the volume is \(\frac{1}{3}\pi h (R^2 + r^2 + Rr)\); this article gives the volume and surface-area formulas for both cone and pyramid frustums, with six worked examples.

## What Is A Frustum?

A **frustum** is the portion of a solid - a [**cone**](/content/math/geometry/cone/index.html) or a **pyramid** - that is left after the top is **cut off by a plane parallel to the base**. The word comes from Latin for "a piece broken off."

The cut has to be **parallel to the base**. That is what makes the two faces of a frustum - the base and the smaller top - similar shapes (two circles for a cone, two matching polygons for a pyramid). A slanted cut would give a different, harder solid.

Two everyday kinds:

- **Conical frustum** - from cutting a cone. Two circular faces of radii R (bottom) and r (top). Think bucket or lampshade.

- **Pyramidal frustum** - from cutting a pyramid. Two similar polygon faces. Think of a truncated square pyramid, like many building bases or a table lamp base.

The sloping sides connecting the two faces are the **lateral surface**. For a cone frustum, this is a curved band; for a pyramid frustum, it is a set of trapezoidal faces.

## What Is The Volume Of A Frustum?

For a **conical frustum** with bottom radius R, top radius r, and height h (the perpendicular distance between the two faces):

\[V = \frac{1}{3}\pi h (R^2 + r^2 + Rr)\]

The symbols:

- R = radius of the **larger (bottom)** circular face.
- r = radius of the **smaller (top)** circular face.
- h = perpendicular **height** between the two faces.
- \(\pi \approx 3.14159\).

## What Is the Surface Area of a Frustum?

First the **slant height** l of a conical frustum - the length up the sloping side, not the vertical height:

\[l = \sqrt{h^2 + (R - r)^2}\]

Then the areas:

- **Curved (lateral) surface area:** \(CSA = \pi l (R + r)\)
- **Total surface area:** \(TSA = \pi l (R + r) + \pi R^2 + \pi r^2\)

The total surface area is the curved band **plus** the two circular ends.

For a **pyramidal frustum**, each sloping face is an [isosceles trapezoid](/content/math/geometry/isosceles-trapezoid/index.html), and the lateral surface area is the sum of those trapezoid areas. The total surface area adds the two polygon end-faces.

## Examples of Frustum

### Example 1

**Find the slant height of a conical frustum with R=5 cm, r=2 cm, h=4 cm.**

\[l = \sqrt{h^2 + (R - r)^2} = \sqrt{4^2 + (5 - 2)^2} = \sqrt{16 + 9} = 5 \text{ cm}\]

Final answer: slant height 5 cm.

### Example 2

**Find the volume of a conical frustum with R=6 cm, r=3 cm, h=7 cm.**

\[V = \frac{1}{3} \cdot 3.14\cdot 7 \cdot (36 + 9 + 6 \cdot 3) = \frac{1}{3}  \cdot 3.14 \cdot 7 \cdot 63 = 461.6 \text{ cm}^3\]

Final answer: about 461.6 cm³.

### Example 3

**Find the curved surface area of the frustum from Example 1.**

\[CSA = \pi l (R + r) = 3.14 \cdot 5 \cdot (5 + 2) = 109.9 \text{ cm}^2\]

Final answer: about 109.9 cm².

### Example 4

**Find the total surface area of the same frustum.**

\[TSA = 109.9 + 3.14 \cdot 25 + 3.14 \cdot 4 = 200.96 \text{ cm}^2\]

Final answer: about 200.96 cm².

### Example 5

**A bucket is a conical frustum, open at the top, with bottom radius 10 cm, top radius 15 cm, and height 12 cm.**

\[V = \frac{1}{3} \cdot 3.14 \cdot 12 \cdot (225 + 100 + 150) = 5966 \text{ cm}^3\]

Final answer: roughly 5.97 litres.

### Example 6

**A truncated square pyramid has a bottom square of side 8 cm, a top square of side 4 cm, and height 9 cm.**

\[V = \frac{1}{3}\cdot 9 \cdot \left(64 + 16 + \sqrt{64 \cdot 16}\right) = 336 \text{ cm}^3\]

Final answer: 336 cm³.

## Why the Frustum Matters

The frustum is the shape you reach for whenever a full cone or pyramid would be impractical, which turns out to be constantly.

## The Mistakes Students Make Most Often

### Mistake 1: Dropping the Rr term in the volume

The correct way is \(V = \frac{1}{3}\pi h (R^2 + r^2 + Rr)\).

### Mistake 2: Using vertical height and slant height incorrectly

### Mistake 3: Mixing up which radius is R and which is r in surface area

## Conclusion

- A **frustum** is a cone or pyramid with its top cut off by a plane parallel to the base.
- Conical-frustum volume: \(\frac{1}{3}\pi h (R^2 + r^2 + Rr)\).
- Slant height \(l = \sqrt{h^2 + (R - r)^2}\); curved surface \(\pi l (R + r)\); total surface area adds \(\pi R^2 + \pi r^2\).
- Use vertical height for volume and slant height for surface area.