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 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, 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.