Frustums - Theory & Concepts

Frustums and Truncated Apex Solids

$Learning Objectives$

Define a frustum and distinguish it from a general truncated solid.
Use similarity to reconstruct the original pyramid or cone from a frustum.
Calculate frustum volume by difference of similar solids and by the direct base-area formula.
Determine lateral and total surface area of regular pyramidal and right circular conical frustums.
Relate vertical height, slant height, and changes in corresponding base dimensions.
Solve inverse capacity and material-quantity problems involving frustums.
Check limiting cases and dimensional consistency of frustum formulas.

$Frustum$

A frustum is the portion of a pyramid or cone remaining after the apex is removed by a plane parallel to the base. Its two parallel bases are similar figures.

$Truncated Solid$

A truncated solid is produced when a solid is cut by a plane. It is a frustum only when the cutting plane is parallel to the original base of a pyramid or cone.

$Geometric Structure of a Frustum$

A frustum retains the similarity of the original pyramid or cone. Corresponding dimensions of the upper and lower bases have one common linear ratio. This similarity allows the removed small solid and the original full solid to be related through linear, area, and volume scale factors.

$Frustum Versus Oblique Cut$

Do not apply frustum formulas to a cone or pyramid cut by a nonparallel plane. A nonparallel cut does not generally produce similar upper and lower sections, so the standard direct frustum formulas no longer follow.

General Frustum Volume

For a frustum of a pyramid or cone with similar parallel bases, volume depends on the two base areas and perpendicular height.

V=\frac{h}{3}\left(A_1+A_2+\sqrt{A_1A_2}\right)

Variables

Symbol	Description	Unit
$V$	frustum volume	cubic units
$A_1$	area of the larger parallel base	square units
$A_2$	area of the smaller parallel base	square units
$h$	perpendicular distance between bases	linear units

$Why the Geometric-Mean Area Appears$

If corresponding base lengths have ratio $k$ , then $A_2=k^2A_1$ and $\sqrt{A_1A_2}=kA_1$ . The middle term therefore carries the linear similarity information needed to reproduce the difference between the full pyramid or cone and the smaller similar solid removed from its apex.

Conical Frustum Volume

For circular bases of radii R and r, the general base-area formula reduces to the standard conical-frustum expression.

V=\frac{\pi h}{3}\left(R^2+Rr+r^2\right)

Variables

Symbol	Description	Unit
$V$	volume	cubic units
$R$	larger base radius	linear units
$r$	smaller base radius	linear units
$h$	perpendicular height	linear units

Conical Frustum Slant Height

For a right circular conical frustum, the axial section forms a right triangle using the difference in radii.

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

Variables

Symbol	Description	Unit
$l$	slant height of the frustum	linear units
$h$	perpendicular height	linear units
$R$	larger radius	linear units
$r$	smaller radius	linear units

Lateral Area of a Conical Frustum

The curved lateral area is pi times the sum of radii times slant height.

A_L=\pi(R+r)l

Variables

Symbol	Description	Unit
$A_L$	lateral curved area	square units
$R$	larger base radius	linear units
$r$	smaller base radius	linear units
$l$	slant height	linear units

Total Area of a Closed Conical Frustum

Total area includes the curved surface and both circular bases.

A_T=\pi(R+r)l+\pi R^2+\pi r^2

Variables

Symbol	Description	Unit
$A_T$	total surface area	square units
$R$	larger radius	linear units
$r$	smaller radius	linear units
$l$	slant height	linear units

$Interactive Frustum Reconstruction and Verification$

Use the lab below to vary $R$ , $r$ , and $h$ . It simultaneously checks the direct frustum formula, difference of the reconstructed full and removed cones, and the prismatoidal $1:4:1$ relation while also reporting slant height and surface area.

Frustum Reconstruction and Verification Lab

Change $R$ , $r$ , and $h$ . The geometry, direct formula, difference-of-cones check, and exact prismatoidal $1:4:1$ check update together. Dimensions use model units $u$ .

Slant height

6.500 u

Removed height

6.000 u

Mid radius

3.750 u

Three independent volume checks

Direct274.8894 u³

Full − removed274.8894 u³

Prismatoidal274.8894 u³

Larger radius

R

5.0 u

Smaller radius

r

2.5 u

Perpendicular height

h

6.0 u

Frustum volume

274.889 u³

Lateral area

153.153 u²

Closed total area

251.327 u²

The reconstructed full-cone height is 12.000 u. As

r\to R

, the reconstruction apex moves arbitrarily far away and the frustum approaches a cylinder.

$Frustum Scaling and Capacity Growth$

A geometrically similar frustum scaled uniformly by $k$ has both radii and its perpendicular height multiplied by $k$ . Its corresponding surface areas scale by $k^2$ , while its volume and geometric capacity scale by $k^3$ . A capacity increase therefore requires the cube root—not the direct ratio—to recover the required linear enlargement.

$Interactive Frustum Scale Explorer$

Select Frustum below, adjust the two radii and perpendicular height, and vary $k$ to compare original and scaled area and capacity.

Volume, Surface Area, and Scale Explorer

Compare common solids, then change the similarity factor to see why corresponding area scales with $k^2$ and volume with $k^3$ . Dimensions use generic model units $u$ .

Base geometry

V=s^3

A=6s^2

Similarity check

A_2/A_1=2.250

V_2/V_1=3.375

Side5.0 u

Similarity scale

k

1.50

Original volume

125.00 u³

Scaled volume

421.88 u³

Area: original → scaled

150.00 → 337.50 u²

$Regular Pyramidal Frustum$

A regular pyramidal frustum is produced by cutting a regular pyramid with a plane parallel to its base. Its upper and lower bases are similar regular polygons, and each lateral face is an isosceles trapezoid.

Lateral Area of a Regular Pyramidal Frustum

The sum of the trapezoidal lateral faces equals one-half the sum of the base perimeters times the common face slant height.

A_L=\frac{1}{2}(P_1+P_2)l

Variables

Symbol	Description	Unit
$A_L$	lateral area	square units
$P_1$	perimeter of larger base	linear units
$P_2$	perimeter of smaller base	linear units
$l$	common slant height of a lateral trapezoid	linear units

Pyramidal Frustum Slant Height

For corresponding base apothems a1 and a2, the vertical height and apothem difference form the right triangle for face slant height.

l=\sqrt{h^2+(a_1-a_2)^2}

Variables

Symbol	Description	Unit
$l$	face slant height	linear units
$h$	perpendicular height	linear units
$a_1$	apothem of larger regular base	linear units
$a_2$	apothem of smaller similar base	linear units

$Difference-of-Similar-Solids Method$

A frustum may be analyzed as the full original pyramid or cone minus the smaller similar pyramid or cone removed from the apex. If the upper-to-lower linear scale factor is $k$ , then the small and full solids have height ratio $k$ and volume ratio $k^3$ . This method is especially useful when a problem provides original or extended heights instead of both base areas.

Similarity Reconstruction

Corresponding linear dimensions of the small removed solid and original solid are proportional to their apex-measured heights.

\frac{r}{R}=\frac{x}{x+h}=k

Variables

Symbol	Description	Unit
$r$	representative upper-base dimension	linear units
$R$	corresponding lower-base dimension	linear units
$x$	height of the removed similar solid	linear units
$h$	height of the frustum	linear units
$k$	upper-to-lower linear scale factor	dimensionless

$Reconstructing the Removed Apex Solid$

STEP-BY-STEP

Choose one pair of corresponding upper and lower linear dimensions, such as radii or polygon side lengths.
Let $x$ be the perpendicular height of the small removed pyramid or cone.
Write the similarity relation $r/R=x/(x+h)$ .
Solve for $x$ and compute the original full height $x+h$ .
Calculate the full and removed volumes, then subtract; or use the result as an independent check of the direct frustum formula.

$Capacity and Material Applications$

Frustums occur in hoppers, transition pieces, tapered bins, retaining or pedestal forms, chimneys, buckets, funnels, architectural features, and earthwork transitions. Capacity uses the interior dimensions, while material quantity may require subtracting an inner frustum from an outer frustum or evaluating only specified surfaces.

$Prismatoidal Cross-Check for a Conical Frustum$

For a conical frustum, radius varies linearly through the height, so cross-sectional area is a quadratic function of position. The prismatoidal formula is therefore exact when the end areas and the true halfway-section area are used. Because the halfway radius is $(R+r)/2$ , the $1:4:1$ calculation provides a strong independent check of the direct formula.

$Limiting-Case Checks$

A formula is easier to trust when its limiting behavior is correct. If $A_2=A_1$ , the general frustum volume becomes $V=A_1h$ , the prism/cylinder formula. If $A_2\to0$ , it becomes $V=A_1h/3$ , the pyramid/cone formula. For a conical frustum, if $r=R$ , the lateral area becomes $2\pi R h$ because $l=h$ , matching a cylinder.

$Common Frustum Errors$

Using slant height instead of perpendicular height in the volume equation.
Treating upper and lower base areas as if they vary linearly with corresponding side length; areas vary with the square of the linear ratio.
Omitting one base when total surface area is requested.
Applying regular-frustum lateral-area formulas to irregular or oblique configurations without a common face slant height.
Calling every cut cone or pyramid a frustum even when the cutting plane is not parallel to the base.

$Frustum Calculation Checklist$

Confirm the two bases are parallel and similar.
Convert diameters to radii before using circular formulas.
Use perpendicular height for volume and slant height for lateral area.
Decide whether the direct formula or difference-of-solids method is more efficient.
For regular pyramidal frustums, use corresponding base apothems to obtain face slant height.
Distinguish interior capacity from exterior material volume and distinguish lateral area from total area.
Test the result against a limiting case, prismatoidal calculation, or difference-of-solids calculation when practical.

Key Takeaways

A frustum is formed only by a cut parallel to the base of a pyramid or cone.
The universal frustum volume is $V=h(A_1+A_2+\sqrt{A_1A_2})/3$ for similar parallel bases.
A conical frustum has $V=\pi h(R^2+Rr+r^2)/3$ and $A_L=\pi(R+r)l$ .
A regular pyramidal frustum has $A_L=(P_1+P_2)l/2$ .
Similarity reconstruction, the prismatoidal formula, and difference of solids provide independent solution methods and checks.
Uniformly similar frustums scale area by $k^2$ and capacity by $k^3$ .
Volume uses perpendicular height; lateral area uses the appropriate slant height.

Previous TopicCylinders and Circular Solids - Examples & Applications

Quiz Me

Next TopicFrustums - Examples & Applications

Prev Next

Quiz Me