Frustum of a pyramid (or cone) is a portion of pyramid (or cone) included between the base and the section parallel to the base not passing through the vertex.

### Formula for Volume of a Frustum

The volume of a frustum is equal to one-third the product of the altitude and the sum of the upper base, the lower base, and the mean proportional between the bases. In symbols

$V = \frac{1}{3}\left( A_1 + A_2 + \sqrt{A_1A_2} \right)h$

Derivation of formula for volume of frustum of pyramid and frustum of cone

## Frustum of a Regular Pyramid

Frustum of a regular pyramid is a portion of right regular pyramid included between the base and a section parallel to the base.

### Properties of a Frustum of Regular Pyramid

• The slant height of a frustum of a regular pyramid is the altitude of the face.
• The lateral edges of a frustum of a regular pyramid are equal, and the faces are equal isosceles trapezoids.
• The bases of a frustum of a regular pyramid are similar regular polygons. If these polygons become equal, the frustum will become prism.

### Elements of a Frustum of Regular Pyramid

a = upper base edge
b = lower base edge
e = lateral edge
h = altitude
L = slant height
A1 = area of lower base
A2 = area of upper base
n = number of lower base edges

### Formulas for Frustum of a Regular Pyramid

Area of Bases, A1 and A2
See the formulas of regular polygon for the formula of A1 and A2

Volume

$V = \frac{1}{3}\left( A_1 + A_2 + \sqrt{A_1A_2} \right)h$

Lateral Area, AL
The lateral area of frustum of regular pyramid is equal to one-half the sum of the perimeters of the bases multiplied by the slant height.

$A_L = \frac{1}{2} n (a + b)L$

The relationship between slant height L, lower base edge b, upper base edge a, and lateral edge e, of the frustum of regular pyramid is given by

$(b - a)^2 + 4L^2 = 4e^2$

## Frustum of a Right Circular Cone

Frustum of a right circular cone is that portion of right circular cone included between the base and a section parallel to the base not passing through the vertex.

### Properties of Frustum of Right Circular Cone

• The altitude of a frustum of a right circular cone is the perpendicular distance between the two bases. It is denoted by h.
• All elements of a frustum of a right circular cone are equal. It is denoted by L.

### Formulas for Frustum of Right Circular Cone

Area of lower base, A1
$A_1 = \pi R^2$

Area of upper base, A2
$A_2 = \pi r^2$

Lateral Area, AL
The lateral area of the frustum of a right circular cone is equal to one-half the sum of the circumference of the bases multiplied by slant height. Expand the derivation of formula for lateral area of frustum of a right circular cone.

Derivation of Formula for Lateral Area of Frustum of a Right Circular Cone

Let C and c, the circumference of lower and upper bases, respectively.

$A_L = \frac{1}{2}(C + c)L$

A more convenient formula is when we substitute C = 2πR and c = 2πr, giving us

$A_L = \pi (R + r) L$

Volume, V
The volume of a frustum of any cone is equal to one-third of the product of the altitude and the sum of the upper base, the lower base, and the mean proportional between the two bases. See the derivation of formula for the volume of any frustum.

For any Frustum, the volume is $V = \frac{1}{3}\left( A_1 + A_2 + \sqrt{A_1 A_2} \right) h$. For frustum of right circular cone, $A_1 = \pi R^2$ and $A_2 = \pi r^2$. Thus,

$V = \frac{1}{3}\left[ \pi R^2 + \pi r^2 + \sqrt{\pi R^2 (\pi r^2)} \right] h$

$V = \frac{1}{3} \left[ \pi R^2 + \pi r^2 + \pi Rr \right] h$

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

Relationship between L, R, h, and r
The relationship between the lower base radius R, upper base radius r, altitude h, and element L can be found using Pythagorean theorem. It is given by
$(R - r)^2 + h^2 = L^2$

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