Volume of Frustum

The volume of frustum is the space occupied by it. Let us understand what is a frustum by taking an example. Let us assume that a cone is divided (or chopped) into two parts by a plane parallel to its base (or perpendicular to its height). Among these two parts, the part that contains the base of the cone is called the frustum of the cone. It is also called a truncated cone.

The volume of a frustum is calculated by the formula V = H/3 (S₁ + S₂ + √(S₁S₂)), where H is the height of the frustum and S₁ and S₂ are the areas of its bases. We will further use this formula to calculate the volume of frustum of cone as well.

The volume of a frustum is the amount of space that is present inside it (or) the quantity of matter that it can hold. It is measured in cubic units such as cm³, m³, in³, etc. When a three-dimensional shape with vertex (or apex) is cut by a plane (that is parallel to the base of the shape) into two parts, the part of the shape that contains the base of the shape is called the frustum of the shape. For example, a square pyramid can be cut into two parts as mentioned above, then one of the parts with the base is called the frustum of a square pyramid. There are different types of frustums like the frustum of cone (or truncated cone), the frustum of a square pyramid (or truncated square pyramid), the frustum of a triangular pyramid (or truncated triangular pyramid), etc. A frustum is determined by:

• Its height.
• Its base radius 1 (radius of one base).
• Its base radius 2 (radius of the other base).

The volume of any frustum (of any shape) can be calculated using its height and the areas of its bases. Let us consider a frustum of height H and base areas \(S_1\) and \(S_2\). Then its volume is calculated using the formula:

Volume of frustum, V = \(\dfrac{H}{3}\left(S_{1}+S_{2}+\sqrt{S_{1} S_{2}}\right)\), where

• H = Height of the frustum (the distance between the centers of two bases of the frustum)
• \(S_1\) = Area of one base of the frustum
• \(S_2\) = Area of the other base of the frustum

The formula which we learned in the previous section can be used to calculate the volume of any frustum and hence it can be used to calculate the volume of the frustum of a cone as well. While solving the problems in geometry, we usually come across with frustum of a cone and here we will see how to derive the formula of the volume of a frustum of a cone. There are two methods for the same. In both the methods, consider a cone of height H + h and base radius R. Also, consider the frustum of a cone of height H with a small base radius 'r' and the large base radius 'R'. Here, L and L + l are the slant heights of the frustum and the cone respectively. Then the volume of the frustum of the cone is,

Volume of frustum of cone = πh/3 [ (R³ - r³) / r ] (OR)

Volume of frustum of cone = πH/3 (R² + Rr + r²)

Note: Here π is a constant whose value is 22/7 (or) 3.141592653...

Method 1 to Derive the Volume of Frustum of Cone Formula

We will use the volume of frustum formula (from the previous section) to derive the volume of the frustum of a circular cone. The base areas (areas of circles) of the frustum of the cone from the above figure are:

\(S_1\) = πR²

\(S_2\) = πr²

Substituting these values in the volume of frustum formula,

V = \(\dfrac{H}{3}\left(S_{1}+S_{2}+\sqrt{S_{1} S_{2}}\right)\)

V = \(\dfrac{H}{3} (\pi R^2 + \pi r^2 + \sqrt{\pi R^2 \cdot \pi r^2})\)

V = \(\dfrac{\pi H}{3}\left(R^{2}+R r+r^{2}\right)\)

We derived one of the formulas of volume of the frustum of the cone in this method. But we can derive both the formulas using the following method.

Method 2 to Derive the Volume of Frustum of Cone Formula

The volume of the full cone is, πR² (H + h) / 3.

The volume of the cone (with apex) that is cut is, πr²h / 3.

We have,

The volume of frustum of the cone, V = The volume of the full cone - The volume of the cone that is cut

V = πR² (H + h) / 3 - πr²h / 3 ... (1)

The triangles OBC and PQC are similar (by AA property of similarity) and thus,

(H + h) / h = R / r ... (2)

H + h = Rh / r ... (3)

Substituting this in (1),

V = πR² · (Rh / r) - πr²h / 3

V = πh/3 [ (R³ - r³) / r ]

We derived one formula of volume of the frustum of the cone. Now we will derive another formula from this.

From (2),

(H / h) + 1 = R / r

H / h = (R / r) - 1

H / h = (R - r) / r

Reciprocating on both sides,

h / H = r / (R - r)

h = (H r) / (R - r)

Substituting this in the above formula,

V = (π / 3) [ (H r) / (R - r) ] [ (R³ - r³) / r ]

Using one of the algebraic formulas, a³ - b³ = (a - b) (a² + ab + b²). By applying this formula to R³ - r³,

V = (π / 3) [ (H r) / (R - r) ] [ (R - r) (R² + Rr + r²) / r ]

V = πH/3 (R² + Rr + r²)

Hence, we derived the second formula of the volume of frustum of the cone as well.

Note: The relation between the slant height (L), height (H), and base radii R and r of the frustum of the cone (using the Pythagoras theorem) is, L² = H² + (R - r)². We may need to use this while solving the problems related to the volume of the frustum of a cone.

What Is the Volume of Frustum of Cone?

A frustum of a cone is a part of a cone when its upper/lower part (with apex) is cut by a plane that is parallel to its base. The volume of frustum of cone is the amount of space that is inside the frustum. It is calculated by the formula: V = πH/3 (R² + Rr + r²), where r and R are the radii of the bases of the frustum and H is its height.

What Is the Volume of the Frustum of a Rectangular Pyramid Formula?

The frustum of a rectangular pyramid is a portion of it that is left after its top portion (with apex) is chopped by a plane that is parallel to the base rectangle. The volume of a rectangular pyramid is the amount of space that is occupied by it and is given by the formula:

V = \(\dfrac{H}{3}\left(S_{1}+S_{2}+\sqrt{S_{1} S_{2}}\right)\), where

• H = Height of the frustum
• \(S_1\) = The area of the one base rectangle of the frustum
• \(S_2\) = The area of the other base rectangle of the frustum

What Is the Volume of Frustum Formula?

The volume of a frustum of any shape can be calculated by using its height 'H' and its base areas \(S_1\) and \(S_2\). The formula to calculate the volume (V) of the frustum is,

V = \(\dfrac{H}{3}\left(S_{1}+S_{2}+\sqrt{S_{1} S_{2}}\right)\)

What Is the Volume of Square Frustum Formula?

The square frustum is a frustum that is formed when a square pyramid's portion (containing apex) is cut by a plane that is parallel to the base (square). The volume of a square frustum is calculated by the formula:

V = \(\dfrac{H}{3}\left(S_{1}+S_{2}+\sqrt{S_{1} S_{2}}\right)\), where

• H = Height of the frustum
• \(S_1\) = The area of the one base square of the frustum
• \(S_2\) = The area of the other base square of the frustum

What Is the Volume of the Frustum of a Cone Formula Using Slant Height?

Consider a frustum of radii 'R' and 'r', and height 'H' which is formed by a cone of base radius 'R' and height 'H + h'. The relation between the height (H), slant height (L), and base radii R and r of the frustum of the cone is, L² = H² + (R - r)². We can use this formula to calculate one of the unknown values among 'r', 'R', and 'H' and then we can find the volume (V) of the frustum using the formula:

V = πH/3 (R² + Rr + r²)

What Is the Volume of the Frustum of a Cone Formula?

There are two formulas that are used to calculate the volume of a frustum of a cone. Consider a frustum of radii 'R' and 'r', and height 'H' which is formed by a cone of base radius 'R' and height 'H + h'. Its volume (V) can be calculated by using:

• V = πh/3 [ (R³ - r³) / r ] (OR)

• V = πH/3 (R² + Rr + r²)