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A frustum is a Latin word that means ‘piece cut off’. When a solid (generally a cone or a pyramid) is cut in such a manner that the base of the solid and the plane cutting the solid are parallel to each other, part of the solid which remains between the parallel cutting plane and the base is known as a frustum of that solid. Some common examples of a frustum of a cone are the shade of a table lamp, bucket, glass tumbler, etc.

The surface area of a frustum is basically the sum of areas of its faces and is measured in square units like the area of any other shape. i.e., it is measured in $m^{2}$, $cm^{2}$, $ft^{2}$, $in^{2}$, etc.

Let’s learn how to find the surface area of a frustum of a cone and its methods and formulas.

## Frustum of a Cone – A 3D Solid Shape

The frustum of a cone is the part of the cone without a vertex when the cone is divided into two parts with a plane that is parallel to the base of the cone.

### Properties of a Frustum of a Cone

The following properties of a frustum of a cone will help you to identify a sphere easily.

- The frustum of a cone doesn’t contain the vertex of the corresponding cone but contains the base of the cone.
- The frustum of a cone is determined by its height and two radii (corresponding to two bases).
- The height of the frustum of a cone is the perpendicular distance between the centers of the two bases of the frustum.
- If the cone is a right circular cone, then the frustums formed from it also would be right-circular.

### Slant Height of a Frustum of a Cone

Consider a frustum of a cone of height $h$ cut and radii of upper(smaller) and lower(larger) bases as $r$ and $R$ respectively.

Now consider a right $\triangle \text{ABC}$, right-angled at $\text{B}$, where, $\text{AB} = h$ (height of a frustum), $\text{BC} = \text{R} – \text{r}$ and $\text{AC} = l$.

Using Pythagorean formula, we get $l = \sqrt{\text{AB}^{2} + \text{BC}^{2}} = h^{2} + \left(\text{R} – \text{r} \right)^{2} $.

## Surface Area of Frustum of a Cone

The area occupied by the surface/boundary of a frustum of cone is known as the surface area of frustum of a cone. It is always measured in square units. As it has two flat bases, thus it has a total surface area as well as a curved surface area.

The surface area of frustum a cone is measured as the “number of square units” ($cm^{2}$, $m^{2}$, $in^{2}$, $ft^{2}$, etc.). There are two types of surface areas of frustum a cone

- Curved Surface Area
- Total Surface Area

### Curved Surface Area of Frustum of a Cone

The curved surface area of a frustum of a cone is the area of its curved face.

Let’s derive the formula for the curved surface area of the frustum of a cone.

#### Method 1 to Derive the Curved Surface Area of Frustum of a Cone Formula

We will use the surface area of the frustum formula (from the previous section) to derive the surface area of the frustum of a cone. The base circumferences (circumference of circles) of the frustum of the cone from the above figure are:

$C_{1} = 2 \pi R$ and $C_{2} = 2 \pi r$

Substituting these values in the curved surface area of the frustum formula,

CSA = $\frac {1}{2} \times \left(C_{1} + C_{2} \right) \times L$

CSA = $\frac {1}{2} \times \left(2 \pi \times R + 2 \pi r \right) × L$

CSA = $ \pi L \left(R + r \right)$

#### Method 2 to Derive the Curved Surface Area of Frustum of a Cone Formula

The curved surface area of the full cone is, $\pi R \left(L + l \right)$.

The curved surface area of the cone (with apex) that is cut is $\pi rl$.

The curved surface area (CSA) of a frustum of the cone = The curved surface area of the full cone – The curved surface area of the cone that is cut, therefore,

CSA = $ \pi R \left(L + l \right) – \pi rl$ ———————————– (1)

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

$\frac {L + l}{l} = \frac {R}{r}$ ———————————————- (2)

$L + l = \frac {Rl}{r}$ ——————————————————— (3)

Substituting this in (1),

CSA = $\pi R \frac{Rl}{r} – \pi rl$

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

From (2), we get $\frac {L}{l} + 1 = \frac {R}{r} => \frac {L}{l} = \frac {R}{r} – 1 => \frac {L}{l} = \frac {R – r}{r}$

Reciprocating on both sides,

$\frac {l}{L} = \frac{r}{R – r} => l = \frac {Lr}{R – r}$

Substituting this in the above formula,

CSA = $\pi \frac {Lr}{R – r} \times \frac {R^{2} – r^{2}}{r}$

Using one of the algebraic formulas, $a^{2} – b^{2} = \left(a – b \right) \left(a + b \right)$. By applying this formula to $R^{2} – r^{2}$ we get

CSA = $ \pi \frac {Lr}{R – r} \times \frac {\left(R – r \right) \left(R + r \right)}{r}$

**$\text{CSA} = \pi L \left(R + r \right)$**

### Total Surface Area of Frustum of a Cone

The total surface area of a frustum of a cone or a pyramid is the sum of the areas of all its faces. The total surface area of frustum of a cone is obtained by adding

- Curved surface area of a frustum of a cone
- Area of an upper (smaller) circular base
- Area of a lower (larger) circular base

As seen above, the curved surface area of the frustum of a cone is $\pi l \left(R + r \right)$.

Area of an upper (smaller) circular base = $\pi r^{2}$ and area of a lower (larger) circular base = $\pi R^{2}$.

Therefore, the total surface area of a frustum of a cone is

$\text{TSA} = \pi l \left(R + r \right) + \pi r^{2} + \pi R^{2}$

**$\text{TSA} = \pi l \left(R + r \right) + \pi \left(r^{2} + R^{2} \right)$**

### Examples

**Ex 1:** Find the total surface area of the frustum of a right circular cone of height $20 in$, large base radius to be $25 in$, and slant height to be $29 in$. Express the answer in terms of $\pi$.

The large base radius of the frustum is $R = 25 in$

Let its small base radius be $r$.

The height of the frustum of the cone is $H = 20 in$

Its slant height is $L = 29 in$

We know that, $L^{2} = H^{2} + \left(R – r \right)^{2}$

Therefore, $29^{2} = 20^{2} + \left(25 – r \right)^{2} =>841 = 400 + \left(25 – r \right)^{2} => 441 = \left(25 – r \right)^{2}$

Taking square root on both sides,

$21 = 25 – r => r = 4$

Thus, the total surface area of the given frustum of a right circular cone is,

TSA = $\pi L \left(R + r \right) + \pi \left(R^{2} + r^{2} \right)$

= $\pi \times 29 \times \left(25 + 4 \right) + \pi \times \left(25^{2} + 4^{2} \right) = 1482 \pi$

Therefore, the TSA of a frustum of the cone = $1482 \pi in^{2}$.

**Ex 2:** A cone is cut by a plane horizontally. The radius of the circular top and base of the frustum is $10m$ and $3m$, respectively. The height of the frustum is $24m$. If the height of the cone is 28m, then find the lateral surface area of a frustum.

Let radii are $r_{1} = 10m$ and $r_{2} = 3m$

Height, $h = 24m$

First, we need to find the slant height of frustum, by the formula:

$l = \sqrt{(r_{1} + r_{2})^{2} + h^{2}} = \sqrt{ \left(10 – 3 \right)^{2} + 24^{2}} = \sqrt{49 + 576} = \sqrt{625} = 25 m$

Lateral surface of frustum = $\pi \left(r_{1} + r_{2} \right)l$

LSA = $\pi \left(10+3 \right) \times 25 = 325 \pi$ sq.m.

## Conclusion

The surface area of a frustum of a cone is the area occupied by the curved surface of the frustum and the area of the two flat circular bases in a three-dimensional space. In the case of a frustum of a cone, there are two types of surface area, viz, curved surface area and total surface area.

## Practice Problems

- Find the surface area of a frustum of a cone, whose larger and smaller radius is $12 cm$ and $8 cm$. The height of the cone is $3 cm$.
- Find the surface area of a frustum of a cone, whose larger and smaller radius is $1.5 cm$ and $0.2 cm$. The slant height of the cone is $2.5 cm$ (Use $\pi = 3.14$)
- Find the surface area of a frustum of a cone, whose larger and smaller radius is $25 cm$ and $12 cm$. The slant height of the cone is $30 cm$ (Use $\pi = 3.14$)

## Recommended Reading

- Volume of a Prism(Formula, Derivation & Examples)
- Volume of a Sphere – Formula, Derivation & Examples
- Volume of a Cone(Formula, Derivation & Examples)
- Volume of a Cylinder(Formulas, Derivation & Examples)
- Volume of Cuboid – Formulas, Derivation & Examples
- Volume of a Cube – Derivation, Formula & Examples
- Surface Area of a Pyramid(Definition, Formula & Examples)
- Surface Area of a Prism(Definition, Formulas & Examples)
- Surface Area of a Sphere(Definition, Formulas & Examples)
- Surface Area of a Cone(Definition, Formulas & Examples)
- Surface Area of a Cylinder(Definition, Formulas & Examples)
- Surface Area of a Cone(Definition, Formulas & Examples)
- Surface Area of A Cube (Definition, Formula & Examples)
- Surface Area of Cuboid (Definition, Formula & Examples)
- Area of Rectangle – Definition, Formula & Examples
- Area of Square – Definition, Formula & Examples
- Area of a Triangle – Formulas, Methods & Examples
- Area of a Circle – Formula, Derivation & Examples
- Area of Rhombus – Formulas, Methods & Examples
- Area of A Kite – Formulas, Methods & Examples
- Perimeter of a Polygon(With Formula & Examples)
- Perimeter of Trapezium – Definition, Formula & Examples
- Perimeter of Kite – Definition, Formula & Examples
- Perimeter of Rhombus – Definition, Formula & Examples
- Circumference (Perimeter) of a Circle – Definition, Formula & Examples
- Perimeter of Square – Definition, Formula & Examples
- Perimeter of Rectangle – Definition, Formula & Examples
- Perimeter of Triangle – Definition, Formula & Examples
- What Are 2D Shapes – Names, Definitions & Properties

## FAQs

### What is a frustum of a cone?

When a cone is cut by a plane horizontally, parallel to the base of the cone, then the lower part of the cone is called a frustum.

### What is the formula for the surface area of a frustum of a cone?

There are two types of surface area for the frustum of a cone.

1) Curved surface area (CSA): Its formula is $\text{CSA} = \pi L \left(R + r \right)$

2) Total surface area (TSA): Its formula is $\pi l \left(R + r \right) + \pi \left(r^{2} + R^{2} \right)$

### What is the formula for the slant height of a frustum of a cone?

The formula for the slant height of a frustum of a cone is $l = h^{2} + \left(\text{R} – \text{r} \right)^{2} $.

Where $h$ is the height of a frustum

$R$ is the radius of the larger base

$r$ is the radius of the smaller base