Surface Area of Frustum with Examples and FAQs

What Is the Surface Area of Frustum?

The surface area is of two types:

(1) Curved Surface Area: The curved surface area of a frustum of a cone is the area surrounded by its curved face. In other words, it’s the area that excludes the area of the circular top and bottom surfaces.

(2) Total Surface Area: The total surface area of a frustum of a cone is the sum of the areas enclosed by all its faces.

Formula for Surface Area of Frustum

The figure represents a frustum, where h is its height, l is the slant height and r_{1} \text { and } r_{2} \text { (where } r_{1}>r_{2} \text { ) } the radii of the lower and upper circular bases respectively, of the frustum of a cone.

\text { Slant height, l }=\sqrt{h^{2}+\left(r_{1}-r_{2}\right)^{2}}

The formula for the Curved Surface Area (C.S.A.) of the Frustum is:

\text { C.S.A }=\pi \times\left(\mathbf{r}_{1}+\mathbf{r}_{2}\right) \times \mathbf{l}

The formula for the Total Surface Area(T.S.A.) of the Frustum is:

\text { T.S.A. } =\text { C.S.A. }+\text { Area of the upper face }+\text { Area of the base }

                =\pi\left(r_{1}+r_{2}\right) {\text{l}}+\pi r_{1}^{2}+\pi r_{2}^{2}

 The value of π is taken to be \frac{22}{7} \text { or } 3.14.

Examples

Example 1: Calculate the curved surface area and total surface area of the frustum of a right circular cone of height 12 cm, large base radius to be 20 cm, and smaller base radius to be 15 cm. Find the surface area in terms of π only.

Solution: 

The height of the frustum of the cone, h = 12 cm.

The large base radius of the frustum,r_{1}=20 \mathrm{~cm}.

The small base radius of the frustum, r_{2}=15\mathrm{~cm}.

We know that,

\text { Slant height, } =\sqrt{h^{2}+\left(r_{1}-r_{2}\right)^{2}}

                              =\sqrt{12^{2}+(20-15)^{2}} \mathrm{~cm}

                             =\sqrt{144+(5)^{2}} \mathrm{~cm}=\sqrt{169} \mathrm{~cm}=13 \mathrm{~cm}

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

\text { C.S.A. } =\pi \times\left(r_{1}+r_{2}\right) \times l

               =\pi \times(20+15) \times 13 \mathrm{~cm}^{2}

               =455 \pi \mathrm{cm}^{2}

And, the total surface area of the frustum is,

\text { T.S.A. } =\pi\left(r_{1}+r_{2}\right) l+\pi r_{1}^{2}+\pi r_{2}^{2}

                =\left(455 \pi+\pi \times 20^{2}+\pi \times 15^{2}\right) \mathrm{cm}^{2}

               =(455 \pi+400 \pi+225 \pi) \text{ }\mathrm{cm}^{2}

               =1080 \pi \text{ } \mathrm{cm}^{2}

Hence the curved surface area of the frustum of the right circular cone is 455 \pi \mathrm{cm}^{2} and its total surface area is 1080\pi \text{  } \mathrm{cm}^{2} .

Example 2: If a cone is cut by a plane horizontally, we get a frustum. The radii of the circular top and base of the frustum are 9 m and 4 m, respectively. The slant height of the frustum is 10 m. Then find the curved surface area of the frustum taking the value of π as 3.14.

Solution: 

The large base radius of the frustum,r_{1}=9 m .

The small base radius of the frustum, r_{2}=4m .

Slant height, l = 10 m

\text{Hence the curved surface area of the frustum of the cone is } 408.2 \mathrm{~m}^{2}.