Surface Area of Frustum of Cone

The surface area of frustum of cone can be calculated by the difference between the surface area of the complete cone and the smaller cone (removed from the complete cone). The surface area of the frustum of cone can be calculated using the below diagram, where one needs to sum up the surface areas of the curved surfaces, and the surface areas of the top and bottom surfaces of the frustum of cone.

Similar to the Volume of the frustum of cone, the curved surface area will be also equal to the difference between the surface areas of the bigger cone and the smaller cone. 

In the figure given above, triangles OAB and OCD are similar. Therefore, using the similarity criteria, one can write,

l’ / l = r’ / r…(1)

Since, l’ = l – L, therefore, from equation (1),

(l – L) / l = r’ / r

After cross-multiplication, 

lr – Lr = lr’

l(r – r’) = Lr

l = Lr / (r – r’)…(2)

The curved surface area of a complete cone = πrl

The curved surface area of the smaller cone = πr’l’

Difference between the curved surface areas of complete cone and smaller cone = π (rl – r’l’)

Thus, the curved surface area (CSA) of the frustum of cone = πl (r – r’l’/l)

Use equation (1) to substitute the value of l’/l in the above equation, and simplify,

CSA of the frustum of cone = πl (r – r’×r’/r) = πl (r² – r’²)/r

Now, substitute the value of l from equation (2), and simplify,

CSA of the frustum of cone = πlr/(r – r’)× (r² – r’²)/r = πl (r + r’)

Thus, one can write,

Curved surface area of frustum of cone = πl (r + r’)

Now, let’s calculate the surface area of the top and bottom bases of the frustum of the cone, such that, 

The surface area of the top base of the frustum of cone having a radius r’ = πr’²

The surface area of the bottom base of the frustum of cone having a radius r = πr²

So,

Total surface area of the frustum of cone = Curved surface area of the frustum of cone + surface area of the top base + surface area of the bottom base

Therefore,

The total surface area of the frustum of cone = πl (r + r’) + πr’² + πr² = πl (r + r’) + π (r² + r’²) 

 Thus, the total surface area of the frustum of cone is = πl (r + r’) + π (r² + r’²) 

This formula can be also written as, 

The total surface area of the frustum of cone is = πl (r² – r’²)/r + π (r² + r’²) 

So, one can write,

Total surface area of frustum of cone = πl(r + r’) + π (r² + r’²) 

or

Total surface area of frustum of cone = πl (r² – r’²)/r  + π (r² + r’²) 

Note that, l is the slant height of the smaller cone that can be given as 

L = √ [H² + (r – r’)²]

Read, More

• Volume of  Cone
• Volume of Cylinder
• Volume of Sphere

Frustum of a cone is a special shape that is formed when we cut the cone with a plane parallel to its base. The cone is a three-dimensional shape having a circular base and a vertex. So the frustum of a cone is a solid volume that is formed by removing a part of the cone with a plane parallel to circular base. The frustum is not only defined for cones but can be also defined for the different types of pyramids (square pyramid, triangular pyramid, etc.).

Some of the common shapes of a frustum of cone which we discover in our daily life are buckets, lamp shade, and others. Let us learn more about the frustum of cones in this article.