A three-dimensional shape is a solid shape with height or depth. For example, the sphere, cuboid, cone, etc. are three-dimensional. The surface area of a three-dimensional shape is the sum total of all of the surface areas of each of the sides. Children will like to think of the shape as a birthday present and the surface area as the wrapping gift paper. Cone is a three-dimensional structure that has a circular base. In this topic, we will discuss the surface area of a cone formula with examples. Let us begin learning!

**The Surface Area of a Cone Formula**

**What is a Cone?**

Cones are pyramid-like structures having a circular base. It is a three-dimensional shape and is a smooth base that tapers at a top point called the vertex. While studying about finding the surface area of cones, we consider a right circular cone that has a circular base. We can consider a cone as a triangle that is being rotated about one of its vertices.

**What is Surface Area?**

When we are finding the surface area of a 3-D shape, think of it as unfolding the shape, or flattening it out, and then finding the area of each side. When we add all of these areas up, we have the surface area. In order to find the area of a 3-D shape, we must know how to find the area of the basic shapes that make up the sides of the 3-D shape.

For a solid object, the space covered from all sides is termed as the surface area of the object. By measuring surface area we can measure the area of material required to cover the 3-D object completely. Since it is the computation of the area, therefore its unit is a square meter or square centimeter or likewise. Computation of the surface area depends upon the shape and size.

Source: wikihow.com

**The Formula for Surface Area of a Cone**

Now, we may compute the total curved portion, with the perimeter of the base of the cone. The circumference of the base of the \(cone = 2\pi\; r\), where r is the base of the radius of the circular base.

So the curved surface area = \(\frac{1}{2} \times l \times 2\pi = \pi \; r \; l\)

We already know that cones constitute the right-angled triangle. In a right-angled triangle, Pythagoras Theorem helps us find out the length of the slant side, with the help of formula:

\(l^2 = r^2+ h^2\)

The area of the circular base = \(\pi \times r^2\).

**Area of the curved surface= \(\pi r l\)**

Total Surface Area of the Cone = Area of its circular base + Area of the total curved surface

Thus, the total Surface Area of a Cone = \(\pi r^2+ \pi r l\)

**SA = \(\pi\;r \;(l + r)\)**

r | The radius of the base circle |

l | Slant height |

SA | Total Surface area |

h | Height of cone |

**Solved Examples Surface Area of a Cone Formula**

Q.1: Find the surface area of a cone of radius 14 cm and height 6 cm.

Solution:

Given, r = 14 cm

H = 6 cm

Thus Slant Height of the cone will be,

l = \(\sqrt{r^2 + h^2} \) = \(\sqrt{14^2+ 6^2} \)

l = 15.23 cm

Now, total surface area of the cone = \(\pi\;r \;(l + r)\)

= \(\frac{22}{7} \times 14 \times (15.23 + 14) \)

= \(44 \times 29.23 \)

= \(1286.12\; square \;cm\)

Thus total surface area = 1286.12 square cm.

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