एक लम्ब वृत्तीय शंकु का पृष्ठीय क्षेत्रफल

Surface area of a cone is the total area covered by its surface. The total surface area will cover the base area and lateral surface area of the cone. Cone is defined as a three-dimensional solid structure that has a circular base. A cone can be viewed as a set of non-congruent circular disks that are placed over one another such that the ratio of the radius of adjacent disks remains constant.

Fig.1 Right Circular Cone

Right circular cone shown in Fig. 1 has a vertex at the top, $r$ the base radius , $h$ the height of the cone and $l$ the slant height of the cone.

Curved Surface Area of a Cone = ${\frac {1}{2}}\times l\times 2\pi r=\pi rl$

Here $r$ the base radius , $l$ the slant height of the cone.

Also $l^{2}=r^{2}+h^{2}$ by applying Pythagoras Theorem. Here $h$ is the height of the cone.

Therefore, $l={\sqrt {r^{2}+h^{2}}}$

Total Surface Area of a Cone = $\pi rl+\pi r^{2}=\pi r(l+r)$

Examples

1. Find the curved surface area and the total surface area of a right circular cone whose slant height is $10$ cm and base radius is $7$ cm.

Solution:

Curved surface area = $\pi rl$

= ${\frac {22}{7}}\times 7\times 10=220$ cm²

Total surface area = $\pi r(l+r)$

=${\frac {22}{7}}\times 7\times (10+7)=374$ cm²