CAT 2025 Lesson : Mensuration - Cone, Pyramid & Frustum

3.5 Right Circular Cone

A right circular cone is a circular cone whose tip (or vertex) is perpendicular to the centre of its circular base. This is very often referred to as just cone in the entrance tests.

In the adjacent figure, $\bm{r}$ is the radius of the base. $\bm{h}$ is the height of the cone. $\bm{l}$ is the slant height of the cone, wherein $l = \sqrt{r^{2} + h^{2}}$ Volume of cone =$\dfrac{1}{3}\pi r^{2}h$ Curved Surface Area (CSA) = $\pi rl$ Flat Surface Area (FSA) = $\pi r^{2}$ Total Surface Area (TSA) = $\pi r(l+r)$

3.6 Right Pyramid

Like Right Circular Cones, all the points of a polygonal base are connected to a single tip that in a pyramid. bottom and top of a right prism are identical. The difference between the two is that the circular cone has a circular base while a pyramid has a polygonal base.

By definition, a right pyramid is a figure with a tip (or vertex) which is perpendicular to the centre of its polygonal base.

In the adjacent figure, $\bm{h}$ is the height of the right pyramid. $\bm{l}$ is the slant height of the right pyramid. Volume of right pyramid =$\dfrac{1}{3} \times$ Area of base $\times h$ Lateral Surface Area (LSA) = $\dfrac{1}{2} \times$ Perimeter of base $\times l$ Total Surface Area (TSA) = LSA + Area of Base

3.7 Frustum of a Cone

When a cut parallel to the base of a cone is made, we would get a smaller cone (that contains the tip) and a frustum of the cone that contains the circular base of the initial cone.

In the adjacent figure, $\bm{r}$ is the radius of the smaller circular top. $\bm{R}$ is the radius of the larger circular base. $\bm{h}$ is the height of the frustum $\bm{l}$ is the slant height of the frustum.wherein $l = \sqrt{(R – r)^2 + h^2}$ Volume of frustum=$\dfrac{1}{3} \pi h \times (R^2 +Rr +r^2)$ Curved Surface Area (CSA) = $\pi l(R + r)$ Flat Surface Area (FSA) = $\pi R^2 + \pi r^2$ Total Surface Area (TSA) = $\pi (lR + lr + R^2 + r^2)$