Frustum of cone formulas with examples | Surface area and Volume

Contents

Conical Frustum Formulas – Slant height,Volume, Lateral, Base and Total surface areas

Frustum of A Cone

When cut a lower part of the right circular cone is called a frustum. We need to cut the right circular cone with a to cut it horizontally or parallel to the base of the cone, then the solid shape between the base and plane of the cone is called the frustum of a cone.

Formulas of Right Circular Cone

Volume of a Cone = V $= \frac{1}{3} \times \pi r^2 h$

Base Area of a cone $= \pi r^2$

Lateral surface area a cone $= \pi r \sqrt{h^2 \ + \ r^2}$

Total Surface area of a cone $= \pi r \left ( r + \sqrt{h^2 \ + \ r^2} \right )$

Slant height of a cone $S = \sqrt { h^2 + r^2 }$

Frustum of cone formulas – Volume of a Frustum of Cone

The volume of a frustum can be defined as a the quantity of matter that it can hold in the space that is present inside it .

Frustum of a cone of altitude “h” with a large base radius “R” and small base radius “r” . Slant heights “s” of the frustum and the cone.

Altitude ( h) is the perpendicular distance between the bases of a frustum of a cone.

Volume of a Frustum of a Cone $= \frac{\pi}{3} \times h \left ( R^2 + r^2 + rR \right )$

Frustum volume can be calculated using its height and the areas of its bases. Let us consider a frustum large base areas “B1” and small base area “B2” and height “h” and . Then its volume is calculated by using the following formula

Volume of a Frustum of a Cone $= \frac{h}{3} \time \left ( B1 + B2+ \sqrt{B1B2} \right )$

Generator or Slant height of frustum of cone formula

The distance measured along the lateral face of the frustum is called slant height of the same

The Slant of the Frustum of a Cone (S) $S = \sqrt { h^2 + (R-r)^2 }$

R= The radius of the cone (large base radius)

r= the radius of the Frustum cone (small base radius)

h= height of the Frustum cone

Slant height and Height of the cone which the frustum is a part of that cone.

Slant height of the cone of which the frustum is a part = $\frac{SR}{{R-r}}$

Height of the cone of which the frustum is a part $\begin{aligned} = \frac{h R}{R - r} . \end{aligned}$ = $\frac{hR}{{R-r}}$

Frustum of cone formulas – Surface area of a Frustum of Cone

What Is the Surface Area of Frustum?

The surface area of a frustum area is calculated by the sum of areas of its faces and is measured in square units.

With respect to the frustum, there are having two types of surface areas.

1. Lateral Surface Area (LSA) or Curved Surface Area (CSA) for the frustum of a cone

2. Total Surface Area (TSA) for the frustum of a cone.

The curved surface area or lateral surface area of a frustum of a cone is the area of its curved face.

Frustum of a cone of height “h” with a large base radius “R” and small base radius “r” and “s” slant height of the frustum.

Curved surface area of frustum of cone formula ( LSA or CSA) $= \pi \ S \ (R +r)$

Slant height of the a frustum = $S = \sqrt { h^2 + (R-r)^2 }$

The curved (or lateral) surface area of any frustum can be calculated using the circumferences (or perimeters) of the bases and the slant height.

C1 is the circumferences (or perimeters) of one base of the frustum, C2 is the circumferences (or perimeters) of another base of the frustum and slant heights “s” frustum and the cone respectively. Then

Lateral (or curved) surface area of a frustum $= \frac{1}{2} S (C1 + C2 )$