Question

To calculate the volume of the right circular cone.

Short answer

The volume of the circular cone is \(\frac{1}{3}\pi {r^2}h\).

Step-by-step solution

Given data

The right circular cone with height \(h\) and base radius \(r\).

The dimensions of the right circular cone as shown in Figure 1 .

Refer to Figure 1.

Consider that the right circular cone is obtained by by rotation of the line \(y = \frac{r}{h}x\) about \(x\)-axis.

The region lies between \(a = 0\) and \(b = h\).

Concept used of Right Circular Cone

The volume of a right circular cone can be calculated by the use of the base radius and height of the cone.

Solve to find the volume

The expression to find the volume of the right circular cone as shown below.

\(V = \int_a^b A (x)dx\) …..(1)

Find the area of the right circular cone as shown below.

\(\begin{aligned}{}A(x) = \pi {r^2}\\ = \pi {\left( {\frac{r}{h}x} \right)^2}\\ = \frac{{\pi {r^2}{x^2}}}{{{h^2}}}\end{aligned}\)

Substitute 0 for a, h for \(b\), and \(\frac{{\pi {r^2}{x^2}}}{{{h^2}}}\) for \(A(x)\) in Equation (1).

\(\begin{aligned}{}V = \int_0^h {\frac{{\pi {r^2}{x^2}}}{{{h^2}}}} dx\\ = \frac{{\pi {r^2}}}{{{h^2}}}\left( {\frac{{{x^3}}}{3}} \right)_0^h\\ = \frac{{\pi {r^2}}}{{{h^2}}}\left( {\frac{{{h^3}}}{3} - 0} \right)\\ = \frac{1}{3}\pi {r^2}h\end{aligned}\)

Therefore, the volume of the right circular cone is \(\frac{1}{3}\pi {r^2}h\).