Question

To calculate the volume of the cap of a sphere.

Short answer

The volume of the cap of a sphere is \(\frac{1}{3}\pi {h^2}(3r - h)\).

Step-by-step solution

Given data

The cap of a sphere with radius \(r\) and height \(h\).

Consider that the Equation of circle \({x^2} + {y^2} = {r^2}\) …..(1)

Rearrange Equation (1).

\(\begin{aligned}{l}{x^2} = {r^2} - {y^2}\\x = \sqrt {{r^2} - {y^2}} \end{aligned}\)

The dimensions of the cap of a sphere as shown in Figure 1.

Refer to Figure 1.

Consider that the cap of a sphere is obtained by rotation of the circle \({x^2} + {y^2} = {r^2}\) about \(y\)-axis.

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

Concept used of Sphere

The formula to determine a sphere’ssurfacearea is\(4\pi {r^2}\), its volume is determined by\(\frac{4}{3}\pi {r^3}\)^.

Solve to find the volume

The expression to find the volume of the cap of a sphere as shown below.

\(V = \int_a^b A (y)dy\) ……(2)

Find the area of the cap of a sphere as shown below.

\(\begin{aligned}A(y) &= \pi {r^2}\\ &= \pi {\left( {\sqrt {{r^2} - {y^2}} } \right)^2}\\ &= \pi \left( {{r^2} - {y^2}} \right)\end{aligned}\)

Substitute \(r - h\) for a, r for \(b\), and \(\pi \left( {{r^2} - {y^2}} \right)\) for \(A(y)\) in Equation (2).

\(\begin{aligned}V &= \int_{r - h}^r \pi \left( {{r^2} - {y^2}} \right)dy\\ &= \pi \left( {{r^2}y - \frac{{{y^3}}}{3}} \right)_{r - h}^r\\ &= \pi \left( {{r^2} \times r - \frac{{{r^3}}}{3} - \left( {{r^2}(r - h) - \frac{{{{(r - h)}^3}}}{3}} \right)} \right)\\ &= \pi \left( {{r^3} - \frac{{{r^3}}}{3} - (r - h)\left( {{r^2} - \frac{{{{(r - h)}^2}}}{3}} \right)} \right)\end{aligned}\)

\(\begin{aligned}V &= \pi \left( {\frac{{2{r^3}}}{3} - \frac{{(r - h)}}{3}\left( {3{r^2} - \left( {{r^2} + {h^2} - 2rh} \right)} \right)} \right)\\ &= \frac{1}{3}\pi \left( {2{r^3} - (r - h)\left( {2{r^2} + 2rh - {h^2}} \right)} \right)\\ &= \frac{1}{3}\pi \left( {2{r^3} - 2{r^3} - 2{r^2}h + r{h^2} + 2{r^2}h + 2r{h^2} - {h^3}} \right)\\ &= \frac{1}{3}\pi \left( {3r{h^2} - {h^3}} \right)\end{aligned}\)

Which means, \(V = \frac{1}{3}\pi {h^2}(3r - h)\)

Therefore, the volume of the cap of a sphere is \(\frac{1}{3}\pi {h^2}(3r - h)\).