## Video Lesson: Calculating the Volume of a Spherical Cap

## Volume and Surface Area of a Spherical Cap Calculator

The calculator below calculates the volume and surface area of a spherical cap.

Simply enter the following values:

- ‘h’ is the height of the spherical cap
- ‘r’ is the radius of the sphere
- ‘a’ is the radius of the base of the spherical cap

## What is a Spherical Cap?

**A spherical cap is a region formed by making any plane cut through a sphere. It has a curved spherical top face and a circular face on its base. The resulting shape looks like a cap which is on the top or side of a sphere.**

Examples of spherical caps are shown in blue the image below.

Spherical caps are made whenever the side part of a sphere is cut off with a straight line.

A spherical cap can be described using the values of *‘a’*, *‘h’* and *‘r’*.

*‘a’*is the radius of the circular base of the spherical cap*‘h’*is the height of the cap*‘r’*is the radius of the sphere

A hemisphere is an example of a circular cap in which the plane cut through the sphere passes exactly through the centre of the circle.

## Formula to Find the Volume of a Spherical Cap

**The volume of a spherical cap in terms of ‘r’ and ‘h’ is V = (πh ^{2}/3)(3r-h), where r is the radius of the sphere and h is the height of the spherical cap.**

** **T**he volume of a spherical cap in terms of ‘a’ and ‘h’ is V = (πh/6)(3a^{2}+h^{2}), where a is the radius of the base of the cap and h is the height of the cap. **

*‘a’* is the radius of the circular base of the spherical cap*‘h’* is the height of the cap *‘r’* is the radius of the sphere

If the both the height of the cap, *‘h’* and the radius of the sphere, *‘r’* are known, then use the formula .

If both the radius of the base of the cap, *‘a’* and the height of the cap, *‘h’* are known, then use the formula .

## How to Find the Volume of a Spherical Cap

**To find the volume of a spherical cap, substitute the values of the sphere radius ‘r’, and height of the spherical cap ‘h’, into the formula V = (πh^{2}/3)(3r-h).**

### For example, calculate the volume of the spherical cap with a sphere diameter of 10m and a spherical cap height of 3m.

If the diameter of the sphere is 10m, then the radius is half of this.

The values of:

*‘r’* = 5m

*‘h’* = 3m

are substituted into .

becomes .

Evaluating this further, we obtain , which simplifies to .

The volume is approximately 113m^{3}.

### For example, calculate the volume of the spherical cap with sphere radius 3cm and spherical cap height of 2cm.

The values of:

*‘r’*= 3*‘h’*= 2

are substituted into .

becomes .

Evaluating this, we obtain which is approximately 29.3 cm^{3}.

### For example, calculate the volume of the spherical cap with base radius 5cm and a height of 2cm.

The values of:

*‘a’*= 5*‘h’*= 2

are substituted into the formula .

becomes .

This simplifies to which is approximately 82.7 cm^{3}.

## Derivation of the Formula for the Volume of a Spherical Cap

**The formula for the volume of a spherical cap can be derived by integrating π𝑥^{2} with respect to y between the limits of r and r-h.**

The full derivation of the formula from integrals is shown below.

**Step 1. Form the integral equation**

The centre of the sphere is taken as the origin point, with the y-axis vertically and the x-axis horizontally.

The volume of the spherical cap can be found by finding the sum of the circular discs as shown in red below. Each circular disc has a different radius, which will be of length x.

The area of each circle is therefore *π*x^{2}.

We want to sum the areas of the circles in the y direction. Therefore the volume is found as the integral of .

In the y-direction, the top of the spherical cap is at a y value of r and the base of the cap is at a y value of r-h.

Therefore the volume is given by the integral .

In order to be able to perform the integration, we need to find x in terms of y.

From Pythagoras’ theorem, the radius of the circle is connected to x and y by the equation .

Rearranging for x, we obtain .

Substituting into , we obtain .

This simplifies to .

**Step 2. Integrate and simplify**

We now integrate to obtain .

Substituting the limits of integration, we obtain .

This simplifies to following the algebraic steps shown below.

## Surface Area of a Spherical Cap

**The curved surface area of a spherical cap is given by Area = 2 πrh where r is the radius of the sphere and h is the height of the cap. Alternatively, the curved surface area is also given by Area = π(a^{2}+h^{2}), where a is the radius of the circle on the base of the cap and h is the height of the cap.**

**The base area of the spherical cap is πa^{2} and the total surface area of a spherical cap can be given by SA=2πrh+πa^{2} or SA=π(2a^{2}+h^{2}).**

### For example, calculate the total surface area of a spherical cap with a height of 4 cm and a cap base radius of 5 cm.

We substitute:

*‘a’*= 5*‘h’*= 4

Into the equation to obtain .

This can be evaluated to which is approximately 207 cm^{2}.

### For example, calculate the outer surface area of a spherical cap with a height of 6 cm and a spherical radius of 12 cm.

We substitute:

*‘h’*= 6*‘r’*= 12

Into the equation to obtain .

This can be evaluated as which is approximately 452 cm^{2}.