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Volume of a Sphere – Formula, Derivation & Examples

volume of a sphere

A sphere is a round 3D shape with a curved surface. The most common real-world examples of a sphere are football and the globe. The space occupied by a sphere is called the volume of a sphere. It’s also referred to as the capacity of a sphere and it depends on the radius of the sphere. So, what is the volume of sphere formula? In this article, we are going to look at the formula and also the derivation of the formula for the volume of a sphere. The volume of a sphere is measured in terms of cubic units, such as $cm^{3}$, $m^{3}$, $mm^{3}$, $in^{3}$, $ft^{3}$, etc.

Let’s learn how to find the volume of a sphere and its formula.

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What is the Volume of a Sphere?

The volume of a sphere is the three-dimensional space occupied by a sphere. This volume depends on the radius of the sphere (i.e, the distance of any point on the surface of the sphere from its centre). 

The formula for calculating the volume of a sphere is $\frac {4}{3} \pi r^{3}$.

Derivation of Formula for Volume of a Sphere

The formula for the volume of a sphere was worked out by Archimedes. The procedure to derive the formula for the volume of a sphere is as follows.

 If the radius of a cylinder, a cone, and a sphere is $r$ and they have the same cross-sectional area, their volumes are in the ratio of $1:2:3$. 

Hence, the relation between the volume of a sphere, the volume of a cone, and the volume of a cylinder is given as:

$\text{Volume of Cylinder} = \text{Volume of Cone} + \text{Volume of Sphere}$

$=> \text{Volume of Sphere} = \text{Volume of Cylinder} – \text{Volume of Cone}$

As we know, the volume of cylinder = $\pi r^{2}h$ and volume of cone = one-third of the volume of cylinder = $\frac {1}{3} \pi r^{2}h$.

volume of a sphere

$\text{Volume of Sphere} = \text{Volume of Cylinder} – \text{Volume of Cone}$

$=> \text{Volume of Sphere} = \pi r^{2}h – \frac {1}{3} \pi r^{2}h = \frac {2}{3} \pi r^{2}h$

In this case, $\text{height of cylinder} = \text{diameter of sphere} = 2r$

Hence, volume of sphere is $\frac {2}{3} \pi r^{2} h = \frac {2}{3} \pi r^{2} \times 2r = \frac {4}{3} \pi r^{3}$.

The volume of a Hollow Sphere

Consider a hollow sphere such that the radius of the outer sphere is $R$ and the radius of the inner sphere is $r$, such that the thickness of the layer of a sphere is $R – r$.

volume of a sphere

$\text{Volume of the layer of a sphere} = \text{Volume of Outer Sphere} – \text{Volume of Inner Sphere}$

$ = \frac {4}{3} \pi R^{3} – \frac {4}{3} \pi r^{3} = \frac {4}{3} \pi (R^{3} – r^{3})$

Examples

Ex 1: Find the volume of a sphere of radius $10.5 cm$.

Radius of a sphere $r = 10.5 cm$

Volume of a sphere = $\frac {4}{3} \pi r^{3} = \frac {4}{3} \times \frac {22}{7} \times 10.5^{3} = 4851 cm^{3}$.

Ex 2: If the sphere has a surface area of $256 \pi m^{2}$, what is the volume?

Let the radius of the sphere = $r$

Surface area of a sphere = $4 \pi r^{2}$

Therefore, $4 \pi r^{2} = 256 \pi =>4r^{2} = 256 =>r^{2} = \frac {256}{4} => r^{2} = 64 => r = 8 m$

Volume of sphere = $\frac {4}{3} \pi r^{3} = \frac {4}{3} \pi \times 8^{3} = \frac {4}{3} \pi \times 512 = \frac {2048}{3} \pi m^{3}$

Ex 3: A typical baseball is $76mm$ in diameter. Find the baseball’s volume in cubic centimeters.

Diameter of a baseball $d = 76 mm$

Therefore, radius of a baseball $r = \frac {76}{2} = 38 mm = 3.8 cm$

Volume of sphere = $\frac {4}{3} \pi r^{3} = \frac {4}{3} \times \frac {22}{7} \times 3.8^{3} = 229.94 c.c.$ (Rounded off to $2$ decimal places$.

The volume of the baseball is $229.94 c.c.$.

Ex 4: The radius of an inflated spherical balloon is $7 feet$. Suppose air is leaking from the balloon at a constant rate of $26$ cubic feet per minute. How long will it take for the balloon to be completely deflated?

Radius of inflated spherical balloon $r = 7 ft$

Volume of air inside the balloon = $\frac {4}{3} \pi r^{3} = \frac {4}{3} \times \frac {22}{7} \times 7^{3} = 1437.33 ft^{3}$

Rate of air leakage = $26 ft^{3}$ per minute

Therefore, the time taken to completely deflate the balloon = $\frac {1437.33}{26} = 55.28$ minutes.

Famous Math Competitions for Kids

Sphere – A 3D Solid Shape

A sphere is a 3D solid figure, which is round in shape. From a mathematical perspective, it is a combination of a set of points connected with one common point at equal distances in three dimensions. Some examples of a sphere include a football, a soap bubble, a globe, etc.

volume of a sphere

The important elements of a sphere are as follows:

  • Radius: The length of the line segment drawn between the center of the sphere to any point on its surface. If $O$ is the centre of the sphere and $A$ is any point on its surface, then the distance $OA$ is its radius. 
  • Diameter: The length of the line segment from one point on the surface of the sphere to the other point which is exactly opposite to it, passing through the centre is called the diameter of the sphere. The length of the diameter is exactly double the length of the radius.
  • Circumference: The length of the great circle of the sphere is called its circumference. 

Properties of Sphere

The following properties of a sphere will help you to identify a sphere easily. 

  • It is symmetrical in all directions.
  • It has only a curved surface area.
  • It has no edges or vertices.
  • Any point on the surface is at a constant distance from the center known as radius.
  • A sphere is not a polyhedron because it does not have vertices, edges, and flat faces. A polyhedron is an object that should definitely have a flat face.
  • Air bubbles take up the shape of a sphere because the sphere’s surface area is the least.
  • Among all the shapes with the same surface area, the sphere would have the largest volume. 

Types of Spheres

A sphere can be one of the following types.

  • Solid Sphere: A solid object in the form of a sphere is called a solid sphere. It is more like a sphere filled up with the same material it is made up of.
  • Hollow Sphere: If a solid sphere is cut and taken out of a big solid sphere, leaving behind a thin surface in the form of a spherical shell is called a hollow sphere. It is more like a balloon or ball filled with air.
volume of a sphere

Practice Problems

  1. Find the volume of a sphere of radius $5 cm$.
  2. What is the volume of a sphere with a diameter of $36 mm$?
  3. If the radius of a sphere is doubled, how much its volume increases?
  4. The volume of a sphere is $523 cubic metres. Find the radius of the sphere.
  5. A spherical ball is made up of a thin metal sheet of thickness $5 mm$. If the outer radius of the spherical ball is $42 mm$, then find the volume of the metal in the sphere.

FAQs

What is the volume of a sphere?

The volume of a sphere is the amount of space in it. The volume of a sphere formula for a sphere of radius $r$ is $\frac {4}{3}\pi r^{3}$.

How do you find the volume of a sphere?

The formula used to find the volume of a sphere of $r$ is $\frac {4}{3}\pi r^{3}$.

How the amount of material used in a sphere is calculated?

The formula for the amount of material used in a sphere is $\frac {4}{3} \pi \left(R^{3} – r^{3} \right)$
where $R$ is the outer radius
$r$ is the inner radius
Thickness of material = $R – r$

What is the volume of a sphere using diameter?

Let us consider a sphere of radius $r$ and diameter $d$. The volume of a sphere of radius $r$ is $V = \frac {4}{3} \pi r^{3}$. 
We also know that $r = \frac {d}{2}$. 
By substituting this in the above formula we get $V = \frac {1}{6} \pi d^{3}$

Conclusion

The volume of a sphere is the number of cubic units, occupied by the sphere completely and is calculated by using its radius. If $r$ is the radius of a sphere, then the formula for computing its volume is $\frac {4}{3} \pi r^{3}$.

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