Volume Of Sphere

Spheres

Volume of a Sphere
A sphere is a solid with all its points the same distance from the center. The distance is known as the radius of the sphere. The maximum straight distance through the center of a sphere is known as the diameter of the sphere. The diameter is twice the radius.

The volume of a sphere is the amount of three-dimensional space it occupies.

The following figure gives the formula for the volume of a sphere. Scroll down the page for examples and solutions.

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Formula for the Volume of a Sphere
The volume of a sphere is equal to four-thirds of the product of pi and the cube of the radius.
The formula for the volume (V) of a sphere is:
\(V = \frac{4}{3} πr^3 \)

Where:
V = Volume of the sphere
π (pi) is a mathematical constant, approximately 3.14159.
r = radius of the sphere (the distance from the center of the sphere to any point on its surface)

The formula for the surface area of sphere is:
A = 4πr²

How to calculate the volume of a sphere?

1. Identify the radius (r):
   If you’re given the diameter (d), remember that the radius is half of the diameter: r = d/2.
   If you’re given the surface area (SA), you can find the radius using SA = 4πr², so
   \( r = \sqrt{\frac{SA}{4π}} \)
   
2. Cube the radius (r³) and multiply by 3/4 and π.
   
3. The volume will always be in cubic units.
   

Example:
Calculate the volume of sphere with radius 4 cm.

Solution:
Volume of sphere

We can also change the subject of the formula to obtain the radius given the volume.

Example:
The volume of a spherical ball is 5,000 cm³. What is the radius of the ball?

Solution:

How to use the formula to calculate the volume of a sphere?

Example:
Find the volume of a sphere with a diameter of 14 cm.

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Volume of a hemisphere

What is a hemisphere?

A hemisphere is half a sphere, with one flat circular face and one bowl-shaped face.

How to find the volume of a hemisphere?

The volume of a hemisphere is equal to two-thirds of the product of pi and the cube of the radius.

The volume of a hemisphere is given by the formula:

where r is the radius.

How to find the volume of a hemisphere?

Example:
Find the volume of a fishbowl with a diameter of 33cm.

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How to solve word problems about spheres?
The following video shows how to solve problems involving the formulas for the surface area and volume of spheres.

Example:
A sphere has a volume of 288π. Find its area. Leave you answer in terms of π

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Example:
A ball has a diameter of 18 cm.
a) Sketch a cylinder that fits the ball and label its height and base.
b) What is the volume of the cylinder?
c) What is the volume of the ball?

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Example:
The cylinder is melted down into a sphere of radius r. Find an expression for r in terms of x.

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How to proof the Formula of the Volume of a Sphere?
This video gives a proof for the formula of the volume of a sphere that does not involve calculus. This is an "approximated" proof. You would need to use calculus for a more rigorous proof.

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How to derive the formula of a sphere using calculus?

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How Archimedes derived the volume of a sphere?
To do so, he had to use a formula for the volume of a cone.

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