Volume of A Sphere

Introduction To Volume of A Sphere

The volume of sphere is the measure of space that may be filled by a spherical. If we make a circle on a piece of paper, take a circular disc, paste a thread along its diameter and spin it along the string. This gives us the form of a spherical.

The unit of volume of a sphere is expressed as the (unit)3. The metric units of volume are cubic metres or cubic centimetres whereas the USCS units of volume are, cubic inches or cubic feet. The volume of sphere relies on the radius of the sphere, so altering it alters the volume of the sphere. There are two sorts of spheres, solid sphere, and hollow sphere. The volume of these sorts of spheres is different. We shall study in the next sections about their volumes.

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Derivation of Volume of A Sphere

A cylinder, cone, and sphere with the same cross-sectional area all have the same volume, hence their volumes are equal to 1:2:3. To put it in simple terms, the relationship between the volume of each of these three objects may be summarised as follows:

Volume of Cylinder = Volume of Cone + Volume of Sphere

⇒ Volume of Sphere = Volume of Cylinder - Volume of Cone
As we know, the volume of cylinder = πr²h and volume of cone = one-third of the volume of cylinder = (1/3)πr²h
The volume of Sphere = Volume of Cylinder - Volume of Cone
⇒ Volume of Sphere = πr²h - (1/3)πr²h = (2/3)πr²h
In this case, height of cylinder = diameter of sphere = 2r
Hence, volume of sphere is (2/3)πr²h = (2/3)πr²(2r) = (4/3)πr³

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How To Calculate Volume of Sphere?

It takes up all of the space inside a sphere. Volume of sphere formula may be used to determine spherical's mass. To figure out how big a sphere's volume is, follow these steps:

The radius of the sphere must be determined in the first step of the process.

This is the second step in the process of making a cube of radius.

R3 is multiplied by (4/3) in step 3.

Add the units to your final answer in step four.

Using this method, we can compute the volume of a sphere. Let's start with a simple example.

Example: Find the volume of sphere having a radius of 4 inches.
Solution: As we know, the volume of sphere, V = (4/3)πr³
Here, r = 4 inches
Thus, volume of sphere, V = (4/3)πr³ = ((4/3) × π × 4³) in³
⇒ V = 268.08 in³
∴ The volume of sphere is 268.08 in³.

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Volume of Sphere Examples

Example 1: What is the amount of air that can be held by a spherical ball of diameter 14 inches?
Solution: We need to find the volume of the ball.
The radius of the ball will be half the diameter = 14/2 inches = 7 inches

Using the volume of sphere formula, the volume of the ball is
Volume of the ball = (4/3)πr³ = ((4/3) × (22/7) × 7³) =1436.75 in³
∴ The amount of air that can be held by the spherical ball of diameter 14 inches is 1436.75 cubic inches.

Example 2: Maria has three wax marbles of radii 6 inches, 8 inches, and 10 inches. She melted all the marbles to recast them into a single solid marble. Can you find the radius of the resulting marble?
Solution: Let the radius of 3 marbles be r1r1, r2r2 and r3r3 and the radius of resulting marble is R. Tp calculate the value of R lets form the equation using the volume of sphere.
Volume of resulting marble = Volume of marble 1 + Volume of marble 2 + Volume of marble 3
(4/3)πR³ = (4/3)π(r1)(r1)³ +(4/3)π(r2)(r2)³ + (4/3)π(r3)(r3)³
⇒ R³ = ( (r1(r1)³+ (r2r2)³+ (r3r3)³
⇒ R³ = 6³ + 8³ + 10³ = 1728
⇒ R = 12 inches

∴ The radius of the resulting marble is 12 inches.

Example 3:

Find the volume of the sphere. Round to the nearest cubic meter.

Solution

The formula for the volume of a sphere is

V=43πr3V=43πr3

From the figure, the radius of the sphere is 88 m.

Substitute 88 for rr in the formula.

V=43π(8)3V=43π(8)3

Simplify.

V=43π(512)V=43π(512)

≈2145≈2145

Therefore, the volume of the sphere is about 2145 m32145 m3 .