To derive the formula for volume of a sphere | volume of sphere = 4/3 pi r cube | volume of sphere = 4/3 pi r^3 | Maths made easy with Imran Khan |
Dear viewers, it is my pleasure to deliver you mathematics tutorials in simple and native language so that you can get it easily.
#Maths Made Easy is a channel where you can improve your Mathematics.
The general formula for the volume of sphere in terms of its radius is given as V = (4/3) π r3. Let's say 'd' is its diameter, according to the definition of diameter, we have d = 2r.
The volume of a sphere is the measurement of the space it can occupy. A sphere is a three-dimensional shape that has no edges or vertices. In this short lesson, we will learn to find the volume of a sphere, deduce the formula of volume of a sphere and learn to apply the formulas as well. Once you understand this chapter you will learn to solve problems on the volume of the sphere.
The volume of sphere is the measure of space that can be occupied by a sphere. If we draw a circle on a sheet of paper, take a circular disc, paste a string along its diameter and rotate it along the string. This gives us the shape of a sphere.
The unit of volume of a sphere is given as the (unit)3. The metric units of volume are cubic meters or cubic centimeters while the USCS units of volume are, cubic inches or cubic feet. The volume of sphere depends on the radius of the sphere, hence changing it changes the volume of the sphere. There are two types of spheres, solid sphere, and hollow sphere. The volume of both types of spheres is different. We will learn in the following sections about their volumes.
As suggested by Archimedes, if the radius of the cylinder, cone, and the 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 sphere, volume of cone, and volume of cylinder is given as:
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

Volume of Sphere Formula
The volume of sphere formula can be given for a solid as well as the hollow sphere. In the case of a solid sphere, we only have one radius but in the case of a hollow sphere, there are two radii, having two different values of radius one for the outer sphere and one for the inner sphere.

Volume of Solid Sphere
If the radius of the sphere formed is r and the volume of the sphere is V. Then, the volume of the sphere is given by:
Volume of Sphere, V = (4/3)πr3

Volume of Hollow Sphere
If the radius of the outer sphere is R, the radius of the inner sphere is r and the volume of the sphere is V. Then, the volume of the sphere is given by:
Volume of Sphere, V = Volume of Outer Sphere - Volume of Inner Sphere = (4/3)πR3 - (4/3)πr3 = (4/3)π(R3 - r3)

How to Calculate Volume of Sphere?
The volume of a sphere is the space occupied inside a sphere. The volume of the sphere can be calculated using the formula of the volume of sphere. The steps to calculate the volume of a sphere are:

Step 1: Check the value of the radius of the sphere.
Step 2: Take the cube of the radius.
Step 3: Multiply r3 by (4/3)π
Step 4: At last, add the units to the final answer.
Let us take an example to learn how to calculate the volume of sphere using its formula.

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.

свернуть