FINDING THE VOLUME OF A SPHERE

We can find the volume of a sphere using the volume of a cylinder. 

Cylinder is a solid which has a circular base. 

We know the fact that the volume of any solid is equal to the product of base area and height of the solid.

So, the volume of a right circular cylinder of base radius ‘r’ and height ‘h’ is given by

V  =  (Base Area) x (Height) 

The base of a cylinder is a circle, so for a cylinder,

Base Area  =  πr²

Therefore, 

Volume of a cylinder  =  πr²h cubic units

Consider a right circular cylinder and three right circular cones of same base radius and height as that of the cylinder.

The contents of three cones will exactly occupy the cylinder.

Then,

When we model the volume of a sphere, we will be getting the following result. 

3 x (Volume of a cone)  =  Volume of cylinder

3 x (Volume of a cone)  =  πr²h

Volume of the cone  =  1/3 · πr²h cubic units

Consider a sphere and two right circular cones of same base radius and height such that twice the radius of the sphere is equal to the height of the cones.

Then we can observe that the contents of two cones will exactly occupy the sphere.

Then, 

Volume of sphere   =  2 x (Volume of a cone)

Volume of a sphere  =  2 x (1/3 · πr²h)

Volume of a sphere  =  2/3 · πr²h

A sphere always has a height which is equal to twice the radius.

So, substitute 2r for h.

Volume of sphere  =  2/3 · πr²(2r)

Simplify.

Volume of sphere  =  4/3 · πr³ cubic units

Solved Examples

Example 1 :

Find the volume of the sphere given below. Round your answer to the nearest tenth if necessary. Use the approximate of value of π, that is 3.14. 

Solution : 

Step 1 : 

Write the formula to find volume of a sphere.

V  =  4/3 ·  πr³

Step 2 : 

Substitute the given measures. 

V ≈  4/3 · 3.14 · (2.2)³

Simplify.

V ≈  4/3 · 3.14 · 10.648

V ≈  44.6

So, the volume of the given sphere is about 44.6 cubic cm.

Example 2 :

Find the volume of the sphere given below. Round your answer to the nearest tenth if necessary. Use the approximate of value of π, that is 3.14. 

Solution : 

Step 1 : 

Write the formula to find volume of a sphere.

V  =  4/3 ·  πr³ -----(1)

Step 2 : 

To find the volume, we need the radius of the sphere. But, the diameter is given, that is  42 cm. So, find the radius. 

r  =  diameter / 2

r  =  42/2

r  =  21

Step 3 : 

Substitute π ≈ 3.14 and r = 21 in (1).

V ≈  4/3 · 3.14 · 21³

Simplify.

V ≈  4/3 · 3.14 · 9261

V ≈  38,772.7

So, the volume of the given sphere is about 38,772.7 cubic cm. 

Example 3 :

Find the volume of the sphere given below. Round your answer to the nearest tenth if necessary. Use the approximate of value of π, that is 3.14. 

Solution : 

Step 1 : 

Write the formula to find volume of a sphere.

V  =  4/3 ·  πr³ -----(1)

Step 2 : 

To find the volume, we need the radius of the sphere. But, the height is given, that is  12 cm. 

We know that the height of a sphere equals twice the radius. 

That is, 

h  =  2r

Substitute h  =  12.

12  =  2r

Divide both sides by 2. 

12/2  =  2r/2

6  =  r

Step 3 : 

Substitute π ≈ 3.14 and r = 6 in (1).

V ≈  4/3 · 3.14 · 6³

Simplify.

V ≈  4/3 · 3.14 · 216

V ≈  904.3

So, the volume of the given sphere is about 904.3 cubic cm. 

Apart from the stuff given above, if you need any other stuff in math, please use our google custom search here.

If you have any feedback about our math content, please mail us : 

v4formath@gmail.com

We always appreciate your feedback. 

You can also visit the following web pages on different stuff in math. 

WORD PROBLEMS

HCF and LCM  word problems

Word problems on simple equations 

Word problems on linear equations 

Word problems on quadratic equations

Algebra word problems

Word problems on trains

Area and perimeter word problems

Word problems on direct variation and inverse variation 

Word problems on unit price

Word problems on unit rate 

Word problems on comparing rates

Converting customary units word problems 

Converting metric units word problems

Word problems on simple interest

Word problems on compound interest

Word problems on types of angles 

Complementary and supplementary angles word problems

Double facts word problems

Trigonometry word problems

Percentage word problems 

Profit and loss word problems 

Markup and markdown word problems 

Decimal word problems

Word problems on fractions

Word problems on mixed fractrions

One step equation word problems

Linear inequalities word problems

Ratio and proportion word problems

Time and work word problems

Word problems on sets and venn diagrams

Word problems on ages

Pythagorean theorem word problems

Percent of a number word problems

Word problems on constant speed

Word problems on average speed 

Word problems on sum of the angles of a triangle is 180 degree

OTHER TOPICS 

Profit and loss shortcuts

Percentage shortcuts

Times table shortcuts

Time, speed and distance shortcuts

Ratio and proportion shortcuts

Domain and range of rational functions

Domain and range of rational functions with holes

Graphing rational functions

Graphing rational functions with holes

Converting repeating decimals in to fractions

Decimal representation of rational numbers

Finding square root using long division

L.C.M method to solve time and work problems

Translating the word problems in to algebraic expressions

Remainder when 2 power 256 is divided by 17

Remainder when 17 power 23 is divided by 16

Sum of all three digit numbers divisible by 6

Sum of all three digit numbers divisible by 7

Sum of all three digit numbers divisible by 8

Sum of all three digit numbers formed using 1, 3, 4

Sum of all three four digit numbers formed with non zero digits

Sum of all three four digit numbers formed using 0, 1, 2, 3

Sum of all three four digit numbers formed using 1, 2, 5, 6