Volume and surface area of sphere

In this chapter we will learn the concept of sphere along with the formulas to calculate the volume and surface area.

What is sphere ?

It’s a three dimensional figure with no edges and vertices. Objects like tennis or cricket balls are the best examples to describe the sphere.

In sphere, all the points on the surface are equidistant from the center.

Properties of sphere

(a) It has no vertices or edges

(b) the distance from the center to the surface is constant

(c) It’s a perfectly symmetrical object

(d) Sphere is derived from Greek word ” sphaira” which means ” a ball “

(e) If we cut the sphere from the center, we will get two hemisphere

Volume of sphere

The space enclosed by the sphere is called the volume.

The formula for volume is given as;

Volume = \mathtt{\frac{4}{3} \pi r^{3}}

The volume is measure in cubic units ( \mathtt{cm^{3} ,\ m^{3} ,\ etc\ .\ .} )

Total surface area of sphere

The sphere body is made only of the curved surface.

The formula for total area calculation is given as;

TSA = \mathtt{4\pi r^{2}}

The total surface area is measured in square units ( \mathtt{cm^{2} ,\ m^{2} ,\ etc\ .\ .} )

Memorize the above formulas as they would help to solve below questions.

Questions on volume and surface area of sphere

Question 01
Calculate the volume of sphere with radius 7 cm.

Volume = \mathtt{\frac{4}{3} \pi r^{3}}

Putting the values in formula;

\mathtt{Volume\ =\ \frac{4}{3} \times \frac{22}{7} \times 3^{3}}\\\ \\ \mathtt{Volume\ =113.14\ cm^{3}}

Hence, 113.14 cu. cm is the volume of sphere.

Question 02
Find the total surface area of sphere of radius 21 cm.


TSA = \mathtt{4\pi r^{2}}

Putting the values in the formula;

\mathtt{TSA\ =\ 4\times \frac{22}{7} \times ( 21)^{2}}\\\ \\ \mathtt{TSA\ =\ \ 5544\ cm^{2}}

Hence, 5544 sq cm is the total surface area of sphere.

Question 03
The surface area of sphere is given as 2464 sq. cm. Find the diameter of the sphere.

TSA = \mathtt{4\pi r^{2}}

Putting the values in formula;

\mathtt{2464\ =\ 4\times \frac{22}{7} \times ( r)^{2}}\\\ \\ \mathtt{\frac{2464\times 7}{4\times 22} \ =\ \ r^{2} \ }\\\ \\ \mathtt{196\ =\ r^{2}}\\\ \\ \mathtt{r=\ 14\ cm}

So, diameter of sphere is given as;

D = 2 x r

D = 2 x 14 = 28 cm

Hence, the diameter of given sphere is 28 cm.

Question 04
Two spheres are provided in which the radius of 1st sphere is 4 times that of other sphere. Find the ratio of total surface areas.

Let r1 and r2 are the radius of first and second sphere respectively.

According to question;
r1 = 4. r2

Now calculating, ratio of total surface area;

\mathtt{\frac{TSA_{1}}{TSA_{2}} =\frac{4\pi r_{1}^{2}}{4\pi r_{2}^{2}}}\\\ \\ \mathtt{\frac{TSA_{1}}{TSA_{2}} =\frac{r_{1}^{2}}{r_{2}^{2}}}\\\ \\ \mathtt{\frac{TSA_{1}}{TSA_{2}} =\frac{( 4r_{2})^{2}}{r_{2}^{2}} =\ \frac{16}{1}}

Hence, ratio of surface area is 16 : 1

Question 05
Calculate the volume of water that can be filled in hemisphere of radius 4 meters.

We know that if we cut the sphere in two equal parts we will get hemisphere.

So the volume of hemisphere is half of volume of sphere.

Volume of hemisphere = \mathtt{\frac{1}{2} \times \frac{4}{3} \times \pi r^{3}}

Putting the values;

\mathtt{Volume\ =\ \frac{1}{2} \times \frac{4}{3} \times \pi r^{3}}\\\ \\ \mathtt{Volume\ =\ \frac{1}{2} \times \frac{4}{3} \times \frac{22}{7} \times ( 3)^{3}}\\\ \\ \mathtt{Volume\ =\ 56.57\ \ m^{3}}

Hence, 56.57 cubic meter of water can be filled in the hemisphere

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