In this chapter we will discuss questions related to surface area of sphere with solution.

The formula for surface area of sphere is given as;

Surface Area = \mathtt{Surface\ area\ =\ 4.\pi .r^{2}}

Where, r is the radius of sphere.

## Surface area of sphere – solved examples

**Question 01**

Find the surface area of sphere of radius 14 cm.

**Solution**

r = 14 cm

Surface Area = \mathtt{Surface\ area\ =\ 4.\pi .r^{2}}

Putting the values in formula;

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

Hence, **2464 sq. cm is the surface area of given sphere**.

**Question 02**

Find the radius of sphere, if the surface area is 5544 sq. cm.

**Solution**

Surface area = 5544 sq. cm

\mathtt{4.\pi .r^{2} \ =\ 5544}\\\ \\ \mathtt{4.\frac{22}{7} .r^{2} =\ 5544}\\\ \\ \mathtt{r^{2} \ =\frac{5544\times 7}{88} =\ 441}\\\ \\ \mathtt{r\ =\ 21\ cm}

Hence, **radius of sphere is 21 cm**

**Question 03**

A solid sphere made of iron of radius 10 cm is melted into 8 small spherical balls. Find the surface area of small ball.

**Solution**

Radius of big sphere (R)= 10 cm

Radius of small sphere (r) = ?

Since the material of 8 cm ball is melted into smaller balls, the volume will be the same.

Volume of 10 cm balls = combined volume all 8 balls

\mathtt{\frac{4}{3} .\pi .R^{3} =8\times \frac{4}{3} .\pi .r^{3}}\\\ \\ \mathtt{R^{3} =8.r^{3}}\\\ \\ \mathtt{10^{3} =8.\ r^{3}}\\\ \\ \mathtt{r^{3} =\frac{10^{3}}{2^{3}}}\\\ \\ \mathtt{r=\frac{10}{2} =\ 5\ cm}

Hence,** the radius of smaller balls is 5 cm**.

Surface area of ball = \mathtt{4.\pi .r^{2}}

Putting the values;

\mathtt{Surface\ area\ =\ 4\times \frac{22}{7} \times 5^{2}}\\\ \\ \mathtt{Surface\ area\ =\ 314.3\ cm^{2}}

Hence,** 314.3 sq cm is the surface area of smaller ball**.

**Example 04**

The ratio of volume of sphere is given as 1 : 27. Find the ratio of surface area of the given spheres.

**Solution**

Let radius of first sphere is r1 and radius of 2nd sphere is r2.

Ratio of the volume is given as 1 : 27.

\mathtt{\frac{V_{1}}{V_{2}} =\frac{1}{27}}\\\ \\ \mathtt{\frac{\frac{4}{3} \pi r_{1}^{3}}{\frac{4}{3} \pi r_{2}^{3}} =\frac{1}{27}}\\\ \\ \mathtt{\frac{r_{1}^{3}}{r_{2}^{3}} =\frac{1}{27}}\\\ \\ \mathtt{\frac{r_{1}}{r_{2}} =\frac{1}{3}}

So, **the ratio of the radius is 1 : 3.**

Now, **let’s calculate the ratio of surface area**.

\mathtt{\frac{SA_{1}}{SA_{2}} =\frac{4\pi r_{1}^{2}}{4\pi r_{2}^{2}}}\\\ \\ \mathtt{\frac{SA_{1}}{SA_{2}} =\frac{r_{1}^{2}}{r_{2}^{2}} =\left(\frac{1}{3}\right)^{2}}\\\ \\ \mathtt{\frac{SA_{1}}{SA_{2}} =\frac{1}{9}}

Hence, **1 : 9 is the required ratio.**

**Example 05**

It’s given that the sphere and cube are of same height. Calculate the ratio total surface area of given sphere and cube.

**Solution**

Let “a” be the height of the cube.

Since both sphere and cube are of same height, the diameter of sphere will be ” a” units.

Diameter of sphere = a

Radius of sphere = a/2

Calculating the ratio of total surface areas.

\mathtt{\frac{SA( cube)}{SA( sphere)} =\frac{6a^{2}}{4\pi \left(\frac{a}{2}\right)^{2}}}\\\ \\ \mathtt{\frac{SA( cube)}{SA( sphere)} =\frac{6}{\pi } =\frac{6\times 7}{22}}\\\ \\ \mathtt{\frac{SA( cube)}{SA( sphere)} =\frac{21}{11}}

Hence, **21 : 11 is the required ratio**.

**Example 06**

Consider the sphere of radius 5 cm and a cone of radius 4 cm. It’s given that surface area of sphere if 5 times the curved surface area of cone. Find the height of cone.

**Solution**

Radius of sphere (R) = 5 cm

Radius of cone (r) = 4 cm

**According to question;**

Surface area sphere = 5 ( curved surface area of cone )

\mathtt{4.\pi .R^{2} =5\ ( \ \pi .r.l\ )}\\\ \\ \mathtt{4.( 5)^{2} =( 5) \ ( 4) .\ l}\\\ \\ \mathtt{l\ =\ 5\ cm\ }

Hence, **slant height of the cone is 5 cm.**

Now **let’s calculate the height of cone**.

Applying Pythagoras theorem in triangle OAC.

\mathtt{AC^{2} =AO^{2} +OC^{2}}\\\ \\ \mathtt{5^{2} =\ OA^{2} +4^{2}}\\\ \\ \mathtt{OA^{2} =25-16}\\\ \\ \mathtt{h^{2} \ =\ 9\ \ }\\\ \\ \mathtt{h\ =\ 3\ cm}

Hence, **the height of given cone is 3 cm.**

**Example 07**

The surface area of sphere is given as 154 sq cm. Calculate its volume.

**Solution**

Let r = radius of sphere.

The formula for surface area of sphere is given as;

\mathtt{4\pi r^{2} =\ 154}\\\ \\ \mathtt{4\times \frac{22}{7} \times r^{2} =\ 154}\\\ \\ \mathtt{r^{2} =\ \frac{154\times 7}{4\times 22}}\\\ \\ \mathtt{r^{2} =12.25}\\\ \\ \mathtt{r\ =\ 3.5\ cm}

Hence, **radius of given sphere is 3.5 cm**.

Now **let’s calculate the volume**.

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

\mathtt{Volume\ =\ \frac{4}{3} \times \frac{22}{7} \times ( 3.5)^{3}}\\\ \\ \mathtt{Volume\ =\ \frac{88}{21} \times \frac{7}{2} \times \frac{7}{2} \times \frac{7}{2}}\\\ \\ \mathtt{Volume\ =\ 179.66\ cm^{3}}

Hence, **179.66 cu cm is the required volume**