# Find the surface area of a sphere of diameters.

(i) 14cm

(ii) 21cm

(iii) 3.5cm

Last updated date: 20th Mar 2023

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Answer

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Hint: Assume value of \[\pi =3.14\]. Given is the diameter of the sphere. Find the radius and substitute it in the formula for surface area of the sphere. Simplify it and find the surface area corresponding to each diameter.

Complete step-by-step answer:

Let us assume the value of \[\pi =\dfrac{22}{7}\].

(i) 14cm

Here, the diameter of the sphere is given as 14cm.

Surface area of the sphere is given by the formula \[4\pi {{r}^{2}}\].

But we are given the diameter. To find the radius take half of the given diameter.

\[\begin{align}

& radius=\dfrac{diameter}{2} \\

& r=\dfrac{d}{2}=\dfrac{14}{2}=7cm \\

\end{align}\]

\[\therefore \] radius of the sphere, r = 7cm.

Surface area of area\[=4\pi {{r}^{2}}\]

\[\begin{align}

& =4\times \dfrac{22}{7}\times {{\left( 7 \right)}^{2}} \\

& =4\times \dfrac{22}{7}\times 7\times 7 \\

\end{align}\]

Cancel out the like terms and multiply.

\[=4\times 22\times 7=616c{{m}^{2}}\]

\[\therefore \] Surface area of sphere of radius 14 cm = \[616c{{m}^{2}}\].

(ii) 21cm

The diameter of the sphere is given as 21cm.

Hence, we need to find the radius of the sphere.

Radius\[=\dfrac{Diameter}{2}=\dfrac{21}{2}=10.5\]cm

\[\therefore \] Radius of the sphere, r = 10.5cm.

We know that the surface area of sphere\[=4\pi {{r}^{2}}\]

\[\begin{align}

& =4\times \dfrac{22}{7}\times {{\left( 10.5 \right)}^{2}} \\

& =4\times \dfrac{22}{7}\times 10.5\times 10.5 \\

& =1386c{{m}^{2}} \\

\end{align}\]

\[\therefore \] Surface area of sphere of radius 14 cm =\[1386c{{m}^{2}}\].

(iii) 3.5cm

We are given the diameter of the sphere as 3.5cm.

Hence, we need to find the radius of the sphere.

Radius\[=\dfrac{Diameter}{2}=\dfrac{3.5}{2}=1.75\]cm

\[\therefore \] Radius of sphere, r = 1.75cm.

We know the surface area of sphere\[=4\pi {{r}^{2}}\]

\[\begin{align}

& =4\times \dfrac{22}{7}\times {{\left( 1.75 \right)}^{2}} \\

& =4\times \dfrac{22}{7}\times 1.75\times 1.75 \\

& =38.5c{{m}^{2}} \\

\end{align}\]

\[\therefore \]Surface area of the sphere of radius 3.5cm = \[38.5c{{m}^{2}}\].

Note: The difference between a sphere and circle is that a circle is in 2-dimension, whereas a sphere is a 3-dimensional shape. In a visual perspective it has a 3—dimensional structure that is formed by rotating a disc that is circular with one of the diagonal. Here, read the question carefully, don’t confuse the diameter given to be radius of the sphere. If taking the diameter directly without finding the radius, change the formula of surface area of the sphere to put \[r=\dfrac{d}{2}\].

Surface area of sphere\[=4\pi {{r}^{2}}\]

\[=4\pi {{\left( \dfrac{d}{2} \right)}^{2}}=\dfrac{4\pi {{d}^{2}}}{4}=\pi {{d}^{2}}\]

\[\therefore \] Surface area of sphere \[=\pi {{d}^{2}}\].

Complete step-by-step answer:

Let us assume the value of \[\pi =\dfrac{22}{7}\].

(i) 14cm

Here, the diameter of the sphere is given as 14cm.

Surface area of the sphere is given by the formula \[4\pi {{r}^{2}}\].

But we are given the diameter. To find the radius take half of the given diameter.

\[\begin{align}

& radius=\dfrac{diameter}{2} \\

& r=\dfrac{d}{2}=\dfrac{14}{2}=7cm \\

\end{align}\]

\[\therefore \] radius of the sphere, r = 7cm.

Surface area of area\[=4\pi {{r}^{2}}\]

\[\begin{align}

& =4\times \dfrac{22}{7}\times {{\left( 7 \right)}^{2}} \\

& =4\times \dfrac{22}{7}\times 7\times 7 \\

\end{align}\]

Cancel out the like terms and multiply.

\[=4\times 22\times 7=616c{{m}^{2}}\]

\[\therefore \] Surface area of sphere of radius 14 cm = \[616c{{m}^{2}}\].

(ii) 21cm

The diameter of the sphere is given as 21cm.

Hence, we need to find the radius of the sphere.

Radius\[=\dfrac{Diameter}{2}=\dfrac{21}{2}=10.5\]cm

\[\therefore \] Radius of the sphere, r = 10.5cm.

We know that the surface area of sphere\[=4\pi {{r}^{2}}\]

\[\begin{align}

& =4\times \dfrac{22}{7}\times {{\left( 10.5 \right)}^{2}} \\

& =4\times \dfrac{22}{7}\times 10.5\times 10.5 \\

& =1386c{{m}^{2}} \\

\end{align}\]

\[\therefore \] Surface area of sphere of radius 14 cm =\[1386c{{m}^{2}}\].

(iii) 3.5cm

We are given the diameter of the sphere as 3.5cm.

Hence, we need to find the radius of the sphere.

Radius\[=\dfrac{Diameter}{2}=\dfrac{3.5}{2}=1.75\]cm

\[\therefore \] Radius of sphere, r = 1.75cm.

We know the surface area of sphere\[=4\pi {{r}^{2}}\]

\[\begin{align}

& =4\times \dfrac{22}{7}\times {{\left( 1.75 \right)}^{2}} \\

& =4\times \dfrac{22}{7}\times 1.75\times 1.75 \\

& =38.5c{{m}^{2}} \\

\end{align}\]

\[\therefore \]Surface area of the sphere of radius 3.5cm = \[38.5c{{m}^{2}}\].

Note: The difference between a sphere and circle is that a circle is in 2-dimension, whereas a sphere is a 3-dimensional shape. In a visual perspective it has a 3—dimensional structure that is formed by rotating a disc that is circular with one of the diagonal. Here, read the question carefully, don’t confuse the diameter given to be radius of the sphere. If taking the diameter directly without finding the radius, change the formula of surface area of the sphere to put \[r=\dfrac{d}{2}\].

Surface area of sphere\[=4\pi {{r}^{2}}\]

\[=4\pi {{\left( \dfrac{d}{2} \right)}^{2}}=\dfrac{4\pi {{d}^{2}}}{4}=\pi {{d}^{2}}\]

\[\therefore \] Surface area of sphere \[=\pi {{d}^{2}}\].

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