Formula

Volume and surface area of a sphere

Mensuration • Solids • Sphere

Formula

V = \frac{4}{3} π r^{3}, S = 4 π r^{2}

A sphere of radius

r

has volume

\frac{4}{3} π r^{3}

and surface area

4 π r^{2}

; a solid hemisphere has volume

\frac{2}{3} π r^{3}

, curved surface

2 π r^{2}

and total surface

3 π r^{2}

once its flat circular face is counted.

How to use it

1Halve a stated diameter before cubing or squaring; the error is otherwise a factor of $8$ or $4$ .
2To cost the painting or plating of a spherical vessel, divide the total cost by $4 π r^{2}$ to get the rate per unit area.
3A hemispherical dome closing a cylinder adds $\frac{2}{3} π r^{3}$ to the volume and its own radius to the height.

Find the surface area of a sphere of diameter

21

cm.

Step-by-step

1
$r = \frac{21}{2}$ cm
Why: The formula takes the radius.
2
$S = 4 \times \frac{22}{7} \times (\frac{21}{2})^{2} = 4 \times \frac{22}{7} \times \frac{441}{4} = 1386$ cm $^{2}$

Answer

1386

^{2}

Watch out

The surface of a sphere is

4 π r^{2}

, four times the area

π r^{2}

of its great circle, and the curved surface of a hemisphere is

2 π r^{2}

, not half of

4 π r^{2}

plus nothing — the flat face has to be added separately when the solid is closed.

Builds on

Also known as

volume of a sphere, surface area of a sphere, four thirds pi r cubed, hemisphere volume, 4 pi r squared