AFCAT Numerical Ability · Surface Area and Volume

Sphere and Hemisphere

Surface and volume of a sphere and a hemisphere, solid or hollow, and how both change when the radius changes.

Why this matters

The formulas are short, so the marks go on two things: a solid hemisphere's total surface is 3πr², because the flat face counts, and a ratio of radii must be squared for surface and cubed for volume. With π as 22/7 the fractions get ugly, so keep π until the last line.

Concept 1 of 3: Sphere and hemisphere formulas

A sphere's surface is exactly four circles of its own radius, and its volume is two thirds of the cylinder that just holds it. Halve both for a hemisphere, then remember the flat circle on top when the whole surface is asked.

Definition

Working back: from a surface, divide by $4\pi$ and take the square root; from a volume, multiply by $\tfrac{3}{4\pi}$ and take the cube root. With $\pi = \tfrac{22}{7}$ , radii such as 3.5, 7, 10.5 and 14 give whole numbers. Halve a diameter before you start.

Solid	Curved surface	Total surface	Volume
Sphere of radius r	$4\pi r^2$	$4\pi r^2$	$\tfrac43 \pi r^3$
Solid hemisphere of radius r	$2\pi r^2$	$3\pi r^2$	$\tfrac23 \pi r^3$
Hollow sphere, radii R outside and r inside	$4\pi R^2$ outside	$4\pi (R^2 + r^2)$	$\tfrac43 \pi (R^3 - r^3)$

Practice this conceptself-check · 3 quick reps

The same idea in a practice question:

Practice questionEasy

Example 1 · Surface Area and Volume · Sphere and Hemisphere

The surface area of a sphere is

616 \text{ cm}^2

. If its radius is changed so that the area gets reduced by 75%, then the radius becomes

Four πr², not four thirds

The surface is

4\pi r^2

; the

\tfrac43

belongs to the volume. Mixing them is the commonest slip on this page.

Halve the diameter

A ball 14 cm across has radius 7 cm. Putting 14 into

4\pi r^2

gives a surface four times too big.

Drill 3 more on sphere and hemisphere formulas

Concept 2 of 3: Hemispheres: curved or total surface, solid or hollow

Half a sphere has half its curved surface,

2\pi r^2

. A solid hemisphere also has a flat circular face,

\pi r^2

, so its whole surface is

3\pi r^2

. A bowl is a hollow hemisphere: its metal is the outer half-ball less the inner one.

Definition

Curved surface $2\pi r^2$ (a dome, a bowl's outside); total surface of a solid hemisphere $3\pi r^2$ .
Volume $\tfrac23 \pi r^3$ .
A hollow hemisphere of outer radius $R$ and inner radius $r$ holds metal $\tfrac23 \pi (R^3 - r^3)$ . Its thickness is $R - r$ .
Weight = volume × the mass of one cm³.

Hemisphere of radius r

\text{CSA} = 2\pi r^2,\quad \text{TSA} = 3\pi r^2,\quad V = \tfrac23 \pi r^3

Worked example

A solid hemisphere has radius 7 cm. Find its curved surface, total surface and volume. (Use

\pi = \tfrac{22}{7}

Practice this conceptself-check · 3 quick reps

The same idea in a practice question:

Practice questionModerate

Example 2 · Surface Area and Volume · Sphere and Hemisphere

A hollow hemisphere is made of a sheet of metal 1 cm thick. If the outer radius is 5 cm, what is the weight of the hemisphere (

1 \text{ cm}^3

of the metal weighs 9 g)?

A solid hemisphere's total surface counts the flat face

For radius 7 cm the curved surface is 308 cm² but the total is 462 cm². Read whether the stem says curved or total.

R³ − r³ is not (R − r)³

A bowl with radii 6 and 3 holds

\tfrac23 \pi \times 189

of metal. Cubing the thickness,

3^3 = 27

, gives far too little.

Drill 2 more on hemispheres: curved or total surface, solid or hollow

Concept 3 of 3: Changing the radius of a sphere

All spheres are the same shape. Radii in the ratio k give surfaces in k² and volumes in k³, so any one ratio gives the other two.

Definition

Radii (or diameters) $k$ : surfaces $k^2$ , volumes $k^3$ .
From surfaces, take square roots to get the radii; from volumes, cube roots.
A radius up by $r\%$ multiplies the surface by $\left(1 + \tfrac{r}{100}\right)^2$ and the volume by $\left(1 + \tfrac{r}{100}\right)^3$ .

Two spheres

\dfrac{S_1}{S_2} = \left(\dfrac{r_1}{r_2}\right)^2,\quad \dfrac{V_1}{V_2} = \left(\dfrac{r_1}{r_2}\right)^3

Worked example

Two spheres have volumes in the ratio

27 : 64

. Find the ratio of their surfaces.

Practice this conceptself-check · 3 quick reps

The same idea in a practice question:

Practice questionEasy

Example 3 · Surface Area and Volume · Sphere and Hemisphere

If the surface areas of two spheres are in the ratio of 4 : 25, then the ratio of their volumes is

Becomes 27 times means up by 26 times

Tripling the radius makes the volume 27 times as large, which is an increase of 2,600%. Read whether the stem asks what it becomes or how much it grows.

Square for surface, cube for volume

Spheres with surfaces

1 : 100

have radii

1 : 10

and volumes

1 : 1000

. Cubing the surface ratio, or squaring the volume ratio, is a different question.

Drill 3 more on changing the radius of a sphere

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (2)

Hemispheres: curved or total surface, solid or hollow
Hemisphere of radius r
$\text{CSA} = 2\pi r^2,\quad \text{TSA} = 3\pi r^2,\quad V = \tfrac23 \pi r^3$
Changing the radius of a sphere
Two spheres
$\dfrac{S_1}{S_2} = \left(\dfrac{r_1}{r_2}\right)^2,\quad \dfrac{V_1}{V_2} = \left(\dfrac{r_1}{r_2}\right)^3$

Reference tables (1)

Sphere and hemisphere formulas3 rows

Solid	Curved surface	Total surface	Volume
Sphere of radius r	$4\pi r^2$	$4\pi r^2$	$\tfrac43 \pi r^3$
Solid hemisphere of radius r	$2\pi r^2$	$3\pi r^2$	$\tfrac23 \pi r^3$
Hollow sphere, radii R outside and r inside	$4\pi R^2$ outside	$4\pi (R^2 + r^2)$	$\tfrac43 \pi (R^3 - r^3)$

Watch out for (6)

Four πr², not four thirds
→ Sphere and hemisphere formulas
Halve the diameter
→ Sphere and hemisphere formulas
A solid hemisphere's total surface counts the flat face
→ Hemispheres: curved or total surface, solid or hollow
R³ − r³ is not (R − r)³
→ Hemispheres: curved or total surface, solid or hollow
Becomes 27 times means up by 26 times
→ Changing the radius of a sphere
Square for surface, cube for volume
→ Changing the radius of a sphere

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