Formulas · Mathematics

Surface areas and volumes of solids

8 formulas, each with worked examples. Revision sheet · Practise these formulas

Foundation

Volume of a cuboid

V = l bh

What each letter means
$V$	the volume (the space inside) (volume)
$l$	the length (length)
$b$	the breadth (length)
$h$	the height (length)

A cuboid holds as many unit cubes as fit in one layer (length × breadth) times the number of layers (height).

Why it works

One layer on the base holds l × b unit cubes, and h such layers stack up to the top: l × b × h cubes in all.

When to use it

Boxes, rooms, tanks, bricks: the space inside anything shaped like a box, or how much it holds (1000 cm³ = 1 litre).

How to use it

Put all three lengths in the same unit.
Multiply length, breadth and height.
Give the answer in cubic units (cm³, m³).

To remember: Base area times height.

Other forms

Special case $V = l^{3}$ When a cube (all edges equal, b = h = l).
Rearranged $h = \frac{V}{l b}$ The height (or depth) from the volume and the base.

Worked examples

Story

A water tank is 2 m long, 1.5 m wide and 1 m deep. How many litres does it hold? (1 m³ = 1000 litres)

$V = 2 \times 1.5 \times 1 = 3$
3 m³ is 3 × 1000 = 3000 litres.

Answer: 3

Picture

How many cubes of edge 2 cm fit in a box 10 cm × 6 cm × 4 cm?

Along the edges: 10 ÷ 2 = 5, 6 ÷ 2 = 3 and 4 ÷ 2 = 2 cubes.
$5 \times 3 \times 2 = 30$
Check with volumes: 240 ÷ 8 = 30.

Answer: 30

Direct

Find the volume of a box 12 cm long, 8 cm wide and 5 cm high.

$V = 12 \times 8 \times 5 = 480$

Answer: 480

Reverse

A cuboid has a volume of 360 cm³, a length of 12 cm and a breadth of 6 cm. How high is it?

Try it first, then show the working

$h = \frac{360}{12 \times 6} = \frac{360}{72} = 5$

Answer: 5

Exam

A shed is a cuboid 15 m × 7 m × 8 m with half a cylinder on top (diameter 7 m, length 15 m). Find the volume of air inside (π = 22/7).

Try it first, then show the working

Cuboid: 15 × 7 × 8 = 840 m³.
Half cylinder: 1/2 × 22/7 × 3.5² × 15 = 288.75 m³.
$840 + 288.75 = 1128.75$

Answer: 1128.75 m³

Common mistake: Mixing units (metres with centimetres), or adding the three lengths instead of multiplying.

Foundation

Total surface area of a cuboid

S = 2 (l b + bh + h l)

What each letter means
$S$	the total area of all six faces (area)
$l$	the length (length)
$b$	the breadth (length)
$h$	the height (length)

A cuboid has six rectangular faces in three matching pairs: top and bottom (l × b), front and back (l × h), two sides (b × h).

Why it works

Each pair of opposite faces is the same size, so adding one face of each kind and doubling covers all six.

When to use it

Wrapping, painting or covering a box-shaped object, or the cardboard needed to make a box.

How to use it

Work out lb, bh and hl.
Add them.
Double the sum.

To remember: Three pairs of faces: twice (lb + bh + hl).

Other forms

Special case $S = 6 l^{2}$ When a cube (all edges l).
Same formula, another formThe four walls only (a room without floor and ceiling): 2h(l + b).

Worked examples

Story

A gift box is 20 cm long, 15 cm wide and 10 cm high. How much wrapping paper covers it exactly?

$S = 2 (300 + 150 + 200) = 2 \times 650 = 1300$
It needs 1300 cm² of paper.

Answer: 1300

Picture

Two cubes of edge 4 cm are joined face to face. Find the surface area of the cuboid they make.

The cuboid is 8 cm × 4 cm × 4 cm.
$2 (32 + 16 + 32) = 2 \times 80 = 160$
Check: two cubes have 2 × 96 = 192 cm², minus the two hidden faces, 2 × 16 = 32 cm².

Answer: 160 cm²

Direct

Find the total surface area of a box 10 cm × 6 cm × 4 cm.

$S = 2 (60 + 24 + 40) = 2 \times 124 = 248$

Answer: 248

Reverse

A cuboid is 8 cm long and 5 cm wide, and its total surface area is 184 cm². How high is it?

Try it first, then show the working

$184 = 2 (40 + 5 h + 8 h)$
$92 = 40 + 13 h$
$h = 4$

Answer: 4

Exam

A block is a cube of edge 5 cm with a hemisphere of diameter 4.2 cm fixed on top. Find the total surface area of the block (π = 22/7).

Try it first, then show the working

The cube's six faces: 6 × 5² = 150 cm².
The hemisphere hides a circle of the top face (πr²) but adds its curved surface (2πr²): a net gain of πr² with r = 2.1 cm.
$150 + \frac{22}{7} \times 2. 1^{2} = 150 + 13.86 = 163.86$

Answer: 163.86 cm²

Common mistake: Counting only three faces, or only the four walls (for a room without floor and ceiling, use 2h(l + b) instead).

Class 10

Volume of a cylinder

V = π r^{2} h

What each letter means
$V$	the volume (volume)
$r$	the radius of the circular base (length)
$h$	the height (length)

Like a cuboid, a cylinder's volume is the area of its base times its height; the base is a circle, πr².

Why it works

Stack thin circular slices, each of area πr², up to height h: the volume is πr² × h. Cavalieri's principle says any solid with the same base area at every level has this same volume.

When to use it

Tins, pipes, glasses, wells, tanks, pillars: anything with a round base and straight sides.

How to use it

Find the radius (half the diameter) and the height, in the same unit.
Square the radius and multiply by π.
Multiply by the height.

To remember: Base area times height, and the base is a circle.

Other forms

Rearranged $h = \frac{V}{π r ^{2}}$ The height (or depth) from the volume.

Worked examples

Story

A well has a diameter of 2 m and is 14 m deep. How much earth was dug out? (π = 22/7)

The radius is 1 m.
$\frac{22}{7} \times 1^{2} \times 14 = 44$
44 m³ of earth was dug out.

Answer: 44 m³

Picture

A rectangular sheet 22 cm × 14 cm is rolled along its longer side into a tube of height 14 cm. Find the volume of the tube (π = 22/7).

The 22 cm side becomes the circle: 2πr = 22, so r = 22 ÷ (44/7) = 3.5 cm.
$\frac{22}{7} \times 3. 5^{2} \times 14 = 539$
The tube holds 539 cm³.

Answer: 539 cm³

Direct

Find the volume of a cylinder of radius 7 cm and height 10 cm.

$V = π \times 7^{2} \times 10 = 490 π$
That is about 1540 cm³ with π = 22/7.

Answer: 490π cm³

Reverse

A cylinder of radius 5 cm holds 500π cm³. How tall is it?

Try it first, then show the working

500π = π × 25 × h, so h = 500 ÷ 25 = 20 cm.

Answer: 20 cm

Exam

From a solid cylinder of height 2.4 cm and diameter 1.4 cm, a cone of the same height and diameter is hollowed out. Find the volume left (π = 22/7).

Try it first, then show the working

The cone is one third of the cylinder, so two thirds is left: 2/3 × πr²h with r = 0.7 cm and h = 2.4 cm.
$\frac{2}{3} \times \frac{22}{7} \times 0. 7^{2} \times 2.4 = 2.464$

Answer: 2.464 cm³

Common mistake: Using the diameter instead of the radius (four times too much), or forgetting to square the radius.

Class 10

Curved surface area of a cylinder

S = 2 π r h

What each letter means
$S$	the curved surface area (the side, without the two circles) (area)
$r$	the radius (length)
$h$	the height (length)

Unroll the side of a cylinder and it becomes a rectangle: one side is the height, the other is the distance round the base, 2πr.

Why it works

Cut the curved surface straight down and flatten it. Its width is the circumference of the base, 2πr, and its height is h, so its area is 2πr × h.

When to use it

Labels on tins, painting a pillar, the inside of a pipe, the metal sheet for a drum's side.

How to use it

Find the radius and the height.
Multiply 2 × π × radius × height.
For the total surface, add the two circles: 2πr² more.

To remember: Round the base, times the height.

Other forms

Same formula, another formTotal surface area with both ends: 2πr(h + r).
Rearranged $h = \frac{S}{2 π r}$ The height from the curved area.

Worked examples

Story

A pillar of radius 0.5 m and height 7 m is to be painted all round. How much area is painted? (π = 22/7)

$2 \times \frac{22}{7} \times 0.5 \times 7 = 22$
22 m² is painted.

Answer: 22 m²

Picture

A cylinder''s side is unrolled into a rectangle 44 cm wide and 10 cm tall. Find the radius of the cylinder and its curved surface area (π = 22/7).

The width is the circumference: 2πr = 44, so r = 44 ÷ (44/7) = 7 cm.
The area is the rectangle: 44 × 10 = 440 cm².

Answer: r = 7 cm; 440 cm²

Direct

Find the curved surface area of a cylinder of radius 7 cm and height 10 cm.

$S = 2 \times π \times 7 \times 10 = 140 π$
That is about 440 cm² with π = 22/7.

Answer: 140π cm²

Reverse

A cylinder of radius 3 cm has a curved surface area of 42π cm². How tall is it?

Try it first, then show the working

42π = 2π × 3 × h = 6πh, so h = 7 cm.

Answer: 7 cm

Exam

A bird-bath is a cylinder of height 1.45 m and radius 30 cm with a hemispherical hollow at the top. Find its total surface area (π = 22/7).

Try it first, then show the working

Surface = curved side of the cylinder + inside of the hemisphere = 2πrh + 2πr² = 2πr(h + r).
$2 \times \frac{22}{7} \times 30 \times (145 + 30) = 33000$
That is 33000 cm², or 3.3 m².

Answer: 3.3 m²

Common mistake: Adding the two circular ends when only the side is needed, or leaving them out when the whole surface is asked for.

Class 10

Volume of a cone

V = \frac{1}{3} π r^{2} h

What each letter means
$V$	the volume (volume)
$r$	the radius of the base (length)
$h$	the height (straight up from the centre of the base to the tip) (length)

A cone holds exactly one third of the cylinder with the same base and height.

Why it works

Fill a cone with water or sand and pour it into the matching cylinder: it takes three cones to fill it. In general, any pointed solid (a cone or a pyramid) is one third of base area × height.

When to use it

Ice-cream cones, funnels, heaps of grain or sand, the pointed parts of toys and tents.

How to use it

Find the radius and the perpendicular height (not the slant height).
Work out πr²h, as for a cylinder.
Take one third of it.

To remember: A cone is a third of its cylinder.

Other forms

Same formula, another formThree cones fill the cylinder with the same base and height: 3 × (1/3)πr²h = πr²h.
Rearranged $h = \frac{3 V}{π r ^{2}}$ The height from the volume.

Worked examples

Story

A heap of wheat is a cone of radius 3.5 m and height 12 m. How much wheat is in it? (π = 22/7)

$\frac{1}{3} \times \frac{22}{7} \times 3. 5^{2} \times 12 = 154$
The heap holds 154 m³ of wheat.

Answer: 154 m³

Picture

A cone and a cylinder have the same base radius 6 cm and the same height 10 cm. How many cones of water fill the cylinder?

Cylinder: π × 36 × 10 = 360π cm³. Cone: 1/3 × 360π = 120π cm³.
$360 π \div 120 π = 3$

Answer: 3

Direct

Find the volume of a cone of radius 3 cm and height 4 cm.

$V = \frac{1}{3} \times π \times 9 \times 4 = 12 π$
That is about 37.7 cm³.

Answer: 12π cm³

Reverse

A cone of radius 3 cm holds 30π cm³. How tall is it?

Try it first, then show the working

30π = 1/3 × π × 9 × h = 3πh, so h = 10 cm.

Answer: 10 cm

Exam

A toy is a cone of radius 3.5 cm on a hemisphere of the same radius, 15.5 cm tall in all. Find its volume (π = 22/7).

Try it first, then show the working

The cone's height is 15.5 − 3.5 = 12 cm.
Cone: 1/3 × 22/7 × 3.5² × 12 = 154 cm³. Hemisphere: 2/3 × 22/7 × 3.5³ = 89.83 cm³ (to 2 places).
Together: 154 + 539/6 = 1463/6 cm³, about 243.8 cm³.

Answer: 1463/6 cm³, about 243.8 cm³

Common mistake: Using the slant height l instead of the height h, or forgetting the one third.

Class 10

Curved surface area of a cone

S = π r l

What each letter means
$S$	the curved surface area (without the base) (area)
$r$	the radius of the base (length)
$l$	the slant height: from the tip down the side to the edge of the base (length)

The curved surface of a cone is π times the radius times the slant height. If you know the height instead, the slant height comes from Pythagoras: l = √(r² + h²).

Why it works

Cut the cone's side open and it flattens into a sector of a circle of radius l. Its arc is the base's circumference, 2πr, and a sector's area is half its arc times its radius: ½ × 2πr × l = πrl.

When to use it

Tents, cone-shaped caps and roofs, the paper for a funnel, painting the pointed part of a toy or a rocket.

How to use it

Find the slant height l (from l = √(r² + h²) if you know the height).
Multiply π × r × l.
For the total surface, add the base circle, πr².

To remember: π r l: radius and slant.

Other forms

Same formula, another formWith the height h instead of the slant height: S = πr√(r² + h²), since l = √(r² + h²).
Same formula, another formTotal surface area with the base: πr(l + r).

Worked examples

Story

A conical tent has a base radius of 7 m and a slant height of 10 m. How much canvas covers its side? (π = 22/7)

$\frac{22}{7} \times 7 \times 10 = 220$
It needs 220 m² of canvas.

Answer: 220 m²

Picture

A cone has radius 6 cm and height 8 cm. Find its slant height and curved surface area.

$l = 6^{2} + 8^{2} = 100 = 10$
S = π × 6 × 10 = 60π cm², about 188.5 cm².

Answer: l = 10 cm; 60π cm²

Direct

Find the curved surface area of a cone of radius 5 cm and slant height 13 cm.

$S = π \times 5 \times 13 = 65 π$
That is about 204 cm².

Answer: 65π cm²

Reverse

A cone of radius 4 cm has a curved surface area of 20π cm². Find its slant height and its height.

Try it first, then show the working

20π = π × 4 × l, so l = 5 cm.
h = √(5² − 4²) = 3 cm.

Answer: l = 5 cm, h = 3 cm

Exam

The cone of a toy rocket has base radius 2.5 cm and height 6 cm, and it sits on a cylinder of radius 1.5 cm. Paint covers the cone and the ring of the cone's base left uncovered by the cylinder. Find the painted area (π = 3.14).

Try it first, then show the working

Slant height: √(2.5² + 6²) = 6.5 cm.
Area = πrl + πr² − π(1.5)² = π(2.5 × 6.5 + 6.25 − 2.25) = 20.25π.
$3.14 \times 20.25 = 63.585$

Answer: 63.585 cm²

Common mistake: Using the height h in place of the slant height l.

Class 10

Volume of a sphere

V = \frac{4}{3} π r^{3}

What each letter means
$V$	the volume (volume)
$r$	the radius (length)

A sphere of radius r holds four thirds of π times r cubed. It is two thirds of the cylinder that just contains it.

Why it works

Archimedes showed that a sphere fills exactly two thirds of the cylinder around it (radius r, height 2r): 2/3 × πr² × 2r = 4/3 πr³. He was so proud of it that he asked for a sphere in a cylinder on his tomb.

When to use it

Balls, marbles, globes, drops, and melting one solid into spheres (or spheres into another shape, where the volume stays the same).

How to use it

Find the radius (half the diameter).
Cube it.
Multiply by π and by 4/3.

To remember: Four thirds π r cubed.

Other forms

Special case $V = \frac{2}{3} π r^{3}$ When half a sphere (a hemisphere).
Same formula, another formA sphere is two thirds of the cylinder that just holds it (radius r, height 2r).

Worked examples

Story

A ball has a radius of 10.5 cm. How much air does it hold? (π = 22/7)

$\frac{4}{3} \times \frac{22}{7} \times 10. 5^{3} = 4851$
It holds 4851 cm³ of air.

Answer: 4851 cm³

Picture

How many lead balls of radius 1 cm can be made from a solid lead sphere of radius 4 cm?

Volumes scale with the cube of the radius: (4 ÷ 1)³ = 64.
So 64 small balls, since 4/3 π × 64 ÷ (4/3 π × 1) = 64.

Answer: 64

Direct

Find the volume of a sphere of radius 3 cm.

$V = \frac{4}{3} \times π \times 27 = 36 π$
That is about 113 cm³.

Answer: 36π cm³

Reverse

A sphere has a volume of 288π cm³. Find its radius.

Try it first, then show the working

4/3 π r³ = 288π, so r³ = 216.
r = 6 cm, because 6 × 6 × 6 = 216.

Answer: 6 cm

Exam

A metal sphere of radius 4.2 cm is melted and recast as a cylinder of radius 6 cm. Find the height of the cylinder.

Try it first, then show the working

The volume stays the same: 4/3 π × 4.2³ = π × 6² × h.
$h = \frac{4 \times 4. 2 ^{3}}{3 \times 36} = 2.744$

Answer: 2.744 cm

Common mistake: Squaring instead of cubing the radius (r² is for areas), or using the diameter.

Class 10

Surface area of a sphere

S = 4 π r^{2}

What each letter means
$S$	the area of the whole outside of the sphere (area)
$r$	the radius (length)

The outside of a sphere has exactly the area of four circles of the same radius.

Why it works

Archimedes showed that a sphere's surface equals the curved side of the cylinder that just holds it: 2πr × 2r = 4πr². You can see it by winding string: the string that covers a hemisphere covers two of its flat circles.

When to use it

Leather or paint for a ball, the skin of a fruit, the surface of a planet; half of it (2πr²) for a bowl or a dome.

How to use it

Find the radius.
Square it.
Multiply by 4π.

To remember: Four circles cover a ball.

Other forms

Special caseA hemisphere's curved surface is half of it, 2πr²; with its flat circle the total is 3πr².When half a sphere.
Rearranged $r = \frac{S}{4 π}$ The radius from the surface area.

Worked examples

Story

A football has a radius of 11 cm. How much leather covers it? (π = 22/7)

$4 \times \frac{22}{7} \times 1 1^{2} = \frac{10648}{7}$
That is about 1521 cm².

Answer: 10648/7 cm², about 1521 cm²

Picture

Show that a sphere's surface equals the curved surface of the cylinder that just holds it (radius r, height 2r).

The cylinder's curved surface is 2πr × 2r = 4πr².
That is exactly the sphere's surface, 4πr².

Answer: Both are 4πr²

Direct

Find the surface area of a sphere of radius 7 cm.

$S = 4 \times π \times 49 = 196 π$
That is 616 cm² with π = 22/7.

Answer: 196π cm²

Reverse

A sphere has a surface area of 36π cm². Find its radius.

Try it first, then show the working

4πr² = 36π, so r² = 9 and r = 3 cm.

Answer: 3 cm

Exam

A top (lattu) is a hemisphere of radius 1.75 cm with a cone of slant height 3.7 cm on it. Find the area to colour (π = 22/7).

Try it first, then show the working

Hemisphere's curved surface: 2πr² = 2 × 22/7 × 1.75² = 19.25 cm².
Cone's curved surface: πrl = 22/7 × 1.75 × 3.7 = 20.35 cm².
$19.25 + 20.35 = 39.6$

Answer: 39.6 cm²

Common mistake: Confusing it with the volume (4/3 πr³) or with one circle (πr²).