Surface Area of a Cuboid and Cylinder

Find the surface area of cuboids, cylinders and prisms by adding the area of every face.

In the exam for Cambridge IGCSE Maths 0580, Cambridge O Level Maths 4024, Edexcel IGCSE Maths 4MA1, Edexcel Further Pure Maths 4PM1, IGCSE Additional Maths 0606 · 60 real past-paper questions below

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Surface Area Is the Wrapping

Surface area is the total area of all the outside faces. Think of the paper you need to wrap the solid.

Solid	Faces
Cuboid	6 rectangles in 3 matching pairs
Cylinder	2 circles and 1 curved side
Triangular prism	2 triangles and 3 rectangles

Rule

Find the area of each face, then add them all. The answer is in square units, like $cm^{2}$ .

Cuboid: The Formula

The most common 3-mark question: find the total surface area of a cuboid. Its 6 faces come in 3 matching pairs, so find 3 areas and double.

Formula

A = 2 (l w + l h + w h)

$l$ is the length, $w$ the width and $h$ the height. $l w$ , $l h$ and $w h$ are the areas of the three different faces.

Worked example

1
$l = 9, w = 4, h = 3$ — A cuboid 9 cm by 4 cm by 3 cm. Name the three lengths first. Any length can be l, w or h; the answer comes out the same.
2
$l w = 9 \times 4 = 36$ — The top. The bottom matches it.
3
$l h = 9 \times 3 = 27$ — The front. The back matches it.
4
$w h = 4 \times 3 = 12$ — One end. The other end matches it.
5
$36 + 27 + 12 = 75$ — One face from each pair. This is only half the box.
6
$2 \times 75 = 150 cm^{2}$ — Double it, because every face has a matching partner. The units are cm² because it is an area.

Key idea

Write down each face area, as in lines 2 to 4. Most of the marks are for this working, so a slip at the end still keeps them.

Key idea

Given a net instead? It is the same box opened flat. Find the three different rectangles and use the same formula. If the net has only 5 rectangles, the box has no lid: add the 5 faces, do not double.

Common mistake

Forgetting to double. 75 cm² is only three faces, half the box.

Cuboids and the Painting Question

The 6-mark question: paint the outside of a box, but not the base. List the faces you need, then share by the tin size.

Worked example

1
$5 \times 2 \times 2.5 m, no base$ — The box is 5 m long, 2 m wide and 2.5 m tall. The base sits on the ground, so it is not painted. That leaves 5 faces.
2
$top : 5 \times 2 = 10$ — The top is a 5 by 2 rectangle. Only one, because its partner, the base, is left out.
3
$front, back : 2 \times 5 \times 2.5 = 25$ — The front is 5 long and 2.5 tall. The back is the same size, so times 2.
4
$ends : 2 \times 2 \times 2.5 = 10$ — Each end is 2 wide and 2.5 tall, and there are two ends.
5
$10 + 25 + 10 = 45 m^{2}$ — Add the 5 faces. This is the area to paint.
6
$45 \div 12 = 3.75$ — One tin covers 12 m², so divide to find how many tins you need.
7
$4 tins$ — Round up. 3 tins cover only 36 m², which is not enough.

Common mistake

Rounding 3.75 down to 3 tins. You cannot buy part of a tin, and 3 tins leave some wall bare.

Cubes: Six Equal Squares

A cube is a cuboid with every edge the same length. All 6 faces are equal squares, so one formula does it.

Formula

A = 6 s^{2}

$s$ is the length of one edge, and $s^{2}$ is the area of one square face. Exams often give the surface area or the volume and ask you to go backwards.

Surface area given, find the edge.

Worked example

1
$6 s^{2} = 337.5$ — A cube has surface area 337.5 cm². Put it into the formula.
2
$s^{2} = 337.5 \div 6 = 56.25$ — Divide by 6. This is the area of one face.
3
$s = 56.25 = 7.5 cm$ — Square root to undo the square. A length cannot be negative, so keep only the positive root.

Volume given, find the surface area.

Worked example

1
$s^{3} = 1728$ — A cube has volume 1728 cm³. The volume of a cube is s × s × s.
2
$s = 31728 = 12$ — Cube root to find the edge: 12 × 12 × 12 = 1728.
3
$12^{2} = 144$ — The area of one square face.
4
$6 \times 144 = 864 cm^{2}$ — Six faces, all the same size.

Common mistake

Square rooting 337.5 straight away. Divide by 6 first: 337.5 is all six faces, not one.

Cylinder: Two Circles and a Label

The 2-mark question asks for the curved surface. Unroll it: it is a rectangle as long as the circumference.

Formula

curved = 2 π r h

$r$ is the radius and $h$ the height. $2 π r$ is the circumference, which is the length of the unrolled rectangle; $h$ is its width.

Formula

total = 2 π r h + 2 π r^{2}

The total adds the two circle ends, $π r^{2}$ each. Use it only if the question says total surface area.

Worked example

1
$r = 3.5, h = 10$ — Write down what you know: radius 3.5 cm, height 10 cm. If you are given the diameter, halve it first.
2
$2 π \times 3.5 \times 10 = 70 π$ — The curved side: put r and h into 2πrh. 2 × 3.5 × 10 = 70.
3
$70 π = 219.9 \dots \approx 220 cm^{2}$ — Times by π on the calculator, then round to 3 significant figures. Stop here if they ask for the curved surface only.
4
$2 \times π \times 3. 5^{2} = 24.5 π$ — The two ends. Square the radius first: 3.5² = 12.25, then 2 × 12.25 = 24.5.
5
$70 π + 24.5 π = 94.5 π$ — Total = curved + ends. Both are lots of π, so add the numbers in front: 70 + 24.5.
6
$94.5 π = 296.8 \dots \approx 297 cm^{2}$ — Times by π, then round to 3 s.f. If they ask for the answer in terms of π, stop at 94.5π.

Key idea

Open at one end, like a cup or a tin with no lid? Add one circle, not two. A cylinder rolled from a rectangle of paper: the rectangle's length is the circumference, so $r = length \div 2 π$ .

Common mistake

Adding the ends when they asked for the curved surface only. Read which one they want.

Triangular Prism: Five Faces

The 3-mark question shows a triangular prism. Two triangles at the ends, and three rectangles along the length. Any prism works the same way.

Formula

A = 2 \times end area + perimeter of end \times l

The end is the shape that runs all the way through, and $l$ is the length of the prism. Every rectangle is $l$ long, and together their widths go once round the end shape, so they add to perimeter × l.

Worked example

1
$6^{2} + 8^{2} = 100 = 10$ — The ends are right-angled triangles with sides 6 cm and 8 cm. Use Pythagoras for the sloping side, because questions often leave it out.
2
$2 \times \frac{1}{2} \times 6 \times 8 = 48$ — The two triangle ends. Each is ½ × base × height = 24.
3
$6 \times 12 = 72$ — The prism is 12 cm long. This is the rectangle under the 6 cm side.
4
$8 \times 12 = 96$ — The rectangle under the 8 cm side.
5
$10 \times 12 = 120$ — The sloping rectangle, under the 10 cm side. This is the face most often forgotten.
6
$48 + 72 + 96 + 120 = 336 cm^{2}$ — Add all five faces.

Key idea

Check with the formula: $2 \times 24 + (6 + 8 + 10) \times 12 = 48 + 288 = 336$ . Same answer.

Common mistake

Missing the sloping rectangle. Each side of the triangle makes its own rectangle.

Working Backwards

Sometimes they give the surface area and ask for a missing length. Write the formula, put in what you know, then solve the equation.

A cuboid 12 cm by 5 cm by h cm has surface area 290 cm².

Worked example

1
$2 (12 \times 5 + 12 h + 5 h) = 290$ — Put the lengths into 2(lw + lh + wh). The unknown h is in two of the faces.
2
$2 (60 + 17 h) = 290$ — Tidy inside: 12 × 5 = 60, and 12h + 5h = 17h.
3
$60 + 17 h = 145$ — Divide both sides by 2 to remove the bracket.
4
$17 h = 85$ — Take 60 from both sides.
5
$h = 5 cm$ — Divide by 17. Check: 2(60 + 60 + 25) = 290. Correct.

A closed cylinder with radius 4 cm has total surface area 112π cm².

Worked example

1
$2 π \times 4 \times h + 2 π \times 4^{2} = 112 π$ — Total = curved side + two ends. Put r = 4 into 2πrh + 2πr².
2
$8 π h + 32 π = 112 π$ — Tidy: 2 × 4 = 8, and 2 × 16 = 32.
3
$8 h + 32 = 112$ — Every term has a π, so divide every term by π. This keeps the numbers exact.
4
$8 h = 80$ — Take 32 from both sides.
5
$h = 10 cm$ — Divide by 8.

Key idea

Write the equation down before you solve it. Most of the marks are for setting it up correctly.

Common mistake

Dividing only some terms by π. Divide every term, or none.

Surface Area: Letters (Extended/Higher)

The 3-mark show that question: the sides are letters. Use the volume to swap one letter out.

A cuboid is x by 2x by y and its volume is 400. Show that $A = 4 x^{2} + \frac{1200}{x}$ .

Worked example

1
$x \times 2 x \times y = 400$ — Volume = length × width × height. This links x and y.
2
$2 x^{2} y = 400$ — Tidy the left side: x × 2x = 2x².
3
$x y = \frac{200}{x}$ — Divide both sides by 2x. We want xy because it appears in the faces.
4
$x \times 2 x = 2 x^{2}, x \times y = x y, 2 x \times y = 2 x y$ — The three different faces. Each one has a matching partner.
5
$A = 2 (2 x^{2} + x y + 2 x y)$ — Add one of each face, then double for the partners.
6
$A = 4 x^{2} + 6 x y$ — Multiply out: 2 × 2x² = 4x², and 2 × xy + 2 × 2xy = 6xy.
7
$A = 4 x^{2} + 6 \times \frac{200}{x}$ — Swap xy for 200/x, from line 3.
8
$A = 4 x^{2} + \frac{1200}{x}$ — 6 × 200 = 1200. This is the expression we had to show.

Key idea

In a show that question the answer is already given, so all the marks are for the working. Write every line.

Which Method? Read the Question

Surface area means covering. Volume means filling.

You see	You do
Paint, wrap, cover	add the faces
No base, open top	leave that face out
Curved surface	$2 π r h$ only
Total surface of a cylinder	$2 π r h + 2 π r^{2}$
A cube	$6 s^{2}$
Area given, a length missing	put in, then solve
How much it holds	Volume of a Prism
A cone	Volume and Surface Area of a Cone

Key idea

Tins or rolls of paper: always round up.

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