Surface Area

What you'll learn

What surface area means, and how it is different from just finding one face.
How to find the total surface area of cuboids and prisms.
How to handle triangular, trapezium and L-shaped cross-sections.
How to solve painting problems, including unit conversions and rounding up.

The main idea

Definition

Surface area

Surface area is the total area of all the outside faces of a 3D shape. It is measured in square units, such as cm² or m².

A face is one flat surface of a 3D shape. For example, a cuboid has 6 rectangular faces.

A net is a flat 2D diagram that shows all the faces of a 3D shape opened out. Nets are useful because surface area is just “add up the areas in the net”.

Key Idea

Think of wrapping paper

Surface area is the amount of material needed to cover the outside of a shape, with no gaps and no overlaps.

Useful area facts to remember

Before surface area questions, you need these area formulae:

Rectangle: area = length times width.
Triangle: area = half times base times height.
Trapezium: area = half times the sum of the parallel sides times the height.
Circle: area is $πr2\pi r^2$ , where $r$ is the radius.

You will also need perimeter, which means the distance all the way around a 2D shape.

Cuboids

Definition

Cuboid

A cuboid is a 3D shape with 6 rectangular faces, like a box.

A cuboid has 3 different pairs of matching faces: front/back, top/bottom, and left/right.

Here is how a cuboid and its net are connected.

Cuboid and net showing matching rectangular faces

If the length is $l$ , width is $w$ , and height is $h$ , then:

area=2(lw+lh+wh)\text{total surface area} = 2(lw + lh + wh)

You do not have to memorise this if you understand the net, but it is a useful shortcut.

Example

Finding the total surface area of a cuboid

Suppose a cuboid has length 11 cm, width 6 cm and height 3 cm. Identify the three different rectangle areas: length by width, length by height, and width by height.
Work out the total surface area by doubling each matching pair:
$area=2(11×6)+2(11×3)+2(6×3)=132+66+36=234\begin{aligned} \text{surface area} &= 2(11 \times 6) + 2(11 \times 3) + 2(6 \times 3) \\ &= 132 + 66 + 36 \\ &= 234 \end{aligned}$
The total surface area is 234 cm².

Common Mistake

Only adding the visible faces

In a 3D diagram, you usually cannot see every face. Surface area means every outside face, including the back and bottom faces.

Prisms

Definition

Prism

A prism is a 3D shape with the same cross-section all the way through its length.

The cross-section is the 2D shape you would see if you sliced straight through the prism. In a triangular prism, the cross-section is a triangle. In a trapezium prism, the cross-section is a trapezium.

For any prism:

rectangles\text{surface area} = 2 \times \text{area of cross-section} + \text{area of side rectangles}

The side rectangles wrap around the cross-section. Their combined area is:

prism\text{perimeter of cross-section} \times \text{length of prism}

So a very useful prism formula is:

area=2A+PL\text{surface area} = 2A + PL

where $A$ is the area of the cross-section, $P$ is the perimeter of the cross-section, and $L$ is the length of the prism.

Triangular prisms

A triangular prism has 2 triangular ends and 3 rectangular faces around the sides.

The net below shows why the three side rectangles use the three side lengths of the triangle.

Triangular prism net showing two triangles and three rectangles

Example

Finding the surface area of a triangular prism

A triangular prism has a right-angled triangular cross-section with side lengths 6 cm, 8 cm and 10 cm. The prism length is 9 cm. First find the area of one triangular end:
$12×6×8=24\frac{1}{2} \times 6 \times 8 = 24$
There are 2 triangular ends:
$\times 24 = 48$
Find the perimeter of the triangular cross-section:
$6 + 8 + 10 = 24$
Multiply the perimeter by the prism length to get the area of the 3 rectangles around the side:
$24 \times 9 = 216$
Add the triangular ends and the side rectangles:
$48 + 216 = 264$
The total surface area is 264 cm².

Tip

Fast prism method

For any prism, find the area of the two identical ends first, then add the rectangles around the outside using perimeter times length.

Trapezium prisms

Definition

Trapezium

A trapezium is a quadrilateral with one pair of parallel sides.

For a trapezium prism, the cross-section is a trapezium. You usually need two separate things:

The area of the trapezium for the two end faces.
The perimeter of the trapezium for the rectangular side faces.

Example

Finding the surface area of a trapezium prism

A prism has a trapezium cross-section with parallel sides 10 cm and 16 cm, height 4 cm, and the two sloping sides both 5 cm. The length of the prism is 7 cm. First find the area of one trapezium:
$12(10+16)×4=52\frac{1}{2}(10 + 16) \times 4 = 52$
Double it because there are 2 identical trapezium ends:
$\times 52 = 104$
Find the perimeter of the trapezium:
$10 + 16 + 5 + 5 = 36$
Multiply by the length of the prism to find the rectangular side area:
$36 \times 7 = 252$
Add the two parts:
$104 + 252 = 356$
The total surface area is 356 cm².

Common Mistake

Using the trapezium height in the perimeter

The perpendicular height of a trapezium is used for area, not perimeter, unless it is also an outside side length.

Composite prisms: L-shaped cross-sections

A composite shape is made from simpler shapes joined together. For an L-shaped prism, split the cross-section into rectangles to find its area.

For the perimeter, be careful: you need the outside boundary only. Do not include lines inside the shape where you split it up.

Example

Finding the surface area of an L-shaped prism

An L-shaped cross-section is made from a 7 cm by 4 cm rectangle and a 3 cm by 5 cm rectangle sitting on its left side. The prism length is 12 cm. Find the area of the cross-section:
$\times 4) + (3 \times 5) = 28 + 15 = 43$
Double the cross-section area for the front and back faces:
$\times 43 = 86$
Find the outside perimeter. The outside sides are 7 cm, 4 cm, 4 cm, 5 cm, 3 cm and 9 cm:
$7 + 4 + 4 + 5 + 3 + 9 = 32$
Multiply the perimeter by the prism length:
$32 \times 12 = 384$
Add the front/back area and the side area:
$86 + 384 = 470$
The total surface area is 470 cm².

Painting problems and unit conversions

In real-life questions, you may need to paint several objects. The method is:

Find the surface area of one object.
Multiply by the number of objects.
Divide by the coverage of one pot.
Round up, because you cannot buy part of a pot.

Common Mistake

Square unit conversions

Since 100 cm = 1 m, it follows that 10,000 cm² = 1 m². Do not divide cm² by 100 to get m².

Example

Boxes being painted

A closed rectangular box measures 90 cm by 45 cm by 30 cm. Convert to metres because the paint coverage is in m²: 0.9 m by 0.45 m by 0.3 m.
Find the surface area of one box:
$area=2(0.9×0.45)+2(0.9×0.3)+2(0.45×0.3)=0.81+0.54+0.27=1.62\begin{aligned} \text{surface area} &= 2(0.9 \times 0.45) + 2(0.9 \times 0.3) + 2(0.45 \times 0.3) \\ &= 0.81 + 0.54 + 0.27 \\ &= 1.62 \end{aligned}$
If there are 6 boxes, find the total area:
$\times 1.62 = 9.72$
If one pot covers 5 m², divide:
$9.72 \div 5 = 1.944$
You need 2 pots of paint, because 1 pot is not enough.

Cylinders with a top and bottom

Definition

Cylinder terms

The radius is the distance from the centre of a circle to its edge. The diameter is the distance all the way across the circle through the centre, so diameter = 2 times radius. The circumference is the distance around the circle.

A closed cylinder has:

2 circular faces: top and bottom.
1 curved rectangular face wrapped around the side.

The side face has height $h$ and width equal to the circumference of the circle.

Closed cylinder and net showing two circles and one rectangle

For a closed cylinder:

area=2πr2+2πrh\text{surface area} = 2\pi r^2 + 2\pi rh

Example

Painting closed cylindrical tanks

A closed cylindrical tank has diameter 60 cm and height 1.2 m. Convert the diameter to metres: 60 cm = 0.6 m.
Find the radius:
$0.6 \div 2 = 0.3$
Find the surface area of one closed cylinder:
$area=2π(0.3)2+2π(0.3)(1.2)=0.18π+0.72π=0.90π\begin{aligned} \text{surface area} &= 2\pi(0.3)^2 + 2\pi(0.3)(1.2) \\ &= 0.18\pi + 0.72\pi \\ &= 0.90\pi \end{aligned}$
Use a calculator approximation:
$0.90π≈2.830.90\pi \approx 2.83$
For 3 tanks, multiply by 3:
$\times 2.83 = 8.49$
If one pot covers 3 m², divide and round up:
$8.49 \div 3 = 2.83$
You need 3 pots of paint.

Exam technique

In the exam

Sketch or imagine the net, then tick off every outside face as you include it.
For prisms, use: two identical ends plus perimeter of cross-section times length.
Check units before calculating; paint coverage is often given in m², while diagrams may use cm.
For pots of paint, always round up to the next whole pot.

Self review

Check yourself

Can you explain why a triangular prism has 3 rectangular side faces?
When finding the surface area of a trapezium prism, which measurements are used for area and which are used for perimeter?
If an area is given in cm² and you need m², should you divide by 100 or by 10,000?

Surface Area

What you'll learn

What surface area means, and how it is different from just finding one face.
How to find the total surface area of cuboids and prisms.
How to handle triangular, trapezium and L-shaped cross-sections.
How to solve painting problems, including unit conversions and rounding up.

The main idea

Definition

Surface area

Surface area is the total area of all the outside faces of a 3D shape. It is measured in square units, such as cm² or m².

A face is one flat surface of a 3D shape. For example, a cuboid has 6 rectangular faces.

A net is a flat 2D diagram that shows all the faces of a 3D shape opened out. Nets are useful because surface area is just “add up the areas in the net”.

Key Idea

Think of wrapping paper

Surface area is the amount of material needed to cover the outside of a shape, with no gaps and no overlaps.

Useful area facts to remember

Before surface area questions, you need these area formulae:

Rectangle: area = length times width.
Triangle: area = half times base times height.
Trapezium: area = half times the sum of the parallel sides times the height.
Circle: area is $πr2\pi r^2$ , where $r$ is the radius.

You will also need perimeter, which means the distance all the way around a 2D shape.

Cuboids

Definition

Cuboid

A cuboid is a 3D shape with 6 rectangular faces, like a box.

A cuboid has 3 different pairs of matching faces: front/back, top/bottom, and left/right.

Here is how a cuboid and its net are connected.

Cuboid and net showing matching rectangular faces

If the length is $l$ , width is $w$ , and height is $h$ , then:

area=2(lw+lh+wh)\text{total surface area} = 2(lw + lh + wh)

You do not have to memorise this if you understand the net, but it is a useful shortcut.

Example

Finding the total surface area of a cuboid

Suppose a cuboid has length 11 cm, width 6 cm and height 3 cm. Identify the three different rectangle areas: length by width, length by height, and width by height.
Work out the total surface area by doubling each matching pair:
$area=2(11×6)+2(11×3)+2(6×3)=132+66+36=234\begin{aligned} \text{surface area} &= 2(11 \times 6) + 2(11 \times 3) + 2(6 \times 3) \\ &= 132 + 66 + 36 \\ &= 234 \end{aligned}$
The total surface area is 234 cm².

Common Mistake

Only adding the visible faces

In a 3D diagram, you usually cannot see every face. Surface area means every outside face, including the back and bottom faces.

Prisms

Definition

Prism

A prism is a 3D shape with the same cross-section all the way through its length.

For any prism:

rectangles\text{surface area} = 2 \times \text{area of cross-section} + \text{area of side rectangles}

The side rectangles wrap around the cross-section. Their combined area is:

prism\text{perimeter of cross-section} \times \text{length of prism}

So a very useful prism formula is:

area=2A+PL\text{surface area} = 2A + PL

where $A$ is the area of the cross-section, $P$ is the perimeter of the cross-section, and $L$ is the length of the prism.

Triangular prisms

A triangular prism has 2 triangular ends and 3 rectangular faces around the sides.

The net below shows why the three side rectangles use the three side lengths of the triangle.

Triangular prism net showing two triangles and three rectangles

Example

Finding the surface area of a triangular prism

A triangular prism has a right-angled triangular cross-section with side lengths 6 cm, 8 cm and 10 cm. The prism length is 9 cm. First find the area of one triangular end:
$12×6×8=24\frac{1}{2} \times 6 \times 8 = 24$
There are 2 triangular ends:
$\times 24 = 48$
Find the perimeter of the triangular cross-section:
$6 + 8 + 10 = 24$
Multiply the perimeter by the prism length to get the area of the 3 rectangles around the side:
$24 \times 9 = 216$
Add the triangular ends and the side rectangles:
$48 + 216 = 264$
The total surface area is 264 cm².

Tip

Fast prism method

For any prism, find the area of the two identical ends first, then add the rectangles around the outside using perimeter times length.

Trapezium prisms

Definition

Trapezium

A trapezium is a quadrilateral with one pair of parallel sides.

For a trapezium prism, the cross-section is a trapezium. You usually need two separate things:

The area of the trapezium for the two end faces.
The perimeter of the trapezium for the rectangular side faces.

Example

Finding the surface area of a trapezium prism

A prism has a trapezium cross-section with parallel sides 10 cm and 16 cm, height 4 cm, and the two sloping sides both 5 cm. The length of the prism is 7 cm. First find the area of one trapezium:
$12(10+16)×4=52\frac{1}{2}(10 + 16) \times 4 = 52$
Double it because there are 2 identical trapezium ends:
$\times 52 = 104$
Find the perimeter of the trapezium:
$10 + 16 + 5 + 5 = 36$
Multiply by the length of the prism to find the rectangular side area:
$36 \times 7 = 252$
Add the two parts:
$104 + 252 = 356$
The total surface area is 356 cm².

Common Mistake

Using the trapezium height in the perimeter

The perpendicular height of a trapezium is used for area, not perimeter, unless it is also an outside side length.

Composite prisms: L-shaped cross-sections

A composite shape is made from simpler shapes joined together. For an L-shaped prism, split the cross-section into rectangles to find its area.

For the perimeter, be careful: you need the outside boundary only. Do not include lines inside the shape where you split it up.

Example

Finding the surface area of an L-shaped prism

An L-shaped cross-section is made from a 7 cm by 4 cm rectangle and a 3 cm by 5 cm rectangle sitting on its left side. The prism length is 12 cm. Find the area of the cross-section:
$\times 4) + (3 \times 5) = 28 + 15 = 43$
Double the cross-section area for the front and back faces:
$\times 43 = 86$
Find the outside perimeter. The outside sides are 7 cm, 4 cm, 4 cm, 5 cm, 3 cm and 9 cm:
$7 + 4 + 4 + 5 + 3 + 9 = 32$
Multiply the perimeter by the prism length:
$32 \times 12 = 384$
Add the front/back area and the side area:
$86 + 384 = 470$
The total surface area is 470 cm².

Painting problems and unit conversions

In real-life questions, you may need to paint several objects. The method is:

Find the surface area of one object.
Multiply by the number of objects.
Divide by the coverage of one pot.
Round up, because you cannot buy part of a pot.

Common Mistake

Square unit conversions

Since 100 cm = 1 m, it follows that 10,000 cm² = 1 m². Do not divide cm² by 100 to get m².

Example

Boxes being painted

A closed rectangular box measures 90 cm by 45 cm by 30 cm. Convert to metres because the paint coverage is in m²: 0.9 m by 0.45 m by 0.3 m.
Find the surface area of one box:
$area=2(0.9×0.45)+2(0.9×0.3)+2(0.45×0.3)=0.81+0.54+0.27=1.62\begin{aligned} \text{surface area} &= 2(0.9 \times 0.45) + 2(0.9 \times 0.3) + 2(0.45 \times 0.3) \\ &= 0.81 + 0.54 + 0.27 \\ &= 1.62 \end{aligned}$
If there are 6 boxes, find the total area:
$\times 1.62 = 9.72$
If one pot covers 5 m², divide:
$9.72 \div 5 = 1.944$
You need 2 pots of paint, because 1 pot is not enough.

Cylinders with a top and bottom

Definition

Cylinder terms

A closed cylinder has:

2 circular faces: top and bottom.
1 curved rectangular face wrapped around the side.

The side face has height $h$ and width equal to the circumference of the circle.

Closed cylinder and net showing two circles and one rectangle

For a closed cylinder:

area=2πr2+2πrh\text{surface area} = 2\pi r^2 + 2\pi rh

Example

Painting closed cylindrical tanks

A closed cylindrical tank has diameter 60 cm and height 1.2 m. Convert the diameter to metres: 60 cm = 0.6 m.
Find the radius:
$0.6 \div 2 = 0.3$
Find the surface area of one closed cylinder:
$area=2π(0.3)2+2π(0.3)(1.2)=0.18π+0.72π=0.90π\begin{aligned} \text{surface area} &= 2\pi(0.3)^2 + 2\pi(0.3)(1.2) \\ &= 0.18\pi + 0.72\pi \\ &= 0.90\pi \end{aligned}$
Use a calculator approximation:
$0.90π≈2.830.90\pi \approx 2.83$
For 3 tanks, multiply by 3:
$\times 2.83 = 8.49$
If one pot covers 3 m², divide and round up:
$8.49 \div 3 = 2.83$
You need 3 pots of paint.

Exam technique

In the exam

Sketch or imagine the net, then tick off every outside face as you include it.
For prisms, use: two identical ends plus perimeter of cross-section times length.
Check units before calculating; paint coverage is often given in m², while diagrams may use cm.
For pots of paint, always round up to the next whole pot.

Self review

Check yourself

Can you explain why a triangular prism has 3 rectangular side faces?
When finding the surface area of a trapezium prism, which measurements are used for area and which are used for perimeter?
If an area is given in cm² and you need m², should you divide by 100 or by 10,000?

Surface Area

What you'll learn

The main idea

Useful area facts to remember

Cuboids

Prisms

Triangular prisms

Trapezium prisms

Composite prisms: L-shaped cross-sections

Painting problems and unit conversions

Cylinders with a top and bottom

Recap questions

You've reached the end

Surface Area

What you'll learn

The main idea

Useful area facts to remember

Cuboids

Prisms

Triangular prisms

Trapezium prisms

Composite prisms: L-shaped cross-sections

Painting problems and unit conversions

Cylinders with a top and bottom

Recap questions

You've reached the end

Surface Area

What you'll learn

The main idea

Useful area facts to remember

Cuboids

Prisms

Triangular prisms

Trapezium prisms

Composite prisms: L-shaped cross-sections

Painting problems and unit conversions

Cylinders with a top and bottom

Recap questions

You've reached the end

Surface Area Revision Guide

Surface Area

What you'll learn

The main idea

Useful area facts to remember

Cuboids

Prisms

Triangular prisms

Trapezium prisms

Composite prisms: L-shaped cross-sections

Painting problems and unit conversions

Cylinders with a top and bottom

Recap questions

You've reached the end

Surface Area Revision Guide