Surface Area

What you'll learn

What total surface area means.
How to find the surface area of cuboids, prisms and cylinders.
How to use a cross-section for triangular, trapezium and L-shaped prisms.
How to handle paint questions with m² and whole pots.

Surface area means “all the outside”

A face is an outside surface of a 3D shape. Some faces are flat, like rectangles. Some are curved, like the side of a cylinder.

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.

Flat area facts you need

For Grade 4 surface area, the main flat shapes are:

Rectangle: area is length multiplied by width.
Triangle: $\frac{1}{2}bh$ , where $b$ is the base and $h$ is the perpendicular height.
Trapezium: $\frac{1}{2}(a+b)h$ , where $a$ and $b$ are the parallel sides.

The perpendicular height is measured at 90° to the base, not along a slanted side.

Example

Area of a trapezium face

A trapezium face has parallel sides 12 cm and 6 cm, with perpendicular height 5 cm.

Trapezium face with parallel sides 12 cm and 6 cm and perpendicular height 5 cm shown at right angles to the bases.

Identify the values: $a = 12$ , $b = 6$ and $h = 5$ .
Substitute into the trapezium formula:
$A=12(12+6)×5A=\frac{1}{2}(12+6)\times 5$
Work it out:
$A=12×18×5=45A=\frac{1}{2}\times 18\times 5=45$
The area of the trapezium face is 45 cm².

Cuboids

A cuboid is a box shape with six rectangular faces. Opposite faces are the same size.

For a cuboid, there are three different rectangle areas:

length by width
length by height
width by height

Then each one is doubled.

Key Idea

Cuboid shortcut

For a cuboid, find the three different face areas, double each one, then add them.

Example

Total surface area of a cuboid

A cuboid has length 9 cm, width 4 cm and height 3 cm. Find its total surface area.

Cuboid with length 9 cm, width 4 cm and height 3 cm, showing the three different rectangular face dimensions.

Find the three different rectangular face areas:
$9×4=36,9×3=27,4×3=129\times 4=36,\quad 9\times 3=27,\quad 4\times 3=12$
Double each area because there are matching opposite faces:
$2×36+2×27+2×12=1502\times 36+2\times 27+2\times 12=150$
The total surface area is 150 cm².

Common Mistake

Only counting three faces

A cuboid has six faces, not three. Finding three different areas is useful, but you must double them.

Prisms

Definition

Prism and cross-section

A prism is a 3D shape with the same shape all the way through. The cross-section is the repeated end shape.

For any prism:

area+perimeter×length\text{total surface area}=2\times \text{cross-section area}+\text{perimeter}\times \text{length}

The perimeter is the distance all the way around the cross-section.

Triangular prisms

A triangular prism has two triangular ends and three rectangular side faces.

Example

Surface area of a triangular prism

A triangular prism has a right-angled triangular cross-section with sides 9 cm, 12 cm and 15 cm. The prism is 7 cm long.

Triangular prism with a right-angled triangular cross-section of sides 9 cm, 12 cm and 15 cm, extended 7 cm long.

Find the area of one triangular end:
$12×9×12=54\frac{1}{2}\times 9\times 12=54$
There are two triangular ends:
$2×54=1082\times 54=108$
Find the perimeter of the triangle:
$9 + 12 + 15 = 36$
Find the total area of the rectangular side faces:
$36×7=25236\times 7=252$
Add the end faces and side faces:
$108 + 252 = 360$
The total surface area is 360 cm².

Trapezium prisms

A trapezium prism works in the same way, but the end face is a trapezium.

Example

Surface area of a trapezium prism

A prism has a trapezium cross-section. The parallel sides are 12 cm and 6 cm, the perpendicular height is 4 cm, and the other two sides are both 5 cm. The prism is 8 cm long.

Trapezium prism with trapezium cross-section labelled with its side lengths, perpendicular height, and prism length.

Find the area of one trapezium end:
$12(12+6)×4=36\frac{1}{2}(12+6)\times 4=36$
There are two trapezium ends:
$2×36=722\times 36=72$
Find the perimeter of the trapezium:
$12 + 6 + 5 + 5 = 28$
Find the area of the rectangular side faces:
$28×8=22428\times 8=224$
Add everything:
$72 + 224 = 296$
The total surface area is 296 cm².

Composite prism cross-sections

A composite shape is made from two or more simpler shapes, or from one shape with a part removed.

For an L-shaped prism, you can still use:

area+perimeter×length\text{total surface area}=2\times \text{end area}+\text{perimeter}\times \text{length}

The tricky part is finding the area and perimeter of the L-shaped end.

Example

Surface area of an L-shaped prism

An L-shaped prism is 10 cm long. Its front cross-section fits inside an 8 cm by 7 cm rectangle, with a 3 cm by 4 cm corner removed.

L-shaped prism formed from an 8 cm by 7 cm rectangle with a 3 cm by 4 cm corner removed, extended 10 cm long.

Find the area of the L-shaped end by subtracting the missing rectangle:
$8×7−3×4=56−12=448\times 7-3\times 4=56-12=44$
There are two identical L-shaped ends:
$2×44=882\times 44=88$
Walk around the L shape to find its perimeter:
$8 + 3 + 3 + 4 + 5 + 7 = 30$
Find the total area of the rectangular side faces:
$30×10=30030\times 10=300$
Add the two ends and the side faces:
$88 + 300 = 388$
The total surface area is 388 cm².

Tip

Walk around the outline

For composite prisms, trace the outside edge of the cross-section with your finger. Include the step edges.

Cylinders

Definition

Cylinder words

A cylinder has two equal circular faces and one curved face. The diameter is the full width through the centre of a circle; the radius is half the diameter.

A closed cylinder has a top and a bottom.

For a closed cylinder:

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

Here, $r$ is the radius and $h$ is the height.

Example

Surface area of a closed cylinder

A closed cylinder has diameter 50 cm and height 1.2 m. Find its surface area in m².

Closed cylinder with diameter 50 cm across the circular face and height 1.2 m.

Convert the diameter to metres: 50 cm is 0.5 m, so the radius is 0.25 m.
Find the area of the two circular faces:
$2πr2=2π×0.252=0.125π2\pi r^2=2\pi \times 0.25^2=0.125\pi$
Find the curved surface area:
$2πrh=2π×0.25×1.2=0.6π2\pi rh=2\pi \times 0.25\times 1.2=0.6\pi$
Add them:
$0.125π+0.6π=0.725π≈2.280.125\pi+0.6\pi=0.725\pi\approx 2.28$
The surface area is about 2.28 m².

Common Mistake

Using diameter as radius

If the question gives the diameter, halve it before using the cylinder formula.

Paint and coverage questions

Definition

Paint coverage

Coverage means the area that one pot of paint can cover.

In paint questions:

Find the surface area of one object.
Multiply by the number of objects.
Divide by the coverage of one pot.
Round up to the next whole pot.

Tip

Converting area units

If paint coverage is in m², convert all lengths to metres before finding area. If you already have cm², divide by 10000 to get m².

Example

How many pots for several boxes?

Four closed boxes are each 120 cm long, 50 cm wide and 40 cm high. One pot of paint covers 3 m². Work out how many pots are needed.

Four identical closed cuboid boxes with dimensions 120 cm by 50 cm by 40 cm and one paint pot covering 3 m².

Convert the dimensions to metres: 120 cm is 1.2 m, 50 cm is 0.5 m, and 40 cm is 0.4 m.
Find the surface area of one box:
$2(1.2×0.5)+2(1.2×0.4)+2(0.5×0.4)=2.562(1.2\times 0.5)+2(1.2\times 0.4)+2(0.5\times 0.4)=2.56$
Four boxes need:
$4×2.56=10.244\times 2.56=10.24$
Divide by the coverage per pot:
$10.24÷3≈3.4110.24\div 3\approx 3.41$
You need 4 pots, because you must buy whole pots.

Exam technique

In the exam

Label each face or use the prism formula so you do not miss any surfaces.
Check whether the shape has a top and bottom, especially cylinders and tanks.
Keep units consistent, then round paint pots up, not to the nearest whole number.

Self review

Check yourself

Can you explain why a cuboid has three pairs of equal faces?
For a prism, can you find the area and perimeter of the cross-section separately?
In a paint question, when do you need to round your answer up?

Surface Area

What you'll learn

What total surface area means.
How to find the surface area of cuboids, prisms and cylinders.
How to use a cross-section for triangular, trapezium and L-shaped prisms.
How to handle paint questions with m² and whole pots.

Surface area means “all the outside”

A face is an outside surface of a 3D shape. Some faces are flat, like rectangles. Some are curved, like the side of a cylinder.

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.

Flat area facts you need

For Grade 4 surface area, the main flat shapes are:

Rectangle: area is length multiplied by width.
Triangle: $\frac{1}{2}bh$ , where $b$ is the base and $h$ is the perpendicular height.
Trapezium: $\frac{1}{2}(a+b)h$ , where $a$ and $b$ are the parallel sides.

The perpendicular height is measured at 90° to the base, not along a slanted side.

Example

Area of a trapezium face

A trapezium face has parallel sides 12 cm and 6 cm, with perpendicular height 5 cm.

Trapezium face with parallel sides 12 cm and 6 cm and perpendicular height 5 cm shown at right angles to the bases.

Identify the values: $a = 12$ , $b = 6$ and $h = 5$ .
Substitute into the trapezium formula:
$A=12(12+6)×5A=\frac{1}{2}(12+6)\times 5$
Work it out:
$A=12×18×5=45A=\frac{1}{2}\times 18\times 5=45$
The area of the trapezium face is 45 cm².

Cuboids

A cuboid is a box shape with six rectangular faces. Opposite faces are the same size.

For a cuboid, there are three different rectangle areas:

length by width
length by height
width by height

Then each one is doubled.

Key Idea

Cuboid shortcut

For a cuboid, find the three different face areas, double each one, then add them.

Example

Total surface area of a cuboid

A cuboid has length 9 cm, width 4 cm and height 3 cm. Find its total surface area.

Cuboid with length 9 cm, width 4 cm and height 3 cm, showing the three different rectangular face dimensions.

Find the three different rectangular face areas:
$9×4=36,9×3=27,4×3=129\times 4=36,\quad 9\times 3=27,\quad 4\times 3=12$
Double each area because there are matching opposite faces:
$2×36+2×27+2×12=1502\times 36+2\times 27+2\times 12=150$
The total surface area is 150 cm².

Common Mistake

Only counting three faces

A cuboid has six faces, not three. Finding three different areas is useful, but you must double them.

Prisms

Definition

Prism and cross-section

A prism is a 3D shape with the same shape all the way through. The cross-section is the repeated end shape.

For any prism:

area+perimeter×length\text{total surface area}=2\times \text{cross-section area}+\text{perimeter}\times \text{length}

The perimeter is the distance all the way around the cross-section.

Triangular prisms

A triangular prism has two triangular ends and three rectangular side faces.

Example

Surface area of a triangular prism

A triangular prism has a right-angled triangular cross-section with sides 9 cm, 12 cm and 15 cm. The prism is 7 cm long.

Triangular prism with a right-angled triangular cross-section of sides 9 cm, 12 cm and 15 cm, extended 7 cm long.

Find the area of one triangular end:
$12×9×12=54\frac{1}{2}\times 9\times 12=54$
There are two triangular ends:
$2×54=1082\times 54=108$
Find the perimeter of the triangle:
$9 + 12 + 15 = 36$
Find the total area of the rectangular side faces:
$36×7=25236\times 7=252$
Add the end faces and side faces:
$108 + 252 = 360$
The total surface area is 360 cm².

Trapezium prisms

A trapezium prism works in the same way, but the end face is a trapezium.

Example

Surface area of a trapezium prism

A prism has a trapezium cross-section. The parallel sides are 12 cm and 6 cm, the perpendicular height is 4 cm, and the other two sides are both 5 cm. The prism is 8 cm long.

Trapezium prism with trapezium cross-section labelled with its side lengths, perpendicular height, and prism length.

Find the area of one trapezium end:
$12(12+6)×4=36\frac{1}{2}(12+6)\times 4=36$
There are two trapezium ends:
$2×36=722\times 36=72$
Find the perimeter of the trapezium:
$12 + 6 + 5 + 5 = 28$
Find the area of the rectangular side faces:
$28×8=22428\times 8=224$
Add everything:
$72 + 224 = 296$
The total surface area is 296 cm².

Composite prism cross-sections

A composite shape is made from two or more simpler shapes, or from one shape with a part removed.

For an L-shaped prism, you can still use:

area+perimeter×length\text{total surface area}=2\times \text{end area}+\text{perimeter}\times \text{length}

The tricky part is finding the area and perimeter of the L-shaped end.

Example

Surface area of an L-shaped prism

An L-shaped prism is 10 cm long. Its front cross-section fits inside an 8 cm by 7 cm rectangle, with a 3 cm by 4 cm corner removed.

L-shaped prism formed from an 8 cm by 7 cm rectangle with a 3 cm by 4 cm corner removed, extended 10 cm long.

Find the area of the L-shaped end by subtracting the missing rectangle:
$8×7−3×4=56−12=448\times 7-3\times 4=56-12=44$
There are two identical L-shaped ends:
$2×44=882\times 44=88$
Walk around the L shape to find its perimeter:
$8 + 3 + 3 + 4 + 5 + 7 = 30$
Find the total area of the rectangular side faces:
$30×10=30030\times 10=300$
Add the two ends and the side faces:
$88 + 300 = 388$
The total surface area is 388 cm².

Tip

Walk around the outline

For composite prisms, trace the outside edge of the cross-section with your finger. Include the step edges.

Cylinders

Definition

Cylinder words

A cylinder has two equal circular faces and one curved face. The diameter is the full width through the centre of a circle; the radius is half the diameter.

A closed cylinder has a top and a bottom.

For a closed cylinder:

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

Here, $r$ is the radius and $h$ is the height.

Example

Surface area of a closed cylinder

A closed cylinder has diameter 50 cm and height 1.2 m. Find its surface area in m².

Closed cylinder with diameter 50 cm across the circular face and height 1.2 m.

Convert the diameter to metres: 50 cm is 0.5 m, so the radius is 0.25 m.
Find the area of the two circular faces:
$2πr2=2π×0.252=0.125π2\pi r^2=2\pi \times 0.25^2=0.125\pi$
Find the curved surface area:
$2πrh=2π×0.25×1.2=0.6π2\pi rh=2\pi \times 0.25\times 1.2=0.6\pi$
Add them:
$0.125π+0.6π=0.725π≈2.280.125\pi+0.6\pi=0.725\pi\approx 2.28$
The surface area is about 2.28 m².

Common Mistake

Using diameter as radius

If the question gives the diameter, halve it before using the cylinder formula.

Paint and coverage questions

Definition

Paint coverage

Coverage means the area that one pot of paint can cover.

In paint questions:

Find the surface area of one object.
Multiply by the number of objects.
Divide by the coverage of one pot.
Round up to the next whole pot.

Tip

Converting area units

If paint coverage is in m², convert all lengths to metres before finding area. If you already have cm², divide by 10000 to get m².

Example

How many pots for several boxes?

Four closed boxes are each 120 cm long, 50 cm wide and 40 cm high. One pot of paint covers 3 m². Work out how many pots are needed.

Four identical closed cuboid boxes with dimensions 120 cm by 50 cm by 40 cm and one paint pot covering 3 m².

Convert the dimensions to metres: 120 cm is 1.2 m, 50 cm is 0.5 m, and 40 cm is 0.4 m.
Find the surface area of one box:
$2(1.2×0.5)+2(1.2×0.4)+2(0.5×0.4)=2.562(1.2\times 0.5)+2(1.2\times 0.4)+2(0.5\times 0.4)=2.56$
Four boxes need:
$4×2.56=10.244\times 2.56=10.24$
Divide by the coverage per pot:
$10.24÷3≈3.4110.24\div 3\approx 3.41$
You need 4 pots, because you must buy whole pots.

Exam technique

In the exam

Label each face or use the prism formula so you do not miss any surfaces.
Check whether the shape has a top and bottom, especially cylinders and tanks.
Keep units consistent, then round paint pots up, not to the nearest whole number.

Self review

Check yourself

Can you explain why a cuboid has three pairs of equal faces?
For a prism, can you find the area and perimeter of the cross-section separately?
In a paint question, when do you need to round your answer up?

Surface Area

What you'll learn

Surface area means “all the outside”

Flat area facts you need

Cuboids

Prisms

Triangular prisms

Trapezium prisms

Composite prism cross-sections

Cylinders

Paint and coverage questions

Recap questions

You've reached the end

Surface Area

What you'll learn

Surface area means “all the outside”

Flat area facts you need

Cuboids

Prisms

Triangular prisms

Trapezium prisms

Composite prism cross-sections

Cylinders

Paint and coverage questions

Recap questions

You've reached the end

Surface Area

What you'll learn

Surface area means “all the outside”

Flat area facts you need

Cuboids

Prisms

Triangular prisms

Trapezium prisms

Composite prism cross-sections

Cylinders

Paint and coverage questions

Recap questions

You've reached the end

Surface Area Revision Guide

Surface Area

What you'll learn

Surface area means “all the outside”

Flat area facts you need

Cuboids

Prisms

Triangular prisms

Trapezium prisms

Composite prism cross-sections

Cylinders

Paint and coverage questions

Recap questions

You've reached the end

Surface Area Revision Guide