Key Stage 3 maths · Geometry and measures

KS3 perimeter, area and volume: a guide for parents

In Years 7 to 9 pupils move from using area and volume formulae to understanding where they come from and applying them to less familiar shapes. They work with triangles, parallelograms, trapezia and circles, split composite shapes into simpler parts, find the volume of prisms and cylinders, and use the faces, edges and vertices of 3-D solids to solve problems.

Perimeter, area and volume, explained

Perimeter is the distance around the outside of a shape, area is the amount of flat space inside it, and volume is the amount of space inside a solid. They are measured in different units: for example, centimetres (cm) for perimeter, square centimetres (cm²) for area and cubic centimetres (cm³) for volume. Most pupils have met rectangles, triangles and cuboids in primary school. At Key Stage 3 the emphasis shifts to deriving the formulae, which means being able to explain why they work, not just remember them.

That is why teachers ask pupils to cut and rearrange shapes. A parallelogram can be cut and re-attached to make a rectangle with the same base and height, so its area is base × perpendicular height. A triangle is half of a parallelogram, so its area is ½ × base × perpendicular height. A trapezium is half of a parallelogram made from two copies of itself, which gives ½ × (a + b) × h, where a and b are the parallel sides. The height is always measured at right angles to the base, never along a slanting side.

Circles bring in π, a number a little above 3.14 that links a circle’s diameter to its circumference. The circumference is C = πd, or C = 2πr, and the area is A = πr². Pupils learn to keep the radius (centre to edge) and the diameter (all the way across through the centre) apart, and to use the π button on a calculator, or 3.14 when a question says so. Answers are often rounded to a stated number of decimal places or significant figures, and a semicircle needs its curved edge plus its straight edge when finding the perimeter.

Composite shapes are made by joining or cutting simpler ones, such as an L-shaped room or a rectangle with a semicircle on one end. Pupils either split the shape into parts and add the areas, or take a cut-out away from a larger shape. Finding a missing length by subtracting known ones is a skill in its own right and the source of many errors.

For solids, the key idea is the prism: any 3-D shape with the same cross-section all the way through. Its volume is the area of the cross-section × the length. A cylinder is a prism with a circular cross-section, so its volume is πr²h. Pupils also learn that 1 cm³ holds 1 ml, so 1,000 cm³ is 1 litre. Alongside this, they use the properties of faces, edges and vertices of cubes, cuboids, prisms, cylinders, pyramids, cones and spheres, for example adding up the areas of all the faces to find the surface area of a cuboid or prism, or working out that a square-based pyramid has 5 faces, 8 edges and 5 vertices.

Words your child will hear

Perimeter: The total distance around the outside edge of a 2-D shape.
Area: The amount of surface inside a 2-D shape, measured in square units such as cm².
Circumference: The perimeter of a circle. It is π times the diameter.
Cross-section: The shape you see when you slice through a solid, such as the trapezium at the end of a trapezoidal prism.
Prism: A solid with the same cross-section all the way along its length, such as a cuboid, a triangular prism or a cylinder.
Composite shape: A shape made by joining simpler shapes together, or by cutting one shape out of another.
Face, edge, vertex: The flat surfaces of a solid, the lines where two faces meet, and the corners where edges meet.

What the National Curriculum says

In Key Stage 3, pupils should be taught to:

Derive and apply formulae to calculate and solve problems involving: perimeter and area of triangles, parallelograms, trapezia, volume of cuboids (including cubes) and other prisms (including cylinders)
Calculate and solve problems involving: perimeters of 2-D shapes (including circles), areas of circles and composite shapes
Use the properties of faces, surfaces, edges and vertices of cubes, cuboids, prisms, cylinders, pyramids, cones and spheres to solve problems in 3-D

Objectives from the National Curriculum in England, © Crown copyright, used under the Open Government Licence v3.0.

A worked example

A small tank is a prism 20 cm long. Its cross-section is a trapezium with parallel sides of 7 cm and 11 cm, which are 5 cm apart. Find the volume of the tank and say how many litres it holds when full.

Identify the cross-section: the trapezium at the end. The volume of any prism is the area of its cross-section multiplied by its length, so the first job is the trapezium’s area.
Use the trapezium formula, with the two parallel sides added together and the perpendicular height of 5 cm. Two copies of the trapezium fit together to make a parallelogram with base 18 cm and height 5 cm, which is why we halve it.
```
Area = ½ × (7 + 11) × 5
     = ½ × 18 × 5
     = 9 × 5
     = 45 cm²
```
Multiply the cross-section by the length of the prism.
```
Volume = 45 × 20
       = 900 cm³
```
Convert to litres. 1,000 cm³ is 1 litre, so divide by 1,000.
```
900 ÷ 1,000 = 0.9 litres
```

The tank has a volume of 900 cm³, so it holds 0.9 litres (900 ml).

Common mistakes and how to help

Using the slanting side instead of the perpendicular height

In a triangle, parallelogram or trapezium the slanted side is labelled and the height is shown by a dotted line. Children multiply the numbers they can see most easily, so the slant gets used by mistake.

What helps: Always ask: does this length meet the base at a right angle? Cut a parallelogram out of paper, turn it into a rectangle and see that the slanted side disappears from the working.

Mixing up radius and diameter, or πr² and 2πr

A question gives the diameter but the formula uses the radius, and the two circle formulae look alike. Some pupils also square the diameter instead of the radius, or forget to square at all.

What helps: Make halving the diameter the first line of working. Remember that area is measured in square units, so its formula has a square in it, and circumference is a length, so it does not.

Converting area or volume units as if they were lengths

Because 1 m = 100 cm, many children assume 1 m² = 100 cm² and 1 m³ = 100 cm³. Areas and volumes scale in two or three directions at once.

What helps: Draw a square metre as 100 cm by 100 cm. It holds 10,000 cm². Then do the same with a cube: 1 m³ is 100 × 100 × 100 = 1,000,000 cm³.

Thinking a bigger perimeter always means a bigger area

Length and space feel like the same kind of quantity, so a longer outline seems to enclose more.

What helps: Compare a 6 by 6 square (perimeter 24, area 36) with a 2 by 10 rectangle (perimeter 24, area 20). Same fence, very different fields.

How to help at home

Measure a real room. Ask your child to measure a floor, a rug or a wall and work out its area, then the length of skirting board needed, which is a perimeter. If the space is L-shaped, they have a composite shape to split up.
Find π with a string. Wrap string around a tin or a plate, measure it, then measure straight across the diameter. Divide the circumference by the diameter and see how close the answer is to 3.14.
Compare pizza sizes. Is a 30 cm pizza (measured across) worth twice as much as a 20 cm one? Work out both areas with πr². The larger pizza has 2.25 times the area, not 1.5 times, because area depends on the radius squared.
Check a volume with water. Measure the inside of a straight-sided container, work out its volume in cm³, then fill it from a measuring jug. Since 1 ml is 1 cm³, the numbers should agree closely. Do this over a sink or tray.
Name the solids in the kitchen. Tins, boxes, a funnel and a ball are cylinders, cuboids, a cone and a sphere. Count the faces, edges and vertices of a cereal box together, then talk about how a tin or a ball is different because it has curved surfaces.

Builds on

Year 6: Units, area and volume
Year 6: Angles, circles and nets
Year 6: Four-quadrant coordinates, translation, reflection

Leads on to

Key Stage 4: Enlargement, similarity and vectors
Key Stage 4: Circles, solids and bearings
Key Stage 4: Pythagoras and trigonometry
Key Stage 4: Circle theorems and vector proof
Key Stage 4: Sine rule and cosine rule