Area, perimeter, volume — three ideas that confuse almost every pupil in Years 5 and 6. In this guide we explain the differences, learn all the important formulae and see how they work in real life — from working out the size of a room to the volume of the fish tank in the living room.
The difference between perimeter, area and volume
Before calculating anything — it is important to understand what each idea means. The easiest way to remember: they are three different worlds — 1D, 2D and 3D.
- Perimeter (1D) — the length of the outer boundary of a shape. Like a fence around a garden. Measured in cm, m and so on.
- Area (2D) — the amount of surface inside the shape. Like the carpet covering the floor of a room. Measured in cm², m², hectares.
- Volume (3D) — the amount of space a 3D object takes up. Like the amount of water that fits in a fish tank. Measured in cm³, dm³ (litres), m³.
A concrete example: a typical two-bedroom flat in England is about 70 m² — that is the area. If you want to lay new flooring, you need to know the area. If you want to put a fence around the garden, you need the perimeter.
Area of a rectangle and a square — the basic formulae
The rectangle is the most common shape in our lives — rooms, doors, windows, books. The formula is simple:
| Topic | Formula | Units |
|---|---|---|
| Area of a rectangle | length × width (a × b) | cm² / m² / km² |
| Area of a square | side × side (a²) | cm² / m² |
| Perimeter of a rectangle | 2 × (length + width) | cm / m |
| Perimeter of a square | 4 × side | cm / m |
Example: a bedroom 4 m long and 3 m wide has an area of 4 × 3 = 12 m². A square floor with a side of 5 m is 5² = 25 m².
Units of area worth knowing: 1 m² = 10,000 cm². A hectare (used for land in the UK) = 10,000 m². A square kilometre = 1,000,000 m². You may also meet the acre on older land measurements — about 4,047 m².
Area of a triangle — why do we divide by 2?
The formula for the area of a triangle is: ½ × base × height. But why exactly a half? That is an excellent question — and it has a lovely visual answer.
Example: a triangle with a base of 8 cm and a height of 5 cm has an area of ½ × 8 × 5 = 20 cm².
Note: the height is the line that drops perpendicular (at a right angle) from the vertex to the base — not necessarily a side of the triangle! In a right-angled triangle, the two shorter sides are the base and the height.
Area of a trapezium and a parallelogram
A parallelogram is like a rectangle that has been pushed sideways. Its height is the perpendicular distance between the two parallel sides — not the slanted side.
| Shape | Formula | Explanation |
|---|---|---|
| Parallelogram | base × height (b × h) | height = perpendicular distance between the two parallel sides |
| Trapezium | ½ × (long base + short base) × height | the average of the two parallel sides times the height |
A trapezium example: a field shaped like a trapezium with a long base of 12 m, a short base of 8 m and a height of 5 m. Its area: ½ × (12 + 8) × 5 = ½ × 20 × 5 = 50 m².
Tip: the trapezium is a generalisation of the rectangle! If the two parallel sides are equal (b₁ = b₂), the formula becomes b × h — exactly like a rectangle.
Area of a circle — an introduction for Year 6
In Year 6 pupils learn to name the parts of a circle and meet the idea of a diameter being twice the radius; the wonderful number π (pi) — about 3.14159… — is formally introduced in Year 7, but curious children often want it early. It appears in every calculation involving a circle. The formula for the area of a circle is: A = π × r², where r is the radius (half the diameter).
| Topic | Formula | Units |
|---|---|---|
| Area of a circle | π × r² | cm² / m² |
| Circumference of a circle (the perimeter) | 2 × π × r | cm / m |
| Radius and diameter | r = d ÷ 2 | — |
Example: a round pizza with a diameter of 30 cm — the radius is 15 cm. Its area: π × 15² = 3.14 × 225 ≈ 706 cm². If you ever wondered why a large pizza is so much bigger than a medium — now you know: the radius squared grows fast!
Volume of a cube and a cuboid
On to the three-dimensional world! Volume is the amount of space an object takes up — and it is very relevant to life: how much water fits in a bottle, how many items fit in a box, what size fish tank we need.
| Solid | Formula | Units |
|---|---|---|
| Cube | a × a × a = a³ | cm³ / m³ |
| Cuboid | length × width × height (a × b × c) | cm³ / m³ |
| Link to the base | area of the base × height | the general rule for prisms |
A real-life example — a fish tank: a tank 60 cm long, 30 cm wide and 40 cm high. Its volume: 60 × 30 × 40 = 72,000 cm³ = 72 litres. That is exactly what you need to know to buy the right filter pump!
A cube with a side of 5 cm: its volume = 5³ = 5 × 5 × 5 = 125 cm³. This explains why a bigger ice cube cools a drink so much more — the volume grows with the cube of the side!
Converting units — the most common mistakes
Converting units of area and volume is confusing because you do not multiply by 10 or 100 — you multiply by the square or the cube of the conversion factor. This is where most pupils go wrong!
| Conversion | Rule | Example |
|---|---|---|
| m² → cm² | × 10,000 (because 100² = 10,000) | 3 m² = 30,000 cm² |
| cm² → m² | ÷ 10,000 | 50,000 cm² = 5 m² |
| hectare → m² | × 10,000 | 2 hectares = 20,000 m² |
| m³ → cm³ | × 1,000,000 (because 100³) | 0.5 m³ = 500,000 cm³ |
| litre → cm³ (dm³) | 1 litre = 1,000 cm³ | 5 litres = 5,000 cm³ |
| 1 dm³ → litre | 1 dm³ = 1 litre | trick: dm³ = litre! |
Real-life examples — calculating like an engineer
Geometry does not stay in the classroom — it appears everywhere. Here are some genuine everyday examples:
Example 1: painting a room
A child's bedroom 4 m long, 3 m wide and 2.5 m high. How much paint is needed? Work out the area of the four walls: 2 × (4 × 2.5) + 2 × (3 × 2.5) = 20 + 15 = 35 m². Minus a window (1.5 m²) and a door (2 m²): 35 − 3.5 = 31.5 m². A tin of emulsion covers about 10 m² per coat → you need about 3.2 tins for one coat.
Example 2: a home fish tank
A rectangular tank: 80 cm long, 35 cm wide, 45 cm high. Volume: 80 × 35 × 45 = 126,000 cm³ = 126 litres. If it is filled only to 80% of its height (as is usual): 126 × 0.8 = 100.8 litres — so roughly 100 litres of water.
Example 3: a two-bedroom flat
A flat of 70 m² (typical for a two-bedroom flat). Split it up: living room 20 m², kitchen 10 m², 2 bedrooms × 12 m² = 24 m², bathroom 5 m², hall and cupboards 11 m² — total 70 m². How many 60 × 60 cm tiles (= 0.36 m²) for the whole flat? 70 ÷ 0.36 ≈ 195 tiles. Plus about 10% for cuts and breakages → 214 tiles.
Example 4: a trapezium-shaped garden
A garden with a long base of 15 m, a short base of 9 m and a height of 8 m. Its area: ½ × (15 + 9) × 8 = ½ × 24 × 8 = 96 m². If a bag of grass seed covers 25 m² — you need 96 ÷ 25 = 3.84 → 4 bags.
Formula summary — keep this for before the test
| Shape / solid | Area / volume formula | Units |
|---|---|---|
| Rectangle | a × b | cm² / m² |
| Square | a² | cm² / m² |
| Triangle | ½ × b × h | cm² / m² |
| Parallelogram | b × h | cm² / m² |
| Trapezium | ½ × (b₁ + b₂) × h | cm² / m² |
| Circle (Year 6/7) | π × r² | cm² / m² |
| Cube | a³ | cm³ / m³ |
| Cuboid | a × b × c | cm³ / m³ |
Want to practise? Try our built-in calculator — enter the measurements and get the area and volume at a click. Perfect for checking answers before the test!
Frequently asked questions
What is the difference between area and perimeter?
Perimeter is the length of the outer boundary of a shape (how many metres of fence you need around a garden). Area is the amount of surface inside the shape (how many m² of grass are in the garden). Perimeter is measured in units of length (m, cm) and area in square units (m², cm²).
Why is the area of a triangle half the base times the height?
Because every triangle is exactly half a parallelogram. If you stick two identical triangles together (one upside down), you get a complete parallelogram. So the area of the triangle = ½ × the area of the parallelogram = ½ × base × height.
What is the height of a triangle — and how do you find it?
The height of a triangle is the perpendicular line (at a right angle) that drops from one vertex to the side opposite it (the base). In test questions the height is usually given. In a right-angled triangle, the two perpendicular sides are the base and the height.
How many cm² are there in 1 m²?
10,000 cm². The reason: 1 m = 100 cm, so 1 m² = 100 cm × 100 cm = 10,000 cm². A common mistake is to think you multiply by 100 — but with units of area you multiply by the square of the ratio.
What is the difference between dm³ and a litre?
There is no difference! 1 dm³ (cubic decimetre) = exactly 1 litre. That is a mathematical definition. So the volume of a box that is 10 cm × 10 cm × 10 cm = 1,000 cm³ = 1 dm³ = 1 litre.
What is the formula for the area of a circle and when is it taught?
Area of a circle = π × r² where r is the radius. It is formally taught in Year 7 (Year 6 names the parts of a circle and links radius and diameter). π ≈ 3.14. For example a circle with a radius of 7 cm: its area ≈ 3.14 × 49 ≈ 153.9 cm². It is relevant to working out the areas of round pitches, pools, pizzas and more.
How do you work out the volume of a fish tank?
A fish tank is a cuboid — so its volume = length × width × height. The result in centimetres gives cm³, and every 1,000 cm³ = one litre. For example: 50 × 30 × 40 cm = 60,000 cm³ = 60 litres.
What is a hectare and how many m² are in it?
A hectare is a metric unit of area used for land. 1 hectare = 10,000 m² — a square 100 m by 100 m. It is used in the UK for fields, farms and parks. Example: a field of 3 hectares is 30,000 m². You may also see the older acre — about 4,047 m², so roughly 2.5 acres to a hectare.
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