Area and volume are the two measurements behind most practical projects. How much paint, carpet or turf do you need? That is area. How much concrete, soil, water or gravel? That is volume. The formulas are some of the oldest in mathematics, and most people learned them at school — then forgot which one goes with which shape. This guide brings them back together, with worked examples, the unit conversions that trip people up, and a real construction example from start to finish.
Area and volume: what is the difference?
Area measures a flat surface — how much of it there is. It is always a length multiplied by a length, so its units are squared: square metres (m²), square feet (ft²), square centimetres (cm²).
Volume measures the space inside a solid shape — how much it holds. It is a length times a length times a length, so its units are cubed: cubic metres (m³), cubic feet (ft³), litres.
Area formulas for common shapes
| Shape | Formula | Example | Area |
|---|---|---|---|
| Square | side² | side 5 | 25 |
| Rectangle | length × width | 5 × 4 | 20 |
| Triangle | ½ × base × height | base 6, height 4 | 12 |
| Circle | π × radius² | radius 3 | 28.27 |
| Trapezoid | ½ × (a + b) × height | sides 6 and 10, height 4 | 32 |
| Parallelogram | base × height | base 8, height 3 | 24 |
Two details catch people out. For a triangle, parallelogram or trapezoid, the height is the perpendicular distance — measured at right angles to the base — not the length of the sloping side. And for a circle, the formula uses the radius, which is half the diameter. Using the diameter by mistake gives an answer four times too big.
The area calculator covers all of these shapes, and the circle calculator works from a radius, diameter or circumference.
Where the formulas come from
It is easier to remember a formula when you can see why it works. A rectangle 5 units long and 4 wide can be filled with 5 × 4 = 20 unit squares, which is all “length × width” means. Cut a rectangle along its diagonal and you get two identical triangles, so a triangle is half of base × height. A parallelogram can be turned into a rectangle by slicing a triangle off one end and moving it to the other, which is why its area is also base × height. And a trapezoid is the average of its two parallel sides, multiplied by the height.
The circle is the one shape that cannot be built from squares. Its area formula, π × r², says that a circle holds just over three times the area of a square drawn on its radius. The number π — about 3.14159 — is the ratio of any circle’s circumference to its diameter, and it appears in every formula involving circles, cylinders, cones and spheres.
Step-by-step example: an L-shaped room
Real rooms are rarely perfect rectangles. The trick is to split an irregular shape into simple ones, work out each area, and add them up.
- Divide the L-shaped floor into two rectangles: one 6 m × 4 m, and one 3 m × 2 m.
- Area of the first: 6 × 4 = 24 m².
- Area of the second: 3 × 2 = 6 m².
- Add them: 24 + 6 = 30 m².
The floor area is 30 m².
Any shape made of straight edges can be treated this way. For a shape with a notch cut out of it, work out the whole rectangle and subtract the notch instead — it is often quicker.
Volume formulas for common solids
| Solid | Formula | Example | Volume |
|---|---|---|---|
| Cube | side³ | side 2 | 8 |
| Box (cuboid) | length × width × height | 2 × 1.5 × 1 | 3 |
| Cylinder | π × r² × height | r 0.5, height 2 | 1.571 |
| Cone | ⅓ × π × r² × height | r 1, height 3 | 3.142 |
| Sphere | ⁴⁄₃ × π × r³ | r 1 | 4.189 |
| Pyramid | ⅓ × base area × height | base 3 × 3, height 4 | 12 |
Notice the pattern. A prism or cylinder — any solid with the same cross-section all the way up — is simply its base area multiplied by its height. A cone or pyramid, which tapers to a point, holds exactly one third of the matching prism or cylinder. The volume calculator handles all of these, and the surface area calculator works out the outside area of the same solids, which is what you need for painting or wrapping them.
Units: the part that goes wrong most often
Converting lengths is easy — there are 100 centimetres in a metre. Converting areas and volumes is where mistakes creep in, because the conversion factor is squared or cubed too:
- 1 m = 100 cm, so 1 m² = 100 × 100 = 10,000 cm², and 1 m³ = 100 × 100 × 100 = 1,000,000 cm³.
- 1 m³ = 1,000 litres, and 1 litre = 1,000 cm³.
- 1 ft = 12 in, so 1 ft² = 144 in² and 1 ft³ = 1,728 in³.
- 1 yd = 3 ft, so 1 yd³ = 27 ft³ — the unit concrete and soil are often sold in, in the US.
- 1 m² ≈ 10.764 ft², and 1 m³ ≈ 35.315 ft³.
The safest habit is to convert every measurement to the same unit before multiplying. A slab measured as 4 m by 3 m by 10 cm should become 4 × 3 × 0.1, not 4 × 3 × 10.
A real example: concrete for a patio slab
Suppose you are laying a concrete slab 4 m long, 3 m wide and 100 mm thick.
- Convert the thickness to metres: 100 mm = 0.1 m.
- Volume: 4 × 3 × 0.1 = 1.2 m³.
- Add a waste allowance of 10% for uneven ground, spillage and formwork: 1.2 × 1.10 = 1.32 m³.
Order about 1.32 m³ of concrete.
In imperial units, a 12 ft × 10 ft slab 4 inches thick is 12 × 10 × (4 ÷ 12) = 40 ft³, which is 40 ÷ 27 = 1.48 yd³ before waste. The concrete slab calculator does these conversions, adds the waste and also counts bags if you are mixing it yourself.
Estimating slopes, curves and odd shapes
Not everything divides neatly into rectangles. For a curved flower bed or an irregular pond, a practical approach is to measure its length, then take several widths at even intervals along it, average them, and multiply the average width by the length. The more widths you take, the closer the estimate. For depth — a pond, a trench, a raised bed that slopes — do the same thing with depths: average several measurements and multiply by the surface area to get the volume.
For a sloping site, remember that a slab or bed is only as thick as its thinnest point if you measure there. Measure the depth at each corner and in the middle, and use the average, or the slab will come up short of concrete on the deep side.
Where area and volume show up in everyday life
- Decorating — wall area for paint and wallpaper, floor area for carpet, tiles or laminate.
- Gardening — lawn area for seed or turf, bed volume for soil, compost and mulch.
- Building — slab and footing volume for concrete, wall area for bricks and blocks.
- Home and kitchen — the volume of a fish tank, a water butt or a cake tin.
- Moving and storage — the volume of boxes and the space in a van or storage unit.
Common mistakes
- Mixing units in one calculation. Convert everything to metres, or everything to feet, first.
- Converting areas and volumes with the length factor. 1 m² is 10,000 cm², not 100.
- Using the diameter as the radius. Halve the diameter before squaring.
- Using the slanted side as the height. Height is always measured at right angles to the base.
- Forgetting waste. Materials are cut, spilled and wasted; allow 5–15% depending on the job.
Frequently asked questions
How do I find the area of an irregular shape?
Split it into rectangles, triangles and circles, work out each area, then add them together. For a shape with a piece missing, work out the full shape and subtract the missing part.
How many litres are in a cubic metre?
Exactly 1,000. A litre is the volume of a cube 10 cm on each side, and a cubic metre holds 10 × 10 × 10 of those cubes.
Why does a cone hold a third of a cylinder?
Because it tapers evenly to a point. The result can be proven with calculus, or seen by filling a cone three times to fill a cylinder of the same base and height.
Final thoughts
Almost every area or volume problem comes down to the same three steps: break the shape into simple pieces, put every measurement into the same unit, then apply the right formula and add a sensible allowance for waste. Keep the table above to hand, or let the area and volume calculators do the arithmetic and show you the working.