Educational explanation, not professional design advice. Last reviewed 11 November 2026. Disclaimer
In plain English
Area is the amount of flat surface a shape covers, such as the floor of a room or the face of a plate. Volume is the amount of space a solid object takes up, or the amount a container can hold, such as the concrete in a slab or the water in a tank.
Most real objects can be broken down into a few simple shapes — rectangles, triangles, circles, boxes and cylinders. Calculate each part and add (or subtract) them to get the total.
Technical definition
Area is measured in square units (m², mm², ft²) and volume in cubic units (m³, mm³, ft³). Capacity is often given in litres: 1 L = 1 dm³ = 0.001 m³. When every length of a shape is multiplied by a factor k, its area is multiplied by k² and its volume by k³. Doubling all the dimensions of a tank therefore holds eight times as much.
How it works and where it is used
| Shape | Area | Shape | Volume |
|---|---|---|---|
| Rectangle | L × W | Box (cuboid) | L × W × H |
| Triangle (base, height) | ½ × b × h | Cylinder | π r² h |
| Triangle (three sides) | √[s(s − a)(s − b)(s − c)] | Sphere | 4⁄3 π r³ |
| Circle | π r² = π d² ÷ 4 | Cone | ⅓ π r² h |
| Trapezoid | (a + b) ÷ 2 × h | Prism (any cross-section) | cross-section area × length |
For the triangle with three sides (Heron’s formula), s is half the perimeter. This is very useful on site and for land plots, where the three sides can be measured with a tape without finding a right angle. For a composite shape, split it into simple shapes, calculate each, and add them; for a shape with a hole or cut-out, subtract the hole.
Formula
Acircle = π r² Atriangle = √[s(s − a)(s − b)(s − c)], s = (a + b + c) ÷ 2
Vcylinder = π r² h Vsphere = 4⁄3 π r³ Vcone = ⅓ π r² h
r = radius (half the diameter), h = height, L, W, H = length, width, height, a, b, c = triangle sides. Use the same length unit for every dimension.
Worked example
Example 1 — an L-shaped floor. Split it into two rectangles: 6 m × 4 m = 24 m² and 3 m × 2.5 m = 7.5 m². Total = 31.5 m².
Example 2 — a concrete slab. A slab 5 m × 4 m and 150 mm thick: V = 5 × 4 × 0.150 = 3.0 m³. Convert the thickness to metres first — 5 × 4 × 150 would give 3000, which is wrong by a factor of 1000.
Example 3 — a water tank. A cylindrical tank with an inside diameter of 1.2 m and a height of 1.5 m: V = π × 0.6² × 1.5 = 1.696 m³ = 1696 L, weighing about 1.7 tonnes when full.
Example 4 — a ball float. A sphere of 2 m diameter: V = 4⁄3 × π × 1³ = 4.19 m³.
In the workshop and in practice
Areas are used for flooring, tiling, painting, roofing, plate cutting and land measurement; volumes for concrete, earthworks, tank capacity, packaging and material weight (volume × density = mass). For ordering materials, add an allowance for waste, overlaps and spillage, and check whether the supplier sells by area, by volume or by weight.
Common mistakes and misconceptions
- Mixing units inside one formula. Metres and millimetres together give results that are off by 1000 or more.
- Using the diameter where the radius is needed. π d² instead of π r² gives four times the correct area.
- Using outside dimensions for capacity. A tank’s wall thickness and fittings reduce the volume it can hold.
- Measuring a sloping surface on the plan. A pitched roof or a slope has more area than its plan view.
Related calculators
- Area Calculator — Area and perimeter of rectangles, triangles, trapezoids and ellipses.
- Volume & Surface Area Calculator — Volume, capacity in litres and surface area of boxes, cylinders, spheres and cones.
- Circle Calculator — Area, circumference, radius and diameter — enter any one of them.
Related topics
- SI Units and Safe Unit Conversion — The international system of units, and a reliable way to convert between units without mistakes.
References
- ISO 80000-3: Quantities and units — Space and time (area and volume)
- Spiegel and Liu, Mathematical Handbook of Formulas and Tables