Maths📅 24 March 2025⏱ 6 min read
How to Calculate the Area and Volume of Any Common Shape
Area and volume formulas for every shape you're likely to need — triangles, circles, cylinders, cones, spheres — with worked examples and the reasoning behind the formulas.
JW
James WhitfieldPersonal Finance & Maths WriterJames has written about personal finance, health metrics, and everyday mathematics for over six years. He holds a BSc in Mathematics from the University of Leeds.
Whether you're tiling a floor, filling a pool, or just sitting a maths exam, area and volume calculations come up constantly. Here are all the key formulas with worked examples.
2D Area Formulas
ShapeFormulaVariables
SquareA = s²s = side length
RectangleA = l × wl = length, w = width
TriangleA = ½ × b × hb = base, h = height
CircleA = π × r²r = radius
TrapeziumA = ½ × (a + b) × ha,b = parallel sides
ParallelogramA = b × hb = base, h = perpendicular height
EllipseA = π × a × ba,b = semi-axes
Worked Area Examples
Triangle: Base 8m, height 5m → A = ½ × 8 × 5 = 20m²
Circle: Radius 7cm → A = π × 7² = π × 49 ≈ 153.9cm²
Trapezium: Parallel sides 6m and 10m, height 4m → A = ½ × (6 + 10) × 4 = 32m²
Circumference and Perimeter
Circle circumference = 2 × π × r = π × d
Rectangle perimeter = 2 × (l + w)
Triangle perimeter = a + b + c (sum of all sides)
3D Volume Formulas
ShapeFormulaVariables
CubeV = s³s = side length
Cuboid (box)V = l × w × hl, w, h = dimensions
CylinderV = π × r² × hr = radius, h = height
SphereV = 4/3 × π × r³r = radius
ConeV = 1/3 × π × r² × hr = base radius, h = height
PyramidV = 1/3 × base area × hh = perpendicular height
Worked Volume Examples
Cylinder (swimming pool): Radius 4m, depth 1.5m → V = π × 4² × 1.5 = π × 16 × 1.5 ≈ 75.4m³ = 75,400 litres
Sphere (football): Radius 11cm → V = 4/3 × π × 11³ = 4/3 × π × 1331 ≈ 5,575cm³
Cone: Base radius 3m, height 5m → V = 1/3 × π × 3² × 5 = 1/3 × π × 45 ≈ 47.1m³
Surface Area
Cube: SA = 6s²
Cuboid: SA = 2(lw + lh + wh)
Cylinder: SA = 2πr² + 2πrh
Sphere: SA = 4πr²
Cone: SA = πr² + πrl (l = slant height = √(r² + h²))
Practical Applications
How many litres in a cylindrical tank? Calculate volume in m³ then multiply by 1,000 (1m³ = 1,000 litres).
How much soil for a raised garden bed? It's a cuboid — length × width × depth in metres gives volume in m³. Multiply by 1,000 for litres, or divide by 0.025 to get the number of standard 25-litre bags.
Concrete for a circular post hole? Cylinder formula — π × radius² × depth. Most pre-mix bags cover about 0.01m³ each.