Volume Calculator — Cube, Sphere, Cylinder & More

3D Shape Volume Guide

Volume Formulas

Cube: V = a³. Cuboid: V = l × w × h. Cylinder: V = πr²h. Sphere: V = (4/3)πr³. Hemisphere: V = (2/3)πr³. Cone: V = (1/3)πr²h. Square pyramid: V = (1/3) × base area × h = (1/3)a²h. Triangular prism: V = (1/2) × base × height × length = (1/2)bhl. Torus: V = 2π²Rr² (R = major radius from centre to tube centre, r = tube radius). The factor of 1/3 appears in both cone and pyramid — they share the principle that a cone/pyramid is 1/3 of the volume of the cylinder/prism with the same base and height.

Surface Area Formulas

Cube: SA = 6a². Cuboid: SA = 2(lw + lh + wh). Cylinder: SA = 2πr² + 2πrh = 2πr(r+h). Sphere: SA = 4πr². Hemisphere: SA = 3πr² (2πr² curved + πr² flat). Cone: SA = πrl + πr² = πr(l+r) where l = √(r²+h²) is the slant height. Square pyramid: SA = a² + 2al where l = slant height of triangular face = √(h²+(a/2)²). The slant height is always measured along the face, not the vertical height — a common error.

Units and Conversions

Volume units: 1 m³ = 1,000,000 cm³ = 1,000 L. 1 cm³ = 1 mL. 1 L = 1,000 cm³. 1 m³ = 1,000 L. Surface area units are always the square of the linear unit. When dimensions are given in cm: volume in cm³, surface area in cm². Converting: if sides change by a factor of k, volume changes by k³ and surface area by k². Example: doubling all sides of a sphere: radius × 2. Surface area × 4. Volume × 8. This 'scaling law' explains why large animals have lower surface-area-to-volume ratios — relevant to he

Real-World Applications

Cylinders: water tanks, pipes, silos, cans. A cylinder with r=h is the optimal shape for a can (minimises material per unit volume). Spheres: storage tanks for liquids and gases (lowest surface area per volume). Planets and droplets are spherical for the same reason. Cones: ice cream cones, funnels, dunce caps (sorry). Pyramids: structurally very stable — the Great Pyramid of Giza has withstood 4,500 years. Prisms: structural beams (I-beams have cross-section designed for bending resistance). Pr