Volume Calculator — Cube, Sphere, Cylinder & More

3D Shape Volume Guide

What do I need to know about 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.

What's the key thing to understand about 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.

What do I need to know about 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 heat loss, since smaller animals lose heat faster relative to their body mass and need proportionally more energy to stay warm.

What's the key thing to understand about 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). Prisms with a consistent cross-section, like extruded aluminium window frames, are manufactured this way specifically because the volume and material cost scale predictably with length.