This volume calculator finds the volume of six common solids — cube, rectangular prism (box), cylinder, sphere, cone, and square pyramid — along with the surface area of each. Pick a shape, enter its dimensions, and both values are computed instantly in cubic and square units.
Volume measures the three-dimensional space a solid occupies, so the answer is always in cubic units: enter centimeters and you get cm³, enter feet and you get ft³. Only the dimensions your chosen shape actually needs are used — a sphere needs just its radius, while a box needs length, width, and height.
Volume Formulas for Every Shape
Each solid has its own volume formula:
- Cube: V = s³ (side cubed)
- Rectangular prism (box): V = l × w × h
- Cylinder: V = πr²h (base circle area × height)
- Sphere: V = (4/3)πr³
- Cone: V = (1/3)πr²h
- Square pyramid: V = (1/3)a²h (a = base side)
Notice the pattern: prisms and cylinders are "base area × height," while cones and pyramids hold exactly one third of the matching prism or cylinder with the same base and height. The cylinder volume formula πr²h is the one people search for most — square the radius, multiply by pi, then multiply by the height.
Surface Area Formulas
Surface area is the total area of every face or curved surface, measured in square units:
- Cube: SA = 6s²
- Rectangular prism: SA = 2(lw + lh + wh)
- Cylinder: SA = 2πr(r + h) — two circular ends plus the wrapped side
- Sphere: SA = 4πr²
- Cone: SA = πr(r + l), where the slant height l = √(r² + h²)
- Square pyramid: SA = a² + 2a√(h² + (a/2)²)
Surface area matters when you are painting, wrapping, or insulating an object, while volume matters when you are filling it. A tank with r = 1 m and h = 2 m holds πr²h ≈ 6.28 m³ of water but needs 2π(1)(3) ≈ 18.85 m² of paint.
Worked Examples: Cylinder, Cone, and Sphere
Cylinder with radius 3 and height 8:
- V = πr²h = π × 3² × 8 = π × 9 × 8 = 72π ≈ 226.19 cubic units.
Cone with the same radius 3 and height 8:
- V = (1/3)πr²h = 72π ÷ 3 = 24π ≈ 75.40 cubic units — exactly one third of the cylinder.
Sphere with radius 6:
- V = (4/3)πr³ = (4/3) × π × 216 = 288π ≈ 904.78 cubic units.
And a quick box check: a 5 × 4 × 10 rectangular prism has V = 5 × 4 × 10 = 200 cubic units and surface area 2(20 + 50 + 40) = 220 square units.
Frequently Asked Questions
What is the formula for the volume of a cylinder?
The cylinder volume formula is V = πr²h: square the radius, multiply by π (about 3.14159), then multiply by the height. It works because a cylinder is a circle (area πr²) extended straight upward through height h. A can with radius 4 cm and height 10 cm holds π × 16 × 10 ≈ 502.65 cm³.
How do you find the volume of a sphere?
Use V = (4/3)πr³. Cube the radius, multiply by π, then multiply by 4/3. A sphere with radius 3 has volume (4/3) × π × 27 = 36π ≈ 113.10 cubic units. If you are given the diameter, halve it first to get the radius before cubing.
Why is a cone one third the volume of a cylinder?
A cone with the same base radius and height as a cylinder holds exactly one third of its volume — that is why the formulas are πr²h and (1/3)πr²h. You can demonstrate it physically: filling a cone with water and pouring it into the matching cylinder takes exactly three pours. The same 1/3 relationship holds between pyramids and prisms.
What units is volume measured in?
Volume is always in cubic units: cubic centimeters (cm³), cubic meters (m³), cubic inches, or cubic feet, depending on what unit you measured the dimensions in. For liquids, 1,000 cm³ equals 1 liter, and 1 m³ equals 1,000 liters. Always use the same unit for every dimension before multiplying.
What is the difference between volume and surface area?
Volume measures the space inside a solid (cubic units) — how much it can hold. Surface area measures the total area of its outside faces (square units) — how much material covers it. A cube with 2 cm sides has a volume of 8 cm³ but a surface area of 24 cm², and the two quantities are not interchangeable.