Volume Calculator

Volume formulas — jump to a shape

Cube · Rectangular prism · Cylinder · Sphere · Cone · Pyramid

Volume of a cube

V = side³. Surface area = 6·side². Space diagonal = side·√3. All six faces identical squares.

Example: a 4 m cube has volume 64 m³, surface area 96 m², and a space diagonal of 4√3 ≈ 6.93 m.

Volume of a rectangular prism (box)

V = length × width × height. Surface area = 2·(lw + lh + wh). Space diagonal = √(l² + w² + h²).

Example: a 6 × 4 × 3 box has volume 72, surface area 108, and a space diagonal of √61 ≈ 7.81.

Volume of a cylinder

V = π·r²·h — base area times height. Surface area = 2·π·r·(r + h) (two circular caps plus the lateral rectangle when unrolled).

Example: a cylinder with r = 3, h = 8 has volume 72π ≈ 226.2 and surface area 66π ≈ 207.3.

Volume of a sphere

V = ⁴⁄₃ · π · r³. Surface area = 4·π·r² — exactly four times the great-circle area, a classical result by Archimedes.

Example: a sphere with r = 5 has volume ⁴⁄₃·π·125 ≈ 523.6 and surface area 4π·25 ≈ 314.2.

Volume of a cone

V = ⅓ · π · r² · h — one-third the matching cylinder. Surface area = π·r·(r + ℓ), where ℓ = √(r² + h²) is the slant height (distinct from the perpendicular height h used in volume).

Example: a cone with r = 3, h = 4 has slant 5 (the 3-4-5 right triangle), volume 12π ≈ 37.7, surface area 24π ≈ 75.4.

Volume of a pyramid (rectangular base)

V = ⅓ · length · width · height — one-third the matching box. Surface area = base + four triangular faces (each face needs its own slant height, computed via Pythagoras).

Example: a pyramid with a 6 × 6 base and height 4 has volume 48 and slant heights of 5 along each base edge.

Why prisms use base × height and pyramids / cones use ⅓ × base × height

A cone or pyramid always has exactly one-third the volume of the prism or cylinder that shares its base and height. Geometrically: three identical pyramids can be assembled into a cube of the same base and height. That's why the formulas share the same base × height skeleton — prisms / cylinders get the whole thing; cones / pyramids get a third.

Slant height vs. perpendicular height

For cones and pyramids, “height” in the volume formula is always the perpendicular distance from the apex to the base — not the slant. The calculator asks for the perpendicular height and computes the slant for you, using the Pythagorean theorem: ℓ² = h² + r² for a cone.

Common mistakes to avoid

• Cubing units the wrong way. Volume is in cubic units (m³, ft³, cm³). A 2m cube has volume 8 m³, not 8 m².
• Using the diameter in π·r². The formula uses radius, not diameter. Halve the diameter first.
• Confusing slant height with perpendicular height. Volume always uses perpendicular height. Surface area for cones/pyramids needs the slant height as a separate ingredient.
• Forgetting the ⅓ on cones and pyramids. Very easy to accidentally use the cylinder / prism formula and triple the answer.