Volume Guide — Twelve 3D Shapes, Built Step-by-Step

What volume measures

Volume is 3D space — how much a shape holds or occupies. Always in cubic units (m³, ft³, cm³).

Area is for 2D shapes; volume is for 3D. A 2×3×4 box contains 24 unit cubes, so its volume is 24.

Volume always has three linear dimensions multiplied together — explicitly (l·w·h for a prism) or via π·r² and a length (cylinder, sphere, cone). That's why the units are always cubic.

Surface area is a companion measurement — the total area of the outer skin. It has units of area (m², ft²), not volume. Gift wrapping uses surface area; filling with water uses volume.

See each formula built — the "How it's built" tab

Every shape in the calculator has a short animation that shows where its formula comes from in plain language.

Pick a shape and switch to the "How it's built" tab. After a short pause it auto-plays — captions narrate the construction one step at a time, slow enough to read.

Cylinders are stacks of identical circles. Cones and pyramids stack circles or rectangles that shrink toward the apex — and the animation puts them next to a matching cylinder/box with a "× 3 →" label so the one-third rule is visible, not just stated.

Capsules assemble in three pieces (cylinder body, top hemisphere, bottom hemisphere). Frustums show a complete cone first, then the tip is cut and lifted away. Spheres slice into pancake circles — calculus does the exact sum.

Cube — V = side³

All edges equal. Volume is one edge length cubed.

A cube is the simplest 3D shape: every edge has the same length, every face is an identical square. Stack "side" identical squares of area side² on top of each other, and you fill the cube — that's why the volume is side³.

Surface area is 6·side² (six identical square faces). The space diagonal — corner-to-opposite-corner through the body — is side·√3, by applying the Pythagorean theorem twice.

Rectangular prism (box) — V = l × w × h

Three perpendicular dimensions multiplied. The shape of every shipping box.

A rectangular prism is just a box. Multiply length × width × height — those are the three perpendicular edges, and their product is the volume. The unit cube is the special case where all three are equal.

Surface area is 2·(lw + lh + wh): the box has three pairs of identical opposite faces. Space diagonal = √(l² + w² + h²) — the corner-to-corner distance through the inside.

Triangular prism — V = ½ × b × th × L

A triangle dragged along a straight line. Roof trusses, Toblerone, ramps.

Compute the triangle's area first (½ × base × triangle-height), then multiply by the prism length L. The calculator assumes an isosceles triangular cross-section, which covers most real-world cases (gable roofs, wedge ramps).

Surface area = 2 × triangle area + (b + 2·slant)·L, where the slant side is √((b/2)² + th²) by the Pythagorean theorem on the isosceles triangle.

Cylinder — V = π × r² × h

Stack identical circles up to height h. Cans, pipes, tanks, drinking glasses.

A cylinder is a stack of identical circles. Each circle has area π·r²; multiply by how tall the stack is (h) and you have the volume. The "How it's built" tab shows the stacking visually.

Surface area = 2·π·r·(r + h) = two circular caps (each π·r²) plus the lateral side, which unrolls into a rectangle of width 2π·r (the circumference) and height h. Doubling the radius quadruples the volume — be careful with this when scaling tank designs.

Hollow cylinder (pipe) — V = π × (R² − r²) × L

A cylinder with a smaller cylinder removed from the middle. Pipes, tubes, washers, rings.

Take the outer cylinder volume (π·R²·L) and subtract the inner hole (π·r²·L). What's left is the wall — that's the pipe. The wall thickness is R − r.

Surface area is more involved: 2π·R·L (outer side) + 2π·r·L (inner side) + 2·π·(R² − r²) (the two ring-shaped end caps). Plumbing calculations for fluid flow use the inner radius (r), not R, because the fluid flows in the hole.

Capsule — V = π·r²·L + ⁴⁄₃·π·r³

A cylinder with a hemisphere stuck on each end. Propane tanks, pills, submarine hulls.

A capsule splits cleanly into a cylinder of length L plus two hemispheres of the same radius. Two hemispheres make a full sphere, so the volume is cylinder + sphere: π·r²·L + ⁴⁄₃·π·r³.

Note: the cylindrical length L excludes the rounded ends. The total length end-to-end is L + 2r. For very long thin capsules the cylinder term dominates; for nearly-spherical "stubby" capsules the sphere term dominates.

Sphere — V = ⁴⁄₃ × π × r³

The sphere is the one shape whose volume formula genuinely needs calculus to derive.

Slice the sphere into thin horizontal pancake circles. At height y the circle has radius √(R² − y²) — biggest at the equator, shrinking to zero at the poles. The volume is the sum of all these pancake areas times their thickness, which calculus computes exactly: ⁴⁄₃·π·r³.

Archimedes worked this out around 250 BC using a "lamp shade" comparison between a sphere and its enclosing cylinder, without modern calculus. Surface area is 4·π·r² — exactly four times the great-circle area πr². The slider above lets you check that ratio for any radius (it's always exactly 4).

Doubling the radius multiplies the volume by 8 (volume grows as r³) and the surface area by 4 (surface grows as r²). This is why a 2× bigger ball has 8× the inflation cost but only 4× the wrapping paper.

Hemisphere — V = ⅔ × π × r³

Exactly half a sphere. Bowls, domes, igloos, half-pipes.

A hemisphere is a sphere sliced exactly in half. Volume is half the sphere's: ½ × ⁴⁄₃·π·r³ = ⅔·π·r³. That part is intuitive.

Surface area, however, is NOT half the sphere's — it's 3·π·r², not 2·π·r². The curved half is 2·π·r² (half of the sphere's 4πr²), but you also need to count the flat circular disc at the cut, which adds π·r². So the total skin is 3·π·r².

Ellipsoid — V = ⁴⁄₃ × π × a × b × c

A sphere stretched independently along three axes. Eggs, footballs, planets.

Replace r³ in the sphere formula with the product of three semi-axes a·b·c. The sphere is the special case a = b = c = r. The "How it's built" tab shows the morph: starts as a sphere, stretches along a, then b, then c.

Surface area is famously messy — there is no closed-form formula in terms of elementary functions. The calculator uses the Knud Thompson approximation (with exponent p = 1.6075), which is accurate to about 1% for normal ellipsoids. For exact results you'd need elliptic integrals.

Cone — V = ⅓ × π × r² × h

Stack circles that shrink to a point. One-third the volume of the matching cylinder.

A cone is like a cylinder, but the circular cross-sections shrink linearly toward the apex. The average shrunk area works out to exactly one-third of the base, so the volume is ⅓ × cylinder volume = ⅓·π·r²·h. The "How it's built" tab shows three identical cones fitting inside the matching cylinder.

Surface area uses the slant height ℓ = √(r² + h²), where h is the perpendicular height. SA = π·r·(r + ℓ) — circular base plus the lateral cone surface (which unrolls into a sector of a disc of radius ℓ).

Frustum (truncated cone) — V = ⅓ × π × h × (r₁² + r₂² + r₁·r₂)

A cone with the tip chopped off. Buckets, lampshades, paper cups, dam walls.

The frustum formula is the difference of two cone volumes: the big cone (radius r₂, full height H by similar triangles) minus the small cone you sliced off the top (radius r₁, height H − h). After algebra, this collapses to the clean form ⅓·π·h·(r₁² + r₂² + r₁·r₂).

Two limiting cases worth remembering: when r₁ = r₂ the formula collapses to π·r²·h (a cylinder); when r₁ = 0 it collapses to ⅓·π·r²·h (a cone). The slider embed above lets you verify both. The slant height for surface area is ℓ = √(h² + (r₂ − r₁)²) — Pythagoras on the slope of the side.

Pyramid (rectangular base) — V = ⅓ × l × w × h

Stack rectangles that shrink to a point. One-third the volume of the matching box.

A pyramid is to a rectangular box what a cone is to a cylinder: the rectangular cross-sections shrink linearly toward the apex, so the volume is one-third of the matching box. Three identical pyramids fit inside the box exactly — the "How it's built" tab shows it visually.

Surface area is base + four triangular faces. The pyramid has TWO slant heights — one along the length-direction edges (√(h² + (w/2)²)) and one along the width-direction edges (√(h² + (l/2)²)) — because the base is rectangular, not square. The calculator computes both for you.

The one-third rule — cones, pyramids, and their prism counterparts

A cone has exactly ⅓ the volume of the cylinder with the same base and height. Same rule holds for pyramids vs prisms.

Given a cylinder of radius r and height h, its volume is π·r²·h. A cone with the same base and same height has volume ⅓·π·r²·h. Fill the cone with sand three times, pour each into the cylinder — it fills exactly.

Same for pyramids vs rectangular prisms. A pyramid with l·w base and height h has volume ⅓·l·w·h — a third of the l·w·h box with the same base and height.

Geometric proof sketch: three identical pyramids with square base s and height s can be arranged inside a cube of side s without overlap, filling it exactly. So each pyramid is ⅓ of s³. The slider above lets you confirm the 3:1 ratio for any radius and height.

Slant vs perpendicular height

Volume uses perpendicular height. Surface area of cones and pyramids needs slant height instead.

For a cone: perpendicular height h is apex → centre of base. Slant height ℓ is apex → any point on the base circle. They relate via ℓ = √(r² + h²) (Pythagorean theorem on the right triangle formed by h, r, and ℓ).

For a pyramid with rectangular base l × w: there are two slant heights — one along the length direction and one along the width direction. Each is also Pythagorean: √(h² + (w/2)²) and √(h² + (l/2)²).

Volume always uses perpendicular height. Surface area typically needs slant heights. The calculator asks for perpendicular height and derives slant values for you.

Five common volume mistakes

Volume errors usually boil down to unit confusion, diameter/radius swaps, or missing the ⅓.

1. Wrong units. Volume is cubic — m³, not m². A 5m × 5m × 5m room has volume 125 m³.

2. Diameter vs radius. π·r² uses radius. If you were given diameter 10, use r = 5 (not 10).

3. Slant in place of perpendicular height. Volume of a cone uses perpendicular h, not slant ℓ.

4. Forgetting the ⅓. Cone/pyramid volume is one-third of the prism/cylinder equivalent. Triple-check.

5. Confusing lateral with total surface area. Lateral = side only (no caps). Total = lateral + base(s). The calculator reports total SA; lateral is in the extras.