Volume Calculator
Volume and surface area for 12 common 3D shapes — with an animated 'How it's built' view that shows where each formula comes from in plain language.
Shape & Dimensions
Answer
What do these terms mean?
- Volume
- How much space the shape holds, in cubic units (e.g. cm³, m³).
- Surface area
- The total area of the outer skin — like how much wrapping paper you'd need.
- Lateral area
- The area of the side surface only — does NOT include the top or bottom caps.
- Base area
- The area of the flat base (a circle for cylinders/cones, a rectangle for boxes/pyramids).
Show your work
- A cylinder is a stack of identical circles. Its volume is the area of one circle times how tall the stack is.V = π × r² × h→ r = 3, h = 8
- First, find the area of one circular base.π × 3² = π × 9= 28.274334
- Surface area = two circular caps + the lateral side (which unrolls into a rectangle of width 2π·r and height h).2 × π × 3 × (3 + 8)= 207.345115
Volume Calculator — every common 3D shape, with an animated build view
A volume calculator computes how much space a 3D shape holds (cubic units) and its outer surface area (square units). This one supports twelve common shapes — cube, rectangular prism, triangular prism, cylinder, hollow cylinder (pipe), capsule, sphere, hemisphere, ellipsoid, cone, frustum (truncated cone), and rectangular pyramid.
Beyond the answer, every shape includes a “How it's built” view: a short animation that shows where each formula comes from — a cylinder as stacked circles, a cone as shrinking circles, a capsule as a cylinder with a hemisphere on each end. Captions narrate the math in plain language. The aim is to help you understand the formula, not just memorise it.
The “How it's built” view — bridging plane shapes to 3D volume
Pick any shape, switch to the How it's built tab, and press Play (or wait — it auto-plays after a short pause). Each animation walks you through the construction and explains the formula one step at a time:
- Cylinder: flat circles stack up to height
h. Volume = π·r² × h. - Cone & Pyramid: shrinking circles / rectangles taper to a point. Three of them fill the matching cylinder / box exactly — that's where the ⅓ comes from.
- Capsule: cylinder body, then a hemisphere drops on top, then another rises from below. Volume = cylinder + full sphere.
- Frustum: a complete cone is drawn first, the tip is sliced and lifted away — what's left is the frustum. The closed-form formula and the “full cone − tip” identity are both shown.
- Sphere: the sphere is sliced into thin pancake circles whose radii follow √(R²−y²). We're honest that the exact sum is calculus — Archimedes' ⁴⁄₃·π·r³ formula is the result.
- Ellipsoid: starts as a sphere, then stretches independently along axes a, b, and c.
Volume formula reference — all 12 shapes
| Shape | Volume formula | Common use |
|---|---|---|
| Cube | V = side³ | Dice, sugar cubes |
| Rectangular prism | V = l · w · h | Boxes, rooms |
| Triangular prism | V = ½ · b · th · L | Roof trusses, Toblerone |
| Cylinder | V = π · r² · h | Cans, pipes, tanks |
| Hollow cylinder | V = π · (R² − r²) · L | Pipes, tubes, washers |
| Capsule | V = π·r²·L + ⁴⁄₃·π·r³ | Propane tanks, pills |
| Sphere | V = ⁴⁄₃ · π · r³ | Balls, planets, bubbles |
| Hemisphere | V = ⅔ · π · r³ | Bowls, domes, igloos |
| Ellipsoid | V = ⁴⁄₃ · π · a·b·c | Eggs, footballs |
| Cone | V = ⅓ · π · r² · h | Ice cream, traffic cones |
| Frustum | V = ⅓·π·h·(r₁² + r₂² + r₁·r₂) | Buckets, lampshades |
| Pyramid | V = ⅓ · l · w · h | Egyptian pyramids, roof apex |
The ⅓ rule — why cones and pyramids divide by three
A cone with the same base radius and same height as a cylinder has exactly one-third the cylinder's volume. The same rule holds for pyramids vs rectangular boxes. The intuition: a prism has a constant cross-section; a cone or pyramid tapers to a point, and the average shrunk cross-section is one-third of the base.
The “How it's built” view shows this concretely — the built cone sits next to its matching cylinder with a “× 3 →” label and a one-line caption: three cones of sand fill the cylinder exactly.
Slant height vs. perpendicular height
Volume formulas always use the perpendicular height — the straight-line distance from apex to base. Slant height is along the slanted surface and is needed for surface-area calculations on cones and pyramids. The two are related by Pythagoras: for a cone, ℓ = √(r² + h²).
The classic 3-4-5 cone is the easy example: a cone with r = 3 and perpendicular height 4 has slant 5, volume 12π, and surface area 24π. The calculator asks for perpendicular height and derives the slant for you.
Five common volume mistakes
- Wrong units. Volume is cubic — m³, not m². A 5m × 5m × 5m room has volume 125 m³.
- Diameter vs radius. π·r² uses radius. If you were given diameter 10, use r = 5.
- Slant for perpendicular height. Volume of a cone uses the perpendicular height h, not slant ℓ.
- Forgetting the ⅓. Cone and pyramid volumes are one-third of the matching prism / cylinder. Triple-check.
- Confusing lateral with total surface area. Lateral = side only. Total = lateral + base(s). The calculator reports total SA; lateral is in the extras.
Frequently asked questions
What 3D shapes does this volume calculator support?
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What 3D shapes does this volume calculator support?
▾Twelve shapes — cube, rectangular prism, triangular prism, cylinder, hollow cylinder (pipe), capsule, sphere, hemisphere, ellipsoid, cone, frustum (truncated cone), and rectangular pyramid. Each one returns volume, surface area, and the relevant extras (slant heights, diagonals, lateral and base areas).
What is the "How it's built" view?
▾
What is the "How it's built" view?
▾A short animation that shows where each formula comes from. Cylinders are built by stacking flat circles; cones are stacks of shrinking circles; cubes are stacks of squares; capsules are a cylinder with a hemisphere on each end. Captions narrate the math step-by-step in plain language so you understand the formula, not just memorise it.
Why does a cone have one-third the volume of a cylinder?
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Why does a cone have one-third the volume of a cylinder?
▾Three identical cones, each with the same base radius and height as a cylinder, will fill that cylinder exactly when poured in (e.g. with sand). The "How it's built" view shows this side-by-side. The same one-third rule applies to pyramids vs rectangular boxes.
How is a frustum's volume calculated?
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How is a frustum's volume calculated?
▾A frustum is a cone with the tip chopped off. Its volume equals the full cone's volume minus the small tip you removed: V = ⅓·π·h·(r₁² + r₂² + r₁·r₂), where r₁ is the top radius, r₂ the bottom radius, and h the perpendicular height between them.
Why is a sphere's volume ⁴⁄₃·π·r³?
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Why is a sphere's volume ⁴⁄₃·π·r³?
▾Imagine slicing a sphere into many thin horizontal pancakes. Each pancake is a circle with radius √(R² − y²), biggest at the equator and shrinking toward the poles. Adding all these circles up — what calculus calls integration — gives ⁴⁄₃·π·r³. Archimedes worked this out around 250 BC using a clever lamp-shade argument.
Can I use this calculator for fluid capacity (litres or gallons)?
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Can I use this calculator for fluid capacity (litres or gallons)?
▾Yes — compute the volume in cm³ or in³ and convert: 1 litre = 1000 cm³, 1 US gallon = 231 in³. The calculator is unit-agnostic, so put in cm and you get cm³; put in inches and you get in³.
What is the difference between slant height and perpendicular height?
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What is the difference between slant height and perpendicular height?
▾Volume formulas always use perpendicular height — the straight line from apex to base. Slant height is the distance along the slanted surface (apex to a point on the base edge). Surface area for cones and pyramids needs the slant height; the calculator computes it for you using the Pythagorean theorem.
Why do prisms use base × height but cones and pyramids use ⅓ × base × height?
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Why do prisms use base × height but cones and pyramids use ⅓ × base × height?
▾A prism (cylinder, box) has a constant cross-section all the way up — every horizontal slice is the same shape and area, so the volume is just slice-area × height. A cone or pyramid tapers to a point: the cross-sections shrink linearly, and the average shrunk area works out to exactly one-third of the base area.
Want the full reference?
The Volume Guide walks through every formula with examples, the ⅓ rule, the slant-vs-perpendicular distinction, plus interactive embeds you can play with inline.
Read the Volume Guide →