Pick a shape, type in its dimensions, and the 3D model redraws as you type while the volume and surface area recalculate instantly. Rotate the model, turn on the dimension labels to see which measurement is which, and read each formula with your own numbers filled in.
Enter its dimensions and the model updates instantly.
Choose one of twelve shapes: box, cube, sphere, hemisphere, cylinder, cone, cone frustum, square pyramid, triangular prism, torus, ellipsoid or capsule. Enter the measurements in any length unit from millimetres to yards. The calculator returns the volume, the total surface area and the extra values you usually need, such as slant height or lateral area, plus the capacity in litres and US gallons.
The model is drawn to the same proportions as your numbers, so a tall, thin cylinder looks tall and thin. That makes the picture a quick way to catch a swapped radius and height before you rely on the result.
If you only need a volume for one of the basic shapes, the Volume Calculator is a lighter page. Use this one when you want to see the shape, work with less common solids such as the torus or frustum, or check surface area as well.
| Shape | Volume | Total surface area |
|---|---|---|
| Box | l × w × h | 2(lw + lh + wh) |
| Cube | a³ | 6a² |
| Sphere | (4/3)πr³ | 4πr² |
| Hemisphere | (2/3)πr³ | 3πr² |
| Cylinder | πr²h | 2πr(r + h) |
| Cone | (1/3)πr²h | πr(r + s), where s = √(r² + h²) |
| Cone frustum | (πh/3)(r₁² + r₁r₂ + r₂²) | π(r₁ + r₂)s + πr₁² + πr₂² |
| Square pyramid | (1/3)a²h | a² + 2al, where l = √(h² + (a/2)²) |
| Triangular prism | (√3/4)a²L | (√3/2)a² + 3aL |
| Torus | 2π²Rr² | 4π²Rr |
| Ellipsoid | (4/3)πabc | approximate, see the note below |
| Capsule | πr²h + (4/3)πr³ | 2πrh + 4πr² |
The triangular prism here has an equilateral triangle as its base, so one side length (a) describes the whole triangle. For the capsule, h is the length of the straight middle section, not the overall length.
Water tank (cylinder). Radius 1.5 m and height 3 m give V = π × 1.5² × 3 ≈ 21.21 m³. Since 1 m³ is 1,000 litres, the tank holds about 21,206 litres, or roughly 5,602 US gallons. Painting the whole outside, top and bottom included, covers S = 2π × 1.5 × (1.5 + 3) ≈ 42.41 m².
Square pyramid. A base side of 6 and a height of 8 give a face slant height of √(8² + 3²) ≈ 8.544. The volume is (1/3) × 6² × 8 = 96 and the total surface area is 36 + 2 × 6 × 8.544 ≈ 138.53.
Planter (cone frustum). A planter with a bottom radius of 5 cm, a top radius of 3 cm and a height of 8 cm has a slant height of √(2² + 8²) ≈ 8.246 cm. Its volume is about 410.5 cm³, close to 0.41 litres, and its total surface area is about 314.06 cm².
Torus. With R = 6 and r = 2, the volume is 2π² × 6 × 2² ≈ 473.74 and the surface area is 4π² × 6 × 2 ≈ 473.74. The two numbers match only because r = 2: in general the volume divided by the surface area equals r/2.
Capsule. A pill-shaped capsule with a 3 mm radius and an 8 mm straight section is 14 mm long overall. Its volume is π × 3² × 8 + (4/3)π × 3³ ≈ 339.29 mm³, or about 0.34 millilitres.
People were measuring solids long before they wrote formulas down. The Moscow Mathematical Papyrus, an Egyptian text from around 1850 BCE, works out the volume of a truncated square pyramid, which belongs to the same family as the cone frustum in this calculator.
The Greeks turned these rules into proofs. Democritus is credited with noticing that a cone holds one third of the cylinder with the same base and height, and Eudoxus supplied a rigorous proof using the method of exhaustion. Euclid’s Elements, Book XII, collects results of this kind.
Archimedes went further in On the Sphere and Cylinder, written around 225 BCE. He showed that a sphere has two thirds of the volume and two thirds of the surface area of the cylinder that just contains it. You can check this with the formulas above: for a sphere of radius r, the enclosing cylinder has volume 2πr³ and total surface area 6πr². Archimedes considered this his finest result and asked for the figure to be carved on his tomb, which the Roman statesman Cicero later found overgrown in Syracuse.
The torus formula follows from a theorem usually credited to Pappus of Alexandria in the 4th century CE and rediscovered by Paul Guldin in the 17th: the volume of a solid of revolution equals the area of the shape being turned multiplied by the distance its centre travels. A circle of area πr² whose centre travels 2πR gives 2π²Rr².
In 1635 Bonaventura Cavalieri published his principle that two solids with equal cross-sections at every height have equal volumes, a stepping stone toward integral calculus. The ellipsoid is where simple formulas run out: its volume is easy, but its exact surface area needs elliptic integrals, which is why an approximation is used here.
How do I calculate the volume and surface area of a 3D shape? Choose the shape, enter its dimensions in one length unit, and read the results. The calculator applies the standard formula for that shape and shows it with your numbers filled in, so you can follow every step.
What is the difference between lateral surface area and total surface area? Lateral area covers only the sides of a solid, such as the curved wall of a cylinder. Total surface area adds the flat bases or ends. For a cylinder the total is 2πrh for the side plus 2πr² for the two circular ends.
How do I convert the volume to litres or gallons? The results include the capacity in litres and US gallons for the unit you chose. By hand, one cubic metre is 1,000 litres, one litre is 1,000 cubic centimetres, and one US gallon is about 3.785 litres.
What is a torus and how is its volume found? A torus is a doughnut-shaped solid. If R is the distance from the centre of the hole to the centre of the tube and r is the tube radius, the volume is 2π²Rr² and the surface area is 4π²Rr. R must be larger than r for a ring shape.
Is the surface area of an ellipsoid exact? No. An ellipsoid has no simple exact surface area formula, so this calculator uses Knud Thomsen’s approximation, which is accurate to about 1%. The volume of an ellipsoid is exact.
Can I enter the diameter instead of the radius? Yes. For any shape with round measurements, switch “Enter round shapes as” to Diameter. The calculator halves the value for you and converts the numbers you already typed, so the shape stays the same.
What is a cone frustum? A cone frustum is a cone with its tip cut off by a plane parallel to the base. Buckets, lampshades and flower pots are common examples. Enter the bottom radius, the top radius and the straight vertical height, not the slanted side.
Does the 3D viewer work on a phone? Yes. Drag with one finger to rotate and pinch to zoom. If a browser cannot run WebGL, the calculator still works and only the picture is missing.