Volume Calculator | 3D Shapes
Calculate the volume of cubes, spheres, cylinders, cones, and other 3D shapes.
What Is the Volume Calculator | 3D Shapes?
The Volume Calculator computes both volume and surface area for 10 common 3D shapes: sphere, hemisphere, cylinder, cone, cube, rectangular prism, triangular pyramid, ellipsoid, capsule, and triangular prism. Select a shape, enter the required dimensions, and get instant results with a step-by-step formula. A multi-unit conversion table shows the volume in 8 units simultaneously (mm³ through km³, plus liters, mL, in³, ft³).
Formula
V = (4/3)πr³V = πr²hV = (1/3)πr²hV = s³V = l × w × hV = (1/3) × l × w × hV = (4/3)π·a·b·cHow to Use
- ›Select the 3D shape from the dropdown at the top.
- ›Enter the required dimensions, only the fields for the selected shape are shown.
- ›Optionally select a display unit (default: cm³/m²).
- ›Click Calculate to see volume, surface area, step-by-step working, and a full unit conversion table.
- ›Use the Reset button to clear all inputs.
Example Calculation
Example 1, Sphere (basketball)
Radius r = 12 cm
Example 2, Cylinder (water tank)
Radius r = 0.5 m, Height h = 2 m
Example 3, Ellipsoid (rugby ball)
Semi-axes a = 14 cm, b = c = 8 cm
Understanding Volume | 3D Shapes
Volume and Surface Area, The Basics
Volume measures the amount of three-dimensional space enclosed by a solid, expressed in cubic units (cm³, m³, L). Surface area measures the total area of the solid's outer surface, expressed in square units (cm², m²). Both are fundamental to engineering, manufacturing, biology, and everyday problem-solving.
- ›Volume determines capacity, how much liquid, gas, or material a container holds.
- ›Surface area governs heat transfer, reaction rates, and material cost.
- ›The ratio SA/V (surface-to-volume ratio) decreases as size increases, key to biology (why cells stay small) and heat loss (why small animals lose warmth faster).
Shape Reference Table
| Shape | Volume | Surface Area | Key dimension |
|---|---|---|---|
| Sphere | (4/3)πr³ | 4πr² | radius r |
| Hemisphere | (2/3)πr³ | 3πr² | radius r |
| Cylinder | πr²h | 2πr(r+h) | radius r, height h |
| Cone | (1/3)πr²h | πr(r+√(r²+h²)) | radius r, height h |
| Cube | s³ | 6s² | side s |
| Rect. Prism | l·w·h | 2(lw+lh+wh) | length l, width w, height h |
| Pyramid (rect.) | (1/3)l·w·h | lw+ls₁+ws₂ | base l×w, height h |
| Ellipsoid | (4/3)π·a·b·c | Thomsen approx. | semi-axes a, b, c |
| Capsule | πr²(4r/3+h) | 2πr(2r+h) | radius r, cylinder height h |
| Triang. Prism | ½·b·h·L | 2A_tri+3·face | base b, height h, length L |
Volume Unit Conversion Reference
| Unit | Equivalent in cm³ | Equivalent in liters | Common use |
|---|---|---|---|
| 1 mm³ | 0.001 | 0.000001 | Droplet, small dose |
| 1 cm³ | 1 | 0.001 | = 1 mL; medicine, chemistry |
| 1 mL | 1 | 0.001 | Fluid volume (medicine) |
| 1 L | 1,000 | 1 | Everyday fluid measure |
| 1 m³ | 1,000,000 | 1,000 | Large tanks, concrete |
| 1 in³ | 16.387 | 0.01639 | US/UK engineering |
| 1 ft³ | 28,316.85 | 28.317 | HVAC, lumber |
| 1 US gal | 3,785.41 | 3.785 | US fuel, liquid measure |
Real-World Applications
- ›Construction, Calculating concrete volume for foundations, columns, and slabs.
- ›Fluid systems, Pipe and tank sizing based on required flow capacity.
- ›Packaging, Minimizing material (surface area) while maximizing product volume.
- ›Pharmaceuticals, Dosage calculations based on capsule or vial volume.
- ›3D printing, Estimating material usage from model geometry before printing.
Frequently Asked Questions
Why does a cone have exactly 1/3 the volume of a cylinder?
A cone with the same base radius and height as a cylinder encloses exactly one-third the volume. This was proved by Archimedes using the method of exhaustion and confirmed by calculus: integrating circular cross-sections from tip to base gives V = π∫₀ʰ (rx/h)² dx = πr²h/3.
How do I convert volume units?
- ›1 m³ = 1,000 L = 1,000,000 cm³ = 1,000,000 mL
- ›1 L = 1,000 mL = 1,000 cm³
- ›1 ft³ = 28.317 L = 0.02832 m³
- ›1 in³ = 16.387 cm³ = 16.387 mL
- ›1 US gallon = 3.785 L; 1 UK gallon = 4.546 L
What is the surface area of a sphere and why is it 4πr²?
Archimedes showed that a sphere has the same surface area as the curved surface of its circumscribed cylinder (excluding caps). The cylinder has circumference 2πr and height 2r, giving SA = 2πr × 2r = 4πr². This elegant result was considered one of Archimedes' greatest discoveries.
What is the Knud Thomsen approximation for ellipsoid surface area?
There is no closed-form formula for an ellipsoid's surface area. The Knud Thomsen approximation SA ≈ 4π × [(aᵖbᵖ + aᵖcᵖ + bᵖcᵖ)/3]^(1/p) with p ≈ 1.6075 is accurate to within 1.061% for all ellipsoids and is widely used in practice.
Why is volume proportional to the cube of linear dimensions?
Volume occupies three-dimensional space, so it scales with the cube of any linear dimension. Double the radius of a sphere and its volume increases 8× (2³). This cube law is why large animals need proportionally stronger skeletons, their volume (and weight) grows faster than their cross-sectional area (strength).