Geometry

Volume Calculator | 3D Shapes

Calculate the volume of cubes, spheres, cylinders, cones, and other 3D shapes.

Instant Results100% FreeAny DeviceNo Sign-up

Shape

Radius

Display unit

Overview

What Is the Volume Calculator?

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).

How it works

Volume Calculator Formula and Method

SphereV = (4/3)πr³SA = 4πr²

CylinderV = πr²hSA = 2πr(r+h)

ConeV = (1/3)πr²hSA = πr(r + √(r²+h²))

CubeV = s³SA = 6s²

Rect. PrismV = l × w × hSA = 2(lw+lh+wh)

PyramidV = (1/3) × l × w × hSA = lw + l·slant + w·slant

EllipsoidV = (4/3)π·a·b·cSA ≈ Knud Thomsen approx.

Instructions

How 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.

Real example

Volume Calculator Example

Example 1, Sphere (basketball)

Radius r = 12 cm

V = (4/3) × π × 12³ = (4/3) × π × 1728 = 7,238.2 cm³ SA = 4 × π × 144 = 1,809.6 cm²

Example 2, Cylinder (water tank)

Radius r = 0.5 m, Height h = 2 m

V = π × 0.5² × 2 = π × 0.25 × 2 = 1.571 m³ = 1,571 L SA = 2π × 0.5 × (0.5 + 2) = 7.854 m²

Example 3, Ellipsoid (rugby ball)

Semi-axes a = 14 cm, b = c = 8 cm

V = (4/3) × π × 14 × 8 × 8 = 2,981.1 cm³

Guide

Understanding Volume

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.

Answers

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).

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