Volume Calculator
Calculate the volume of cubes, spheres, cylinders, cones, and other 3D shapes.
What Is the Volume Calculator?
The Volume Calculator computes the volume of common 3D shapes. Volume measures the amount of space enclosed inside a solid, expressed in cubic units. Select a shape, enter the required dimensions, and get the volume with the step-by-step formula.
Formula
How to Use
Select the 3D shape from the dropdown. Enter the required dimensions — radius for spheres, radius and height for cylinders and cones, side length for cubes, and length/width/height for rectangular prisms and pyramids. Click Calculate to see the volume.
Example Calculation
Sphere: r = 6 V = (4/3)×π×6³ = (4/3)×π×216 = 904.779 cubic units Cylinder: r = 3, h = 10 V = π×3²×10 = π×9×10 = 282.743 cubic units Rectangular Prism: l=4, w=5, h=3 V = 4×5×3 = 60 cubic units
Understanding Volume
Volume calculations are essential in science, engineering, and everyday life. From calculating medication dosages (drug concentration × volume) to estimating how much concrete is needed for a foundation, volume is one of the most applied geometric measurements.
Archimedes discovered many volume formulas through an ingenious method of comparing solids to known volumes — a precursor to integral calculus by 1800 years. His "Eureka!" moment came from realizing that the volume of an irregularly shaped object equals the volume of water it displaces.
In calculus, volumes of revolution are computed by rotating a 2D region around an axis, using the disk/washer method (V = π∫[f(x)]² dx) or the shell method (V = 2π∫x·f(x) dx). All the formulas in this calculator can be derived by these integration methods.
Frequently Asked Questions
Why does a cone have 1/3 of a cylinder's volume?
A cone with the same base and height as a cylinder holds exactly 1/3 of the cylinder's volume. Cavalieri's principle and calculus (integration of circular cross-sections) both confirm this. Three identical cones fill exactly one cylinder.
What is the difference between volume and capacity?
Volume is the geometric measure of space in cubic units (cm³, m³). Capacity is the practical measure of how much a container can hold (liters, gallons). 1 liter = 1000 cm³.
Why is the sphere volume formula V = (4/3)πr³?
Archimedes discovered that a sphere has exactly 2/3 the volume of its circumscribed cylinder. Using the cylinder volume πr²(2r) = 2πr³, the sphere is 2/3 of that = 4πr³/3.
What is the volume of a hemisphere?
A hemisphere is exactly half a sphere, so V = (1/2)(4/3)πr³ = (2/3)πr³.
How do I convert between volume units?
1 m³ = 1,000 liters = 1,000,000 cm³ = 1,000,000 mL. 1 foot³ ≈ 28.317 liters. For conversion between units, scale the linear measurements first, then apply the volume formula.