Volume Calculator

Last Verified: 2026-07-12

Calculate the volume of spheres, cylinders, cones, capsules, and boxes with live dimensions and worked formulas.

Measurement Unit
Radius (r)5 cm
Calculated Volume
523.5988 cm³
r = 5
How is this calculated?
Formula: V = (4/3) × π × r³
Substitute: V = (4/3) × π × 5³
Calculate: V = (4/3) × π × 125
Result: V ≈ 523.5988
Assumptions & Sources
Assumptions:
  • All shapes are mathematically perfect geometric solids.
  • Inputs and outputs use consistent, uniform dimensional units.
Sources:
  • Calculus: Early Transcendentals, Cengage Learning — Integration derivations for spheres, cones, pyramids, and spherical caps.
  • Thomas' Calculus, Pearson — Standard definitions of volume, cylinder, and hollow washer methods.
  • NIST Digital Library of Mathematical Functions (dlmf.nist.gov).

About the Volume Calculator

Volume is the measure of the three-dimensional space occupied by a solid object, liquid, or gas. Expressed in cubic units (such as cubic meters, cubic centimeters, or cubic inches) or capacity metrics (such as liters or gallons), volume is crucial in chemistry, manufacturing, fluid dynamics, and packaging logistics. Mathematically, volume calculations depend on the geometric structure of the solid: regular prisms and cylinders are calculated by multiplying their two-dimensional base area by their height; pyramids and cones represent exactly one-third of the volume of their corresponding flat-topped shapes; and spheres represent two-thirds of the volume of a cylinder that encloses them. Accurate volume calculation enables precise density, weight, and capacity modeling across engineering and daily life.

Mathematical Formula & Logic

Volume formulas define the capacity of three-dimensional solids based on their linear dimensions: 1. Cube: Volume = Side³ 2. Rectangular Prism (Box): Volume = Length × Width × Height 3. Cylinder: Volume = π × Radius² × Height 4. Sphere: Volume = (4/3) × π × Radius³ 5. Cone: Volume = (1/3) × π × Radius² × Height 6. Pyramid: Volume = (1/3) × Base_Area × Height

Step-by-Step Example

Calculate the volume of a cylinder with a base radius of 4 cm and a height of 10 cm: 1. Identify the variables: - Radius (r) = 4 cm - Height (h) = 10 cm 2. Apply the cylinder volume formula: Volume = π × r² × h Volume = π × 4² × 10 3. Simplify the expression: Volume = π × 16 × 10 Volume = 160 × π ≈ 502.6548 4. The volume of the cylinder is approximately 502.65 cubic centimeters (502.65 cm³).

Reference Data & Values

solidparametersvolume formulaexample calculation
CubeSide (s)s³s=4 cm → Volume = 64 cm³
Rectangular Prisml, w, hl × w × hl=6, w=4, h=5 → Volume = 120
Cylinderr, hπr²hr=4, h=10 → Volume ≈ 502.65
SphereRadius (r)(4/3)πr³r=3 → Volume ≈ 113.10
Coner, h(1/3)πr²hr=3, h=6 → Volume ≈ 56.55

Frequently Asked Questions

The volume formula depends on the shape of the object. For a rectangular prism (box), multiply length times width times height (V = l * w * h). For a cylinder, multiply the circular base area by the height (V = pi * r^2 * h). For other shapes, use the dedicated volume formulas.
Measuring volume depends on the state of matter. For liquids, use measuring cups or graduated cylinders. For regular solids, measure dimensions (length, radius, height) and use geometric formulas. For irregular solids, use the fluid displacement method.
Volume is always measured in cubed units (e.g., cubic meters, cubic inches) because it quantifies three-dimensional space. Area is measured in squared units because it measures two-dimensional flat surfaces.
To calculate volume, identify the 3D shape, measure the required inputs (like radius, height, or side length), ensure all dimensions use the same unit, and apply the corresponding formula (e.g., V = s^3 for a cube).
The SI unit for volume is the cubic meter (m³). Commonly used metric units include cubic centimeters (cm³), milliliters (mL), and liters (L). Common US imperial units include cubic inches, cubic feet, cubic yards, fluid ounces, pints, quarts, and gallons.
The volume of a liquid can be read directly using graduated vessels like measuring cups, flasks, pipettes, or graduated cylinders. Liquid volume is typically expressed in liters, milliliters, fluid ounces, or gallons.
The official SI unit of volume is the cubic meter (m³). It is a derived unit based on the SI base unit of length, the meter.
Volume is an extensive property, meaning its value scales with the quantity of matter in the system. For example, doubling the amount of water doubles its volume, whereas intensive properties like density remain unchanged.