Volume Calculator

Volume is the measure of the three-dimensional space enclosed within a solid shape. It answers the question: "How much space does this object occupy?" or "How much can this container hold?"

Volume is expressed in cubic units (e.g., $\text{cm}^3$, $\text{m}^3$, $\text{ft}^3$) or in capacity units (liters, gallons).

Why Volume Matters

• Engineering: sizing tanks, pipes, and containers
• Medicine: calculating dosages and organ sizes
• Shipping: determining cargo space and packaging
• Cooking: measuring ingredients
• Construction: estimating concrete, gravel, or fill

Units of Volume

Volume Formulas for Common 3D Shapes

Cube

All sides are equal:

$V = s^3$

Example: A cube with side $s = 5$ has volume $V = 5^3 = 125$ cubic units.

Sphere

A perfectly round 3D shape:

$V = \frac{4}{3}\pi r^3$

Example: A sphere with radius $r = 6$ has volume $V = \frac{4}{3}\pi(6)^3 = \frac{4}{3}\pi(216) = 288\pi \approx 904.78$ cubic units.

Cylinder

A cylinder is essentially a circle extruded to height $h$:

$V = \pi r^2 h$

This is simply the base area ($\pi r^2$) times the height ($h$).

Example: A cylinder with $r = 3$ and $h = 10$ has volume $V = \pi(3)^2(10) = 90\pi \approx 282.74$ cubic units.

Cone

A cone is one-third of a cylinder with the same base and height:

$V = \frac{1}{3}\pi r^2 h$

Example: A cone with $r = 4$ and $h = 9$ has volume $V = \frac{1}{3}\pi(4)^2(9) = \frac{1}{3}\pi(144) = 48\pi \approx 150.80$ cubic units.

Relationship Between Shapes

• A cone is exactly $\frac{1}{3}$ the volume of a cylinder with the same base radius and height
• A sphere has the same volume as a cone with height equal to $4r$ and base radius equal to $r$ (since $\frac{4}{3}\pi r^3 = \frac{1}{3}\pi r^2 (4r)$)
• A hemisphere is exactly $\frac{2}{3}$ of the cylinder that encloses it

• Confusing radius and diameter — always check whether you are given the radius or diameter. If given the diameter, divide by 2 before using volume formulas.
• Forgetting the $\frac{1}{3}$ factor for cones and pyramids — a cone is NOT the same volume as a cylinder. The $\frac{1}{3}$ factor accounts for the tapering.
• Using slant height instead of perpendicular height — for cones and pyramids, the formula requires the vertical (perpendicular) height, not the slant height along the surface.
• Cubing vs. squaring errors — for a sphere, the radius is cubed ($r^3$); for a cylinder, the radius is squared ($r^2$) then multiplied by height. Mixing these up gives very wrong answers.
• Unit conversion errors — when converting cubic units, remember to cube the linear conversion factor. For example, $1\,\text{m}^3 = (100\,\text{cm})^3 = 1{,}000{,}000\,\text{cm}^3$, not $100\,\text{cm}^3$.