Volume Calculator — 10 Geometric Shapes

Last reviewed: April 2026

All 10 Volume Formulas — Complete Reference Guide

Volume is the measure of three-dimensional space occupied by or enclosed within an object. It is always expressed in cubic units: cubic centimeters (cm³), cubic meters (m³), cubic feet (ft³), liters, or gallons. Below are the exact formulas for every shape in this calculator, with variables defined, worked examples, and the mathematical reasoning behind each formula.

Why Does a Cone Hold Exactly 1/3 of a Cylinder?

One of the most beautiful relationships in geometry: a cone and a cylinder with the same base radius and height are related by a factor of exactly 1/3. This can be proven with calculus (by integrating the area of circular cross-sections from the apex to the base), but it was also demonstrated experimentally by Euclid around 300 BC. Pour water from a cone into a cylinder of the same dimensions — it takes exactly three cones to fill the cylinder. The same 1/3 relationship holds for a triangular pyramid and a triangular prism.

Volume vs. Capacity vs. Surface Area

These three related concepts are often confused. Volume is the total 3D space a solid object occupies, including the material it is made of. Capacity is the volume of the interior space of a hollow container — how much it can hold. For a solid metal ball, volume ≠ capacity (capacity is zero). For a hollow tank, volume ≈ capacity (assuming thin walls). Surface area is the total area of all outer faces — it is a 2D measurement (m², cm²) that describes how much material wraps the outside of the shape.

When Volume Is Not in Cubic Units — Liquids and Gases

Volume is expressed in cubic units for solids (cm³, m³, ft³). For liquids, it is more commonly expressed in liters (L), milliliters (mL), US gallons, or fluid ounces. These are simply alternative names for specific cubic volumes: 1 liter = 1000 cm³ = 1 dm³. For gases, volume depends on pressure and temperature (ideal gas law: PV = nRT). This calculator gives geometric volume — actual liquid or gas volume filling that shape may differ if the shape is a container at specific conditions.

Volume Unit Conversions — Complete Reference

Volume appears in many different units depending on the field: cubic centimeters in chemistry, liters in everyday use, gallons for fuel and fluids, cubic feet for construction and HVAC, and cubic meters for large-scale engineering. Knowing how to convert between these is essential for practical volume calculations.

Cylinder Volume in Liters and Gallons

A common practical need is calculating how many liters or gallons a cylindrical tank holds. The process: calculate V = πr²h in cubic centimeters (using cm), then divide by 1000 for liters or 3785.41 for US gallons. Example: a cylindrical water tank with diameter 60 cm (r=30 cm) and height 120 cm: V = π × 900 × 120 = 339,292 cm³ = 339.3 liters = 89.6 US gallons.

Cubic Feet to Cubic Yards in Construction

Concrete, soil, gravel, and mulch are commonly ordered by the cubic yard (yd³). 1 cubic yard = 27 cubic feet. For a rectangular concrete slab 20 ft × 10 ft × 0.333 ft (4 inches thick): V = 66.7 ft³ ÷ 27 = 2.47 cubic yards. This is the standard calculation for ordering concrete for a patio or driveway. Always add a 10% waste factor for concrete pours.

How Volume Calculations Are Used in Real Life

Construction — Concrete, Soil, and Gravel

Volume calculation is fundamental to construction materials estimation. Concrete for a rectangular foundation, soil for a landscaping project, or gravel for a driveway all require accurate volume calculations to determine how much material to order. The standard approach: calculate volume in cubic feet, convert to cubic yards (divide by 27), then add a 10% waste factor. For cylindrical columns or round footings, use V = πr²h. A cylindrical concrete column 12 inches in diameter and 8 feet tall requires V = π × 0.25 × 8 = 6.28 ft³ = 0.233 cubic yards of concrete.

Pool and Tank Capacity — Water Volume in Liters and Gallons

Swimming pool volume determines the amount of chemicals needed for treatment, the cost of filling the pool, and the pump sizing for filtration. A rectangular pool 12 m × 5 m × 1.5 m depth: V = 90 m³ = 90,000 liters = 23,775 US gallons. A cylindrical hot tub with diameter 2 m and depth 0.9 m: V = π × 1 × 0.9 = 2.827 m³ = 2,827 liters = 747 gallons. Pool chemical dosing is calculated per 1,000 liters or per 10,000 gallons, so accurate volume is essential to avoid over- or under-treatment.

Shipping and Logistics — Box and Container Volume

Freight volume determines shipping costs and container utilization. The rectangular box formula V = l × w × h gives the volume of a shipping carton. A 40-foot shipping container has interior dimensions approximately 12.03 m × 2.35 m × 2.39 m = 67.6 m³ of cargo volume. Freight carriers use the concept of “volumetric weight” (dimensional weight): package volume (cm³) ÷ 5000 gives the chargeable weight in kg. If this exceeds the actual weight, the dimensional weight is billed.

Medicine and Pharmacy — Dosage and Syringe Volume

In medicine, volume is critical for drug dosing. Medication concentrations are expressed in mg/mL, and the dose volume is calculated as (dose in mg) ÷ (concentration in mg/mL). Syringe and vial volumes are in milliliters (mL), where 1 mL = 1 cm³. An IV bag labeled 500 mL contains 500 cm³ of fluid. The flow rate through an IV is measured in mL/hour or drops/minute, and calculating how long a bag will last requires knowing the volume precisely.

Science — Density, Mass, and Moles

Volume connects mass, density, and amount of substance. Density = mass ÷ volume, so volume = mass ÷ density. A block of aluminum weighing 500 g with density 2.7 g/cm³ has volume = 500 ÷ 2.7 = 185.2 cm³. In chemistry, molarity = moles ÷ liters of solution. Volume is the denominator in the most fundamental concentration calculation in analytical chemistry. Archimedes’ famous bathtub discovery used volume displacement to determine the density of the king’s crown without melting it down.