Mathematics · Geometry & Trigonometry · Solid Geometry
Cylinder Volume Calculator
Calculates the volume of a right circular cylinder given its radius and height using the standard geometric formula.
Calculator
Formula
V is the volume of the cylinder (in cubic units). r is the radius of the circular base (distance from the centre to the edge). h is the height (or length) of the cylinder measured perpendicular to the base. π (pi) is the mathematical constant approximately equal to 3.14159.
Source: Euclid, Elements, Book XII; also standardised in ISO 80000-2 mathematical notation.
How it works
A right circular cylinder is a three-dimensional solid formed by two parallel, congruent circular bases connected by a curved lateral surface. The axis of the cylinder is perpendicular to both bases, making it a 'right' cylinder — the most common type encountered in practice. Volume measures the total space enclosed within the solid, expressed in cubic units such as cubic metres (m³), cubic centimetres (cm³), or litres (1 L = 0.001 m³).
The volume formula is derived by multiplying the area of the circular base by the height of the cylinder: V = πr²h. Here, πr² gives the area of a circle with radius r (using the well-known circle area formula A = πr²), and multiplying by the height h effectively stacks an infinite number of infinitesimally thin circular discs along the vertical axis — a concept rigorously justified by Cavalieri's principle and integral calculus. The constant π ≈ 3.14159265 is the ratio of a circle's circumference to its diameter. This calculator also returns the base area, lateral surface area (the curved side: 2πrh), and total surface area (2πr² + 2πrh), which are frequently needed alongside volume in design and materials estimation.
Practical applications span an enormous range. Civil engineers use cylinder volume to determine the concrete required for cylindrical pillars. Mechanical engineers calculate piston displacement in engines using this formula. Food scientists size cylindrical cans to meet volume targets while minimising material cost. Hydrologists estimate the storage capacity of cylindrical tanks and silos. In chemistry and pharmacology, the formula underpins volumetric glassware calibration. Any time a pipe, barrel, rod, drum, column, or tube needs to be sized or measured, this formula is the starting point.
Worked example
Suppose you need to determine how much water a cylindrical storage tank can hold. The tank has a radius of 1.2 m and a height of 3.5 m.
Step 1 — Calculate the base area:
A = πr² = π × (1.2)² = π × 1.44 ≈ 4.5239 m²
Step 2 — Calculate the volume:
V = A × h = 4.5239 × 3.5 ≈ 15.8336 m³
Since 1 m³ = 1,000 litres, the tank holds approximately 15,834 litres of water — useful for procurement and structural load calculations.
Step 3 — Calculate lateral surface area (for cladding or painting):
LSA = 2πrh = 2 × π × 1.2 × 3.5 = 2 × π × 4.2 ≈ 26.3894 m²
Step 4 — Calculate total surface area (including both circular ends):
TSA = 2πr(r + h) = 2 × π × 1.2 × (1.2 + 3.5) = 2 × π × 1.2 × 4.7 ≈ 35.4336 m²
These four results together give a complete geometric picture of the tank, enabling material cost estimation, load analysis, and capacity planning in a single calculation pass.
Limitations & notes
This calculator assumes a right circular cylinder — the axis is perfectly perpendicular to both bases and both bases are perfect circles. It does not apply to oblique cylinders (where the axis is tilted), elliptic cylinders (with elliptical cross-sections), truncated cylinders (cut at an angle), or hollow cylinders such as pipes (which require subtracting the inner cylinder's volume from the outer). For a hollow cylinder, compute the volume of the outer cylinder and subtract the volume of the inner bore: V = π(R² − r²)h, where R is the outer radius and r is the inner radius. Additionally, real-world containers have wall thickness, seams, and end-cap geometry that this idealised formula ignores, so engineering designs should account for these factors with appropriate tolerances. Input units must be consistent — mixing metres for radius and centimetres for height will produce incorrect results. Always confirm the unit system before entering values.
Frequently asked questions
What is the formula for the volume of a cylinder?
The volume of a right circular cylinder is V = πr²h, where r is the radius of the circular base and h is the perpendicular height. For example, a cylinder with radius 3 cm and height 10 cm has a volume of π × 9 × 10 ≈ 282.74 cm³.
How do I convert cylinder volume from cubic metres to litres?
Multiply the volume in cubic metres by 1,000 to get litres, since 1 m³ = 1,000 L. For example, a volume of 0.5 m³ equals 500 litres. To convert cm³ to millilitres, note that 1 cm³ = 1 mL exactly.
How do I calculate the volume of a hollow cylinder or pipe?
For a hollow cylinder with outer radius R and inner radius r, subtract the inner volume from the outer: V = π(R² − r²)h. For example, a pipe with outer radius 5 cm, inner radius 4 cm, and length 100 cm has volume π × (25 − 16) × 100 = 900π ≈ 2,827 cm³.
What is the difference between lateral surface area and total surface area of a cylinder?
The lateral (curved) surface area covers only the side of the cylinder and equals 2πrh — useful for painting or wrapping. The total surface area adds both circular end caps (each πr²), giving 2πr² + 2πrh = 2πr(r + h). Use total surface area when calculating the material needed to fabricate a closed container.
Can this formula be used for an oblique cylinder?
Yes — by Cavalieri's principle, an oblique cylinder with the same base area and perpendicular height as a right cylinder has the same volume. So V = πr²h still holds as long as h is the perpendicular (not slant) height. However, the surface area formulas change for oblique cylinders, so only the volume result from this calculator would remain valid in that case.
Last updated: 2025-01-15 · Formula verified against primary sources.