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Cylinder Calculator

Volume equals pi times radius squared times height

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Calculate Cylinder Volume from Radius and Height

Use this form when you know the base radius and height of a right circular cylinder and need the enclosed volume. It is the standard formula for sizing tanks, pipes, columns, and any cylindrical container.

V = π r² h

Calculate Total Surface Area of a Cylinder

Use this form when you need the area of the curved side plus both circular ends. Common for painting, coating, insulation, and material-takeoff estimates on closed cylinders.

S = 2π r (r + h)

Calculate Lateral Surface Area of a Cylinder

Use this form when only the curved side matters — labels, wraps, banded insulation, the painted side of a tank that already has separate end-cap calculations.

L = 2π r h

Calculate Cylinder Radius from Volume and Height

Use this rearrangement when you know the required volume and a fixed height, and need the base radius. Common when sizing a tank to fit a vertical space.

r = √(V / (π h))

Calculate Cylinder Height from Volume and Radius

Use this rearrangement when the base radius is fixed by the diameter of available stock or footprint, and you need the height that gives a target volume.

h = V / (π r²)

How It Works

This cylinder calculator uses V = π r² h plus the surface-area pair S = 2π r (r + h) and L = 2π r h to solve for any of volume, total surface area, lateral surface area, radius, or height. Pick the unknown with the solve-for toggle, enter the remaining two values with any supported length, area, or volume units, and the calculator converts to SI internally before computing all related quantities — including diameter d = 2r and base circumference C = 2π r — so you can size tanks, pipes, columns, paint coverage, and label wraps from a single page.

Example Problem

A cylindrical water tank has an inner radius of 5 m and a height of 10 m. What volume does it hold?

  1. Identify the measured dimensions: the inner radius is r = 5 m and the inner height is h = 10 m.
  2. Choose the unknown: we want volume, so use V = π r² h.
  3. Square the radius: r² = 5² = 25 m².
  4. Multiply by the height: r² · h = 25 · 10 = 250 m³.
  5. Multiply by π: V = π · 250 ≈ 785.398 m³.
  6. State the result with a more practical unit: 785.398 m³ ≈ 785,398 liters, useful for sizing inlet and outlet flow rates.

Round trip check: from V ≈ 785.398 m³ and h = 10 m, the inverse formula r = √(V / (π h)) recovers r = 5 m, which confirms the forward calculation.

When to Use Each Variable

  • Solve for Volume — when the radius and height are known and you need the enclosed volume of a tank, pipe segment, or solid column.
  • Solve for Total Surface Area — when you need to estimate paint, coating, or material to wrap the entire outside of a closed cylinder including both ends.
  • Solve for Lateral Surface Area — when only the curved side matters — labels around a can, wraps around a pipe, or banded insulation on a tank wall.
  • Solve for Radius — when a target volume and an available height are known, and you need the base radius (or diameter) that produces that volume.
  • Solve for Height — when the base radius is fixed by stock diameter or footprint, and you need the height required to hit a target volume.

Key Concepts

A right circular cylinder is the solid formed by rotating a rectangle around one of its sides — equivalently, two parallel circular bases joined by a perpendicular curved surface. The geometry is fully determined by two numbers: the base radius r and the height h. Every other property — volume, surface area, diameter, base circumference — follows from those two. Lateral surface area is the curved side alone, and total surface area adds the two circular end caps. Because the cap area scales with r² while the lateral area scales with r·h, tall slender cylinders are dominated by lateral area, and short squat ones by their caps.

Applications

  • Tanks and storage vessels: size water tanks, fuel drums, propane cylinders, and process tanks from a target volume and available footprint
  • Piping and conduits: compute internal volume of pipe runs for fluid hold-up, pressure-test fill, or chemical dosing calculations
  • Materials estimation: calculate paint, coating, insulation, or label wrap area for cylindrical structures from columns to soup cans
  • Manufacturing and machining: compute the volume of cylindrical stock to budget material, estimate weight, or size a melt charge

Common Mistakes

  • Using diameter instead of radius in V = π r² h — the formula needs the radius (half the diameter), not the diameter
  • Mixing length units across radius and height (for example r in inches and h in feet) without converting first, which gives a volume that is wrong by orders of magnitude
  • Forgetting that lateral surface area excludes the end caps — total surface area is 2π r² + 2π r h, not just 2π r h
  • Treating inside dimensions and outside dimensions interchangeably; tank capacity uses inside radius and inside height, paint coverage uses outside

Frequently Asked Questions

How do you calculate the volume of a cylinder?

Use V = π r² h. Square the base radius, multiply by the height, then multiply by π. For example, a cylinder with r = 5 m and h = 10 m has V = π · 25 · 10 ≈ 785.4 m³.

What is the formula for cylinder surface area?

Total surface area is S = 2π r (r + h), which expands to 2π r² + 2π r h — the two circular end caps plus the curved side. Lateral surface area (the curved side only) is L = 2π r h.

What is the difference between total surface area and lateral surface area?

Lateral surface area is just the curved side — the part that would unroll into a rectangle of width 2π r and height h. Total surface area adds the two circular end caps, each with area π r², so it is 2π r² greater than the lateral value.

How do you find the radius of a cylinder given volume and height?

Rearrange V = π r² h to r = √(V / (π h)). For example, a 785.4 m³ tank that is 10 m tall has r = √(785.4 / (π · 10)) = √25 = 5 m.

How do you find the height of a cylinder given volume and radius?

Rearrange V = π r² h to h = V / (π r²). For a 785.4 m³ tank with r = 5 m, the height is 785.4 / (π · 25) = 10 m.

What is the diameter of a cylinder?

Diameter is twice the radius: d = 2 r. Real-world specs (pipes, drums, tanks) usually quote diameter, but the volume and surface-area formulas use radius, so divide diameter by two before plugging in.

How is cylinder volume expressed in liters or gallons?

One cubic meter is 1,000 liters or about 264.17 US gallons. The calculator handles the conversion automatically — pick liters or gallons from the volume unit dropdown to read the answer in your preferred unit.

Does the cylinder formula work for hollow cylinders or tubes?

Not directly. A hollow cylinder (a pipe wall) has both an outer radius R and an inner radius r, and its volume is π (R² − r²) h. Use this calculator for the solid-cylinder versions of volume and surface area; for a tube, compute the outer cylinder and inner cylinder separately and subtract.

Reference: Weisstein, Eric W. "Cylinder." MathWorld — A Wolfram Web Resource. https://mathworld.wolfram.com/Cylinder.html

Worked Examples

Tank Sizing

How do you calculate the volume of a cylindrical water tank?

A municipal storage tank has an inner radius of 5 m and inner height of 10 m. Use V = π r² h to find its capacity in cubic meters and liters.

  • Knowns: r = 5 m, h = 10 m
  • Formula: V = π r² h
  • V = π · 25 · 10 = 250π ≈ 785.398 m³
  • Convert to liters: 785.398 m³ × 1000 ≈ 785,398 L

Volume ≈ 785.4 m³ ≈ 785,398 liters

For pressure-test fill or chemical dosing, use the inside dimensions. For paint or insulation, use the outside dimensions instead.

Coatings and Wraps

How much paint is needed to coat the outside of a tall steel column?

A structural column has an outside diameter of 0.6 m (radius 0.3 m) and a height of 4 m. The total exterior including end caps is the total surface area S = 2π r (r + h).

  • Knowns: r = 0.3 m, h = 4 m
  • Formula: S = 2π r (r + h)
  • S = 2π · 0.3 · (0.3 + 4) = 2π · 0.3 · 4.3 ≈ 8.105 m²
  • Lateral area alone (no end caps): L = 2π · 0.3 · 4 ≈ 7.540 m²

Total surface area ≈ 8.10 m²; lateral only ≈ 7.54 m²

If both end caps are welded to other structure and don't need paint, use the lateral area instead of the total surface area.

Inverse Solve

What height is needed for a 100-liter cylinder of fixed radius?

A pharmaceutical tank must hold 100 L (0.1 m³). The vessel diameter is fixed by available stock at 0.4 m (r = 0.2 m). Find the required inside height with h = V / (π r²).

  • Knowns: V = 0.1 m³, r = 0.2 m
  • Formula: h = V / (π r²)
  • h = 0.1 / (π · 0.04)
  • h ≈ 0.1 / 0.12566 ≈ 0.7958 m

Height ≈ 0.796 m (about 79.6 cm)

Real tanks need freeboard above the working volume — add 5–10 % to the calculated height for headspace and a vent allowance.

Cylinder Formulas

All cylinder properties are determined by two dimensions: the base radius r and the height h. From those, the volume, two surface areas, diameter, and base circumference all follow:

Volume equals pi times radius squared times heightTotal surface area equals two pi r times the quantity r plus hLateral surface area equals two pi r hRadius equals the square root of volume divided by pi hHeight equals volume divided by pi r squared
Right circular cylinder with labeled radius r and height hrh

Where:

  • V — enclosed volume (m³, L, gal, ft³, etc.)
  • S — total surface area: 2 circular end caps + curved side (m², ft², in²)
  • L — lateral surface area: curved side only, unrolls into a rectangle of width 2π r and height h
  • r — base radius (m, cm, in, ft, yd)
  • h — height between the two end caps (same units as r)
  • d — diameter (d = 2 r), commonly used to spec real-world cylinders such as pipes and drums
  • C — base circumference (C = 2π r), the perimeter of either end cap

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