The cylinder volume formula (and why it works)

A cylinder is one of the simplest 3D shapes to understand because it’s basically a stack of circles. Imagine a soup can: the top is a circle, the bottom is a circle, and the sides connect them. If you could slice the can into super-thin “coin” layers, each slice would be a circle with the same radius. Stack enough slices and you get the whole cylinder.

That mental model leads directly to the standard formula: V = πr²h. Here’s what each symbol means:

• V = volume (how much space is inside)
• π ≈ 3.14159 (the circle constant)
• r = radius of the circular base
• h = height of the cylinder

The key idea: volume is base area × height. For a cylinder, the base is a circle, so its area is A = πr². Multiply that area by the height and you get the volume: V = A × h = (πr²)h.

If you only know the diameter instead of radius, no problem. Diameter is the full width across the circle, while radius is half of that. So: r = d/2. Substituting into the volume formula gives: V = π(d/2)²h = π(d²/4)h. This calculator automatically does that conversion when you switch to “Diameter + Height.”

The reason cylinder volume changes “fast” is the square on the radius. Height affects volume linearly: double the height, double the volume. But radius affects the base area, and base area goes with r². That’s why even a small increase in radius can create a big increase in capacity — which is exactly why storage tanks and pipes are often designed around diameter constraints.

In pure geometry, radius is just the distance from the center of a circle to its edge. In real objects, it depends on what you’re measuring. If you’re calculating how much water fits inside a pipe, use the inner radius. If you’re calculating how much concrete is needed to pour a cylindrical column, you usually use the outer radius. If a cylinder has thickness (pipe walls, cup walls, insulation), inside and outside radii can be different — and that can change volume dramatically because radius is squared.

• Mixing units: If radius is in cm but height is in m, convert one so they match before calculating.
• Confusing radius vs diameter: Plugging diameter into r makes the answer 4× too large.
• Forgetting cubic units: Volume uses cubic units (cm³, m³, in³), not square units.
• Rounding too early: Keep extra decimals during steps, then round at the end.

A helpful mental check: if your cylinder is a “can,” its volume should feel like “circle area × height.” If the cylinder is short and wide, volume may still be large (big circle). If it’s tall and skinny, volume could be small (tiny circle). This relationship is why radius dominates.

Examples you can copy (with steps)

Example 1: Radius + height (cm)

Suppose r = 7.5 cm and h = 20 cm.
V = πr²h = π(7.5)²(20)
(7.5)² = 56.25
56.25 × 20 = 1125
V = 1125π ≈ 3534.29 cm³

Convert to liters:
1 liter = 1000 cm³ → 3534.29 cm³ ≈ 3.534 L

Example 2: Diameter + height (in)

Suppose d = 10 in and h = 12 in.
Radius r = d/2 = 5 in
V = π(5)²(12) = π(25)(12) = 300π ≈ 942.48 in³

Convert to gallons (approx):
1 US gallon ≈ 231 in³ → 942.48 in³ ≈ 4.08 gallons

Example 3: Big tank (meters)

Suppose r = 1.2 m and h = 3 m.
V = π(1.2)²(3)
(1.2)² = 1.44
1.44 × 3 = 4.32
V = 4.32π ≈ 13.57 m³

Convert to liters (handy for tanks):
1 m³ = 1000 liters → 13.57 m³ ≈ 13,570 liters

• If r or d is 0, volume must be 0.
• If h doubles (same r), volume doubles.
• If r doubles (same h), volume quadruples.
• If a result feels “off,” re-check whether you used radius vs diameter.

What this calculator does behind the scenes

This tool follows the exact same steps you’d do by hand — just faster — and then adds conversions to the units people actually use (liters, gallons, cubic meters). It also formats a share-friendly summary so the result can travel as a screenshot or a quick message.

• Read inputs: radius (or diameter), height, unit.
• If diameter mode: compute radius via r = d/2.
• Compute base area: A = πr².
• Multiply by height: V = A × h.
• Convert volume to m³ → liters → gallons (approx) and other common cubic units.

• 1 L = 1000 cm³
• 1 m³ = 1000 L
• 1 US gallon ≈ 3.785411784 L
• 1 in = 2.54 cm (used for in³ conversion)

For maximum accuracy, measure carefully, keep your units consistent, and round only at the end. If you’re working on something safety-critical (pressure vessels, chemical storage, structural columns), follow engineering standards and use verified measurement tools.

Frequently Asked Questions

• What is the volume of a cylinder?

  It’s the amount of 3D space inside a cylinder. The standard formula is V = πr²h.

• What if I only know diameter?

  Use diameter mode. The calculator converts diameter to radius (r = d/2) automatically.

• What’s the difference between cm³ and m³?

  They’re both volume units. 1 m³ is much larger: 1 m³ = 1,000,000 cm³.

• How do I convert cubic centimeters to liters?

  Divide by 1000 because 1000 cm³ = 1 L.

• How do I find the volume of a hollow cylinder (pipe)?

  Compute the outer cylinder volume minus the inner cylinder volume using two radii: V = π(R² − r²)h.

• Does this work for “tilted” cylinders?

  If the cross-section remains a circle and the height is measured perpendicular to the base, the same formula applies. For skewed shapes, you’ll need a different model.

Cylinder volume at a glance

• Volume: V = πr²h
• Radius from diameter: r = d/2
• Base area: A = πr²
• Units: keep radius + height in the same unit before computing
• Conversions: 1000 cm³ = 1 L · 231 in³ ≈ 1 US gallon

Screenshot this box for homework or quick reference.