Geometry - April 16, 2026
Cylinder Volume Formula: Derivation, Proof, and Why It Works
The cylinder volume formula is not just something to memorize. It comes directly from one geometric idea: volume = base area x height. This guide derives the formula, proves why it works, and adds interactive visuals to make each step concrete.
Quick Answer
The cylinder volume formula is V = pi x r^2 x h. Here, r is the radius of the circular base and h is the perpendicular height. In plain language, the formula says: take the area of one base circle and extend it through the cylinder's height.
If you want the answer instantly, use the cylinder volume calculator. If you want to understand the reasoning first, start with the interactive overview below.
Everything in the formula comes from the shape: a circular base and a perpendicular height.
Derivation
The derivation starts with one standard fact: the area of a circle is pi x r^2. Since a cylinder has a circular base, its base area is the same value.
Next use the general volume rule for solids with constant cross-section: volume = base area x height. Substituting the circle's area gives:
- Base area = pi x r^2
- Volume = base area x height
- Volume = (pi x r^2) x h
- So, V = pi x r^2 x h
If you already know radius and height, the Cylinder Volume Calculator Using Radius follows this exact derivation. If you need all related measurements together, the Cylinder Geometry Calculator is the best companion tool.
The derivation is just circle area multiplied by perpendicular height.
Proof
A clean proof comes from slicing the cylinder into many thin disks. Each disk has the same radius, so each disk has the same area pi x r^2. Stack enough of those equal slices through height h, and you recover the entire cylinder.
That is why the formula works: the cross-sectional area stays constant from top to bottom. The interactive slider below increases the number of slices so you can see the stacking argument more clearly.
More slices make the equal-area stacking idea easier to see.
Why It Still Works for Oblique Cylinders
The formula does not break when a cylinder leans. An oblique cylinder still has parallel, congruent bases. The only thing that matters is the perpendicular height between them, not the slanted side length.
That is why the Oblique Cylinder Calculator still uses the same volume law. The visual below lets you tilt the top base while keeping the volume idea anchored to perpendicular height.
Leaning changes the look, not the rule: volume still equals base area times perpendicular height.
Diameter and Circumference Forms
Real measurements often start with diameter or circumference instead of radius. The formula stays the same in spirit, but the input changes:
- Using diameter: V = pi x d^2 x h / 4
- Using circumference: V = C^2 x h / (4pi)
Use the diameter calculator if you measured straight across, and the circumference calculator if you wrapped a flexible tape around the cylinder.
Radius form is the direct form of the equation.
Worked Example
Let radius = 4 cm and height = 10 cm. Then:
V = pi x 4^2 x 10 = pi x 16 x 10 = 160pi approx 502.65 cm^3
The slider below updates both the geometry and the arithmetic. If you want the final answer in other units, jump to the liters calculator or gallons calculator.
Relevant Calculators
FAQs
What is the formula for cylinder volume?
The formula is V = pi x r^2 x h, where r is the radius and h is the perpendicular height.
Why is the radius squared?
Because the circular base area is pi x r^2. Cylinder volume is base area times height.
Does the formula work for an oblique cylinder?
Yes. It still works if h is the perpendicular distance between the parallel bases.
How do you use diameter instead of radius?
Replace r with d/2. That gives V = pi x d^2 x h / 4.
How do you use circumference instead of radius?
Use r = C / (2pi). That leads to V = C^2 x h / (4pi).