Volume of Cylinder Equation, Derivation & Rearrangements

Core idea

Overview

The volume of a cylinder represents the total three-dimensional space occupied by a solid with two parallel circular bases. It is calculated by taking the area of the base (πr²) and extending it through the perpendicular height (h) of the object.

When to use: Use this formula when calculating the capacity of containers with a uniform circular cross-section, such as pipes, tanks, or jars. It assumes the cylinder is a 'right cylinder', meaning the height is perfectly perpendicular to the base.

Why it matters: This equation is vital in fluid dynamics and mechanical engineering for determining the displacement of pistons or the volume of fuel stored in silos. It helps professionals calculate material costs for manufacturing cylindrical components and estimate the weight of liquid contents.

Symbols

Variables

r = Radius, h = Height, V = Volume

r

Radius

m

h

Height

m

V

Volume

m^{3}

Walkthrough

Derivation

Formula: Volume of a Cylinder

A cylinder is a prism with a circular cross-section. Its volume is the area of the circle base multiplied by the cylinder's height.

The cylinder is a right cylinder (not slanted).

1

Find the Base Area:

Calculate the area of the circular base.

A = π r^{2}

2

Multiply by Height:

Multiply the circular area by the perpendicular height (h) of the cylinder.

V = π r^{2} h

Result

V = π r^{2} h

Source: AQA GCSE Maths — Geometry and Measures

Free formulas

Rearrangements

Solve for $V$

Make V the subject

V = π r^{2} h

V is already the subject of the formula.

Difficulty: 1/5

Solve for $r$

Make r the subject

r = \frac{V}{π h}

To make $r$ the subject of the formula for the volume of a cylinder, first divide both sides by $π h$ , then take the square root of both sides.

Difficulty: 2/5

Solve for $h$

Make h the subject

h = \frac{V}{π r ^{2}}

Rearrange the formula for the volume of a cylinder to solve for its height, h.

Difficulty: 2/5

The static page shows the finished rearrangements. The app keeps the full worked algebra walkthrough.

Visual intuition

Graph

The graph follows a parabolic curve because the radius is squared in the volume formula. Since the radius must be positive, the graph starts at the origin and increases at an accelerating rate as the radius grows. For a student, this shape shows that small changes in radius result in minimal volume gains, while large radius values lead to rapid, exponential increases in total volume. The most important feature is the steepening slope, which demonstrates that the volume is highly sensitive to even minor increases in

Graph type: parabolic

Why it behaves this way

Intuition

Imagine the volume of a cylinder as the area of its circular base (πr2) being uniformly extended or 'swept' upwards through its perpendicular height (h).

V

Volume of the cylinder

The total three-dimensional space enclosed within the cylinder's boundaries, representing its capacity.

π

Mathematical constant, approximately 3.14159

A fundamental constant that scales the square of the radius to give the area of a circle, which forms the base of the cylinder.

r

Radius of the circular base of the cylinder

The distance from the center to the edge of the circular base; a larger radius means a wider cylinder and thus a larger base area.

h

Perpendicular height of the cylinder

The distance between the two parallel circular bases; a greater height means a taller cylinder and thus more 'stacked' area.

Signs and relationships

r^2: The radius is squared because the volume scales with the area of the circular base, which itself depends quadratically on the radius.

Free study cues

Insight

Canonical usage

To calculate the volume of a cylinder, ensuring that the radius and height are expressed in consistent length units.

Common confusion

A common mistake is using different length units for radius and height (e.g., radius in cm and height in m) without converting one to match the other, leading to an incorrect volume value or unit.

Unit systems

$V$ m3 · The unit of volume will be the cube of the length unit used for radius and height (e.g., if r and h are in cm, V will be in cm3; if r and h are in dm, V will be in dm3 or L).

$r$ m · Must be expressed in the same length unit as height (h) for consistent calculation.

$h$ m · Must be expressed in the same length unit as radius (r) for consistent calculation.

One free problem

Practice Problem

Find the volume of a cylindrical silo with a radius of 4 m and a height of 10 m.

Radius4 m

Height10 m

Solve for: $V$

Hint: V = π * r² * h.

The full worked solution stays in the interactive walkthrough.

Where it shows up

Real-World Context

Calculating the volume of a soda can (radius 3cm, height 12cm) to determine how much liquid it holds.

Study smarter

Tips

Always verify that the radius (r) and height (h) are expressed in the same units before starting the calculation.
If you are provided with the diameter, remember to divide it by two to obtain the radius.
For higher precision in engineering contexts, use the π constant on a calculator rather than rounding to 3.14.

Avoid these traps

Common Mistakes

Using diameter instead of radius (diameter = 2r)
Forgetting to square the radius
Forgetting π in the calculation
Mixing units (e.g., radius in cm but height in m)

Keep going

Related Formulas

Common questions

Frequently Asked Questions

A cylinder is a prism with a circular cross-section. Its volume is the area of the circle base multiplied by the cylinder's height.

Use this formula when calculating the capacity of containers with a uniform circular cross-section, such as pipes, tanks, or jars. It assumes the cylinder is a 'right cylinder', meaning the height is perfectly perpendicular to the base.

This equation is vital in fluid dynamics and mechanical engineering for determining the displacement of pistons or the volume of fuel stored in silos. It helps professionals calculate material costs for manufacturing cylindrical components and estimate the weight of liquid contents.

Using diameter instead of radius (diameter = 2r) Forgetting to square the radius Forgetting π in the calculation Mixing units (e.g., radius in cm but height in m)