Volume of Cylinders, Cones & Spheres

Three formulas, one family. A cone is a third of its cylinder, and a sphere is two thirds of it.

Take a cone and a can with the same radius and the same height. Fill the cone with water and pour it into the can.

It does not fill it. Fill the cone again and pour — still not full.

fill the cone, pour it in — three times exactly fills the can pour 1 pour 2 pour 3 same radius, same height three coneloads, not two and not four

Fill it a third time, pour, and the can is exactly full. Not nearly. Exactly.

Three coneloads. Not two, not four — and it works for any radius and any height you choose.

So a cone holds precisely one third of the can that would wrap around it.

That is where the 13 in V = 13 π r2 h comes from. It is not an arbitrary constant to memorise; it is something you can pour. A sphere the width of the can takes two of the three — hence 43 π r3.

Volume of a cylinder, a cone and a sphere three round solids, three formulas worth knowing cold r h r cylinderconesphere V = pi r^2 h V = (1/3) pi r^2 h V = (4/3) pi r^3 a cone holds exactly one third of the cylinder that would wrap around it

The cylinder is the parent

A cylinder is a circle dragged upwards: area of the base times the height, so V = π r2 h.

A cone with the same base and height holds exactly one third as much. Fill a cone with water three times and it fills the cylinder.

A sphere that just fits inside the cylinder holds exactly two thirds of it. That is Archimedes' result, and he was proud enough of it to have it carved on his tomb.

A cylinder, the cone inside it and the sphere inside it, all with radius r and height 2r: the cone holds one third and the sphere two thirds one cylinder, radius r and height 2r — and what fits inside it r 2r cylinderconesphere πr²h = 2πr³⅓πr²h⁴⁄₃πr³ the wholeone thirdtwo thirds cone + sphere = ⅓ + ⅔ = the whole cylinder

Learn one, get three

Cylinder π r2 h. Cone is a third of that. Sphere is 43π r3 — and for a sphere the height is 2r, so it is two thirds of the cylinder around it.

Radius, not diameter

Almost every mistake in this unit is using the diameter where the formula wants the radius.

If you are told the diameter, halve it before you do anything else. Write the radius down.

Since r gets squared or cubed, using the diameter makes the answer wrong by a factor of 4 or 8 — not a near miss.

A cylinder of diameter 10 has radius 5; squaring the diameter by mistake gives 100 instead of 25, four times too big the formula wants the radius — halve the diameter first d = 10 r = 5 h = 6 π × 5² × 6 = 150π ✓ with d by mistake: π × 10² × 6 = 600π 5² 55 1010 r² = 25, but d² = 100 four of the small squares: 600π is 4 times too big squared, the error is ×4; cubed, as in a sphere, it is ×8

Halve it first, in writing

Diameter 10 means r = 5. Write "r = 5" on its own line before substituting. Do not try to remember it mid-calculation.

Working through a calculation

Write the formula. Substitute the numbers. Simplify step by step. Then attach cubic units.

r = 3, h = 10 for a cylinder: V = π(3)2(10) = π(9)(10) = 90π ≈ 282.7.

Leave the answer in terms of π if you can — it is exact, and it shows the working clearly.

Square before multiplying

In π r2 h, deal with r2 first, then multiply by h. Doing (rh)2 by accident is a common slip.

Working backwards

Given the volume and asked for a radius or height, substitute what you know and solve the equation.

Volume = 100π for a cylinder of height 4: 100π = π r2 (4), so r2 = 25 and r = 5.

Divide by π early — it cancels, and the arithmetic gets much easier.

Composite solids

A silo is a cylinder with a cone or half-sphere on top. Find each volume separately and add.

Watch the heights: the cylinder's height is its own, not the total height of the object.

For "how much is left" problems, subtract the smaller volume from the larger one.

Two silos of radius 3: a cylinder of height 8 with a cone of height 4 on top, 84π, and with a half-sphere on top, 90π find each part separately, then add r = 3 84 total 12 cylinder + cone r = 3 83 total 11 cylinder + half-sphere π×3²×8 + ⅓π×3²×4= 72π + 12π = 84π π×3²×8 + ½ × ⁴⁄₃π×3³= 72π + 18π = 90π the cylinder’s height is its own 8, not the total height of the silo

Worked examples

Cylinder from a diameter

A can has diameter 8 cm and height 12 cm. Find its volume in terms of π.

  1. r = 8 ÷ 2 = 4 cm The formula needs the radius. Write it down.
  2. V = π r2 h = π (4)2 (12) Substitute into the cylinder formula.
  3. = π (16)(12) = 192π Square first, then multiply.
  4. ≈ 603.2 cm3 Cubic units, because it is a volume.

192π ≈ 603.2 cm3

Cone

Find the volume of a cone with radius 6 and height 10, in terms of π.

  1. V = 13π r2 h = 13π (6)2 (10) A cone is a third of its cylinder.
  2. = 13π (36)(10) = 360π3 Square, multiply, then take the third.
  3. = 120π The matching cylinder would hold 360π ✓

120π

Working backwards

A sphere has volume 500π3. Find its radius.

  1. 43π r3 = 500π3 Substitute into the sphere formula.
  2. Divide both sides by π: 43r3 = 5003 π cancels straight away.
  3. Multiply by 3: 4r3 = 500, so r3 = 125 Clear the fractions.
  4. r = √[3]125 = 5 Check: 43π(125) = 500π3 ✓

r = 5

Words to know

Cylinder
Two parallel circles joined by a curved surface. V = π r2 h.
Cone
A circle tapering to a point. V = 13π r2 h.
Sphere
A ball. V = 43π r3.
Radius
Half the diameter. Every one of these formulas wants the radius.

Mistakes to watch for

Substituting the diameter where the radius belongs. Since r gets squared or cubed, using 10 instead of 5 makes the answer four or eight times too big — not a near miss. Write "r = 5" on its own line before you start.

Forgetting the 13 for a cone. The answer then comes out three times too large, and it looks entirely reasonable. Pouring three coneloads into the can is the picture that stops this.

Squaring the wrong thing. In π r2 h only the radius is squared, so deal with r2 first and then multiply by h. Doing (rh)2 by accident is an easy slip when the numbers are written close together.

Giving square units for a volume. Volume is a length times a length times a length, so the units are cubic. Writing cm2 for a volume signals that something was multiplied one time too few.

Archimedes proved the sphere is two thirds of its cylinder — and was so pleased that he asked for a sphere inside a cylinder to be carved on his tomb.

He got his wish. Around 75 BC the Roman orator Cicero found the neglected, overgrown grave near Syracuse by looking for that carving.

It is the only monument we know of chosen by a mathematician to commemorate a theorem rather than a life.

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