Volume of Right Cylinders

What a cylinder is

A cylinder is a solid 3-D shape with two parallel congruent circular bases and one curved lateral surface wrapping around between them. The two bases are flat disks of the same radius; the lateral surface is curved, so it's called a "surface," not a "face."

This is the curriculum's reason for using different vocabulary for cylinders and prisms. A prism has lateral faces because each side is a flat rectangle. A cylinder has a lateral surface because the side is a single curved sheet. Same role in the shape, different geometry.

A right cylinder is one where the lateral surface is perpendicular to the bases. Every cylinder in this lesson is a right cylinder.

Two dimensions

A cylinder stretches in just two independent directions: the radius of the circular base, and the height between the two bases.

The volume formula is the same V = Bh as for prisms, but the base is a circle, so B = πr²:

Using π ≈ 3.14, a cylinder with radius r = 3 cm and height h = 5 cm has:

Layer-iteration: stacking πr² disks

The cylinder is the prism with a circular base. The same layer-iteration logic applies: stack h thin circular disks, each one of area πr². Total volume = (area of one disk) × (number of disks stacked):

That picture is V = πr²h made visible. Doubling h doubles the number of disks, doubling the volume. Doubling r quadruples each disk's area (π · (2r)² = 4πr²), and so quadruples the volume.

Radius squared, height linear

The two dimensions don't enter the formula symmetrically. The height appears once, so doubling it doubles the volume. The radius appears squared, so doubling it quadruples the volume.

That asymmetry has real consequences. A storage tank that's twice as wide (same height) holds four times as much. A tank that's twice as tall (same width) holds twice as much. Wider, not taller, is what dominates capacity. That's why most real storage cylinders (grain bins, oil tanks, water reservoirs) tend toward "fat and short" rather than "thin and tall."

See it: stretch radius and height

The widget below has two sliders: radius and height. The volume readout updates live. Try r = 3, h = 4 first to get a feel for the baseline. Then try doubling r alone (set r = 6, h = 4) and notice the volume goes up by a factor of 4. Then return r to 3 and double h (set r = 3, h = 8); the volume goes up by a factor of 2.

The widget's readout shows three things together: the base area B = πr², the layer-iteration form B × h, and the simplified numerical volume. Same formula, three forms.

Units must match

Volume of a cylinder is in cubic units: cm³, m³, L (1 L = 1000 cm³), and so on. Radius and height must be in the same unit before multiplying, same rule as for prisms.

If r = 20 cm and h = 2 m, you have to pick one unit first. Convert height to centimetres (h = 200 cm) and the volume comes out in cm³. Or convert radius to metres (r = 0.20 m) and the volume comes out in m³. Same physical cylinder, two valid expressions:

Multiplying cm × cm × m without converting produces a number that isn't a volume in any unit. The arithmetic still gives an answer, but the answer is nonsense.

Where it shows up in real life

A cylindrical oil tank at a fuel station, or the smaller home oil tank a rural family uses for heating, is a right cylinder. The manufacturer specifies the radius and height, and the capacity in litres follows directly from V = πr²h (then converted to litres via 1 m³ = 1000 L).

A grain silo on an Alberta farm is a tall vertical cylinder. Producers calculate capacity in bushels by computing the cubic metre volume from V = πr²h and converting (1 bushel ≈ 0.0363 m³ for wheat). The radius of the bin and the fill height are what determine how much grain it can hold.

A round straw bale is itself a short, fat cylinder, much shorter than it is wide. A typical bale is roughly 1.5 m in diameter (so r ≈ 0.75 m) and 1.2 m in height, giving a volume near π × 0.5625 × 1.2 ≈ 2.12 m³ of straw. Producers stack and transport bales by the trailer load; knowing the volume tells them how many fit on a truck.

In each case, the cylinder's capacity is a real number a farmer or operator has to measure, store, or transport. V = πr²h is the calculation that turns "tank diameter × tank height" into a purchase order.

Worksheet

These aren't graded. Get them right, get them wrong. Focus on the radius-squared structure and on keeping units consistent.

Going further

V = Bh extends well beyond cylinders and prisms. Whenever a 3-D shape has a constant cross-section running through it (meaning every horizontal slice is the same polygon or circle), its volume equals the area of that cross-section times the length the shape extends. That's the entire pattern of Grade 7 volume work.

In Grade 8, you'll meet pyramids (a tapering version of a prism) and cones (a tapering version of a cylinder). Their volume formulas pick up a factor of one third: V_pyramid = ⅓ × B × h and V_cone = ⅓ × πr² × h. The reason for the one-third is geometric (three congruent pyramids tile a prism), but the underlying Bh shape is still recognizable.

In Grade 9, you'll calculate surface area as well as volume. For a cylinder, the lateral surface unrolls into a rectangle of dimensions (2πr) × h (the circumference times the height), with total surface area 2πr² + 2πrh. The lateral SURFACE, curved rather than flat, is what makes a cylinder different from a prism in its surface-area calculation too.