Every displacement figure on this site comes from one short equation, and it is worth deriving rather than just quoting, because the derivation explains a lot of engine behaviour. A cylinder is a circular prism: the piston sweeps a circle of area A up a height h, so the swept volume is simply A × h. Write the circle's area from its diameter and you have the whole formula. The live calculator above lets you try it on your own numbers as you read.
Building the equation
The area of a circle is usually written πr², but engineers measure bore as a diameter, not a radius. Substitute r = d/2 and πr² becomes π(d/2)² = (π/4)d². That is where the 0.785398 comes from — it is π/4, the constant that turns a diameter straight into an area so you never have to halve the bore yourself.
Why bore is squared and stroke is not
Bore sets the circle's area, and area depends on diameter squared; stroke sets the height, which enters linearly. The practical consequence is large: double the bore and a cylinder's volume roughly quadruples, but double the stroke and it only doubles. It is why a wide bore is such a powerful lever on displacement, and why a small overbore still adds real volume.
Two worked examples, both units
Imperial: a V8 with a 4.00-inch bore and 3.48-inch stroke. 8 × 0.785398 × 4.00² × 3.48 = 349.85 cubic inches — the "350". Multiply by 16.387064 for 5,733 cc.
Metric: a single with a 76 mm bore and 55 mm stroke. 0.785398 × 76² × 55 = 249,500 cubic millimetres; divide by 1,000 for 249.5 cc — the "250". Multiply by the cylinder count for a multi.
The unit discipline the formula demands
Because you multiply bore by stroke, they must share a unit first — mixing millimetres and inches multiplies mismatched scales and produces nonsense. Work in millimetres and the result is cubic millimetres (divide by 1,000 for cc); work in inches and it is already cubic inches. The calculator converts internally so you can enter either, but the principle is the reason the mm/inch toggle converts the value rather than just relabelling it.
What the formula deliberately leaves out
Nothing in the equation mentions valves, fuel, aspiration or the combustion chamber — and that is the point. Displacement is pure geometry, which is exactly why it is stable enough to define racing classes and legislate tax around. Chamber volume belongs to compression ratio, a separate calculation; power belongs to airflow. Keeping displacement geometric is what makes it the one engine number you can trust not to move.
References & standards
- NIST SP 811 — Guide for the use of the SI, including the exact inch and unit-conversion factors.
- SAE J1349 — Engine power test code — the standard behind 'SAE net' crankshaft ratings.
Frequently asked questions
What is the engine displacement formula?
For a reciprocating piston engine, displacement = number of cylinders × π/4 × bore² × stroke, with bore and stroke in the same length unit before you multiply.
Why is bore squared in the formula?
Because the piston sweeps a circular cylinder, and a circle's area is π/4 × diameter². Bore is that diameter, so it enters as a square — doubling bore roughly quadruples per-cylinder volume.
What does the π/4 term represent?
It is the constant that turns a diameter into a circle's area: area = π/4 × d². Writing it this way lets you use the measured bore directly instead of first halving it to a radius.
Is bore the diameter or the radius?
The full diameter. The π/4 form already accounts for the radius internally, so you enter the bore you measure straight across the cylinder.
Where exactly is stroke measured?
From top dead centre to bottom dead centre — the piston's full travel. If you only know the crank throw, stroke is exactly twice that throw.
Why multiply by the number of cylinders?
Because displacement is the total swept volume of the whole engine. One cylinder's volume times the cylinder count gives the sum across every cylinder.
Can I try the formula with my own numbers?
Yes — this page has a live calculator, so you can drop your bore, stroke and cylinder count in and watch the formula trace show the exact substitution.
Does the formula change for a V engine?
No. The layout — inline, V, flat — never enters the maths; you always sum every cylinder's swept volume with the same equation.
Why must bore and stroke share a unit?
Because you are multiplying them; mixing millimetres and inches multiplies mismatched scales and gives a meaningless number. Convert one first so both agree.
How do I get cc versus cubic inches from the formula?
Work in millimetres and divide the mm³ result by 1000 for cc; work in inches and the result is already cubic inches. The tool shows both regardless of input.
Does the formula give combustion-chamber volume?
No — only swept volume. Chamber volume is separate and belongs to compression ratio, which is a different calculation entirely.
What is the per-cylinder version of the formula?
Just drop the cylinder count: π/4 × bore² × stroke gives one cylinder's swept volume, which the tool shows next to the total.
Why do worked examples use a 350 and a 250?
Because they anchor the maths to familiar engines — a 4.00 × 3.48 V8 gives 349.85 CID, and a 76 × 55 single gives about 249.5 cc — so you can see the formula produce a real badge number.
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