Cross sectional area of a cylinder
The cross sectional area of a cylinder, for a cut made square to its axis, is A = πr²: the area of the circle at its base. With the diameter, A = πd² ÷ 4. A cylinder with a 3 in radius has a cross section of π × 3² ≈ 28.27 in², whatever its height.
Contents
A = πr² for a straight cut. With r = 3 in: A = π × 9 = 28.27 in². Slanted cut at 30°: 28.27 ÷ cos 30° = 32.65 in². Lengthwise cut through the axis, h = 8 in: 2 × 3 × 8 = 48 in².
The calculator above handles the three kinds of cut: straight across (a circle), slanted (an ellipse) and lengthwise (a rectangle). Below are the formulas behind each one and two everyday uses: flow in a pipe and stress in a rod.
The three cross sections of a cylinder
A cross section is the flat shape you would see if you sliced the solid with a plane. A cylinder gives three families of shapes, depending on the angle of the cut.
| Cut | Shape | Area | r = 3 in, h = 8 in |
|---|---|---|---|
| Square to the axis | Circle of radius r | πr² | 28.27 in² |
| Tilted by θ from square (between the ends) | Ellipse, semi-axes r and r ÷ cos θ | πr² ÷ cos θ | θ = 30° → 32.65 in² |
| Along the axis | Rectangle 2r × h | 2rh | 48 in² |
| Parallel to the axis, at distance a from it | Rectangle 2√(r² − a²) × h | 2h√(r² − a²) | a = 2 in → 35.78 in² |
When people ask for "the" cross section of a cylinder, they mean the first line: the circle. It is the same at every height, which is what makes a cylinder a cylinder.
Straight cut: A = πr²
Slice a right cylinder anywhere between its ends, square to the axis, and you get a copy of the base. Its area is the area of a circle:
A = πr² = πd² ÷ 4
| Cylinder | Radius or diameter | Cross-sectional area |
|---|---|---|
| Rod or pipe bore | d = 1 in | 0.7854 in² |
| Pipe bore | d = 2 in | 3.1416 in² |
| Can | r = 3 in | 28.274 in² |
| Tank | d = 4 ft | 12.566 ft² |
| Jar | d = 10 cm | 78.540 cm² |
| Pipe bore | d = 50 mm | 1,963.5 mm² |
Doubling the diameter multiplies the area by four: a 2 in bore has four times the open area of a 1 in bore. This square law is why small changes in pipe size matter so much for flow.
The cross-sectional area is also the key to the volume. Multiply the area of the slice by the length over which it repeats and you get V = πr²h; the full method is on the page about how to find the volume of a cylinder, and the cylinder volume calculator does it in one step.
Slanted cut: an ellipse
Tilt the cutting plane by an angle θ away from square, and the slice stretches in one direction. The short axis stays the diameter, 2r. The long axis grows to 2r ÷ cos θ. The area of an ellipse is π × (semi-axis) × (semi-axis), so:
A = π × r × (r ÷ cos θ) = πr² ÷ cos θ
| Tilt θ | Long semi-axis (r = 3 in) | Area |
|---|---|---|
| 0° (square cut) | 3 in | 28.27 in² |
| 30° | 3.46 in | 32.65 in² |
| 45° | 4.24 in | 39.99 in² |
| 60° | 6.00 in | 56.55 in² |
At 60°, cos θ = 0.5, so the area doubles. As θ approaches 90° the ellipse becomes infinitely long, which is why the calculator only accepts angles below 90°. The formula assumes the slanted plane cuts the curved side all the way around; if it runs into one of the flat ends, the slice is only part of an ellipse. This is the shape you see when you cut a pipe or a dowel at an angle, for example to join two pipes at a corner.
Lengthwise cut: a rectangle
Cut the cylinder parallel to its axis and the slice is a rectangle as tall as the cylinder. Through the center, its width is the diameter, so A = 2r × h. For r = 3 in and h = 8 in, that is 6 × 8 = 48 in². In metric, a jar 10 cm across and 15 cm tall cut down the middle shows a 10 × 15 = 150 cm² face.
Off-center, the width is the chord of the circle at that distance a from the axis: 2√(r² − a²). At a = 2 in from the axis of the same cylinder, the width is 2√5 ≈ 4.47 in and the area 35.78 in². Cutting a cylinder lengthwise through the center leaves two half cylinders; their volume and flat face are on the half cylinder volume page.
Cross section of a hollow cylinder (pipe wall)
A tube cut square to its axis shows a ring. Its area is the outer circle minus the inner one:
A = π(R² − r²)
With an outer radius of 3 in and an inner radius of 2 in, A = π(9 − 4) = 15.71 in². That ring is the material that carries load in a tube; the open circle inside, πr², is what carries fluid. The hollow cylinder volume calculator multiplies the ring by the length.
Using the cross-sectional area: flow and stress
Flow in a pipe: Q = A × v
The volume of fluid passing through a pipe each second is the cross-sectional area of the bore times the average speed of the fluid: Q = A × v. Use the inside diameter.
- ImperialA 2 in bore has A = π × 1² = 3.1416 in² = 0.021817 ft². At 5 ft/s, Q = 0.021817 × 5 = 0.10908 ft³/s. Multiply by 7.48052 gal/ft³ and by 60 s: about 48.96 US gallons per minute.
- MetricA 50 mm bore has A = π × 0.025² = 0.0019635 m². At 1.5 m/s, Q = 0.0029452 m³/s, which is 2.945 L/s or about 176.7 L/min.
Turned around, v = Q ÷ A: the same flow forced through a smaller bore moves faster. To find how much water a length of pipe holds rather than how fast it passes, use the pipe volume calculator.
Stress in a rod: σ = F ÷ A
Average normal stress is the force pulling or pushing along a rod divided by its cross-sectional area. This is the general definition; real design also depends on the material, safety factors and the applicable codes.
- ImperialA rod 0.5 in in diameter has A = π × 0.25² = 0.19635 in². Under a pull of 2,000 lbf, σ = 2,000 ÷ 0.19635 ≈ 10,186 psi.
- MetricA rod 12 mm in diameter has A = π × 6² = 113.10 mm². Under 10 kN (10,000 N), σ = 10,000 ÷ 113.10 ≈ 88.42 N/mm², which is 88.42 MPa.
Common mistakes
- Using the diameter as the radius. π × 2² for a 2 in pipe gives 12.57 in², four times the real 3.14 in².
- Confusing the cross section with the surface. The slice πr² is inside the solid. The skin of the cylinder is the surface area of a cylinder, 2πr² + 2πrh, and the curved side alone is the lateral area, 2πrh.
- Outside diameter for flow. The fluid only sees the bore. Use the inside diameter.
- Mixing units in Q = A × v. An area in in² and a speed in ft/s must be put in the same length unit first (144 in² = 1 ft²).
A cross section is a slice, not a skin: its area never depends on the height of a straight cut, only on the radius.
The flat pattern that wraps the outside, rather than the slice through the inside, is shown on the net of a cylinder page.
FAQ
What is the formula for the cross-sectional area of a cylinder?
For a cut square to the axis, A = πr², the area of the circular base. With the diameter, A = πd² ÷ 4. A cylinder 3 in in radius has a cross-sectional area of π × 9 ≈ 28.27 in².
What shape is the cross section of a cylinder?
It depends on the cut. A cut square to the axis gives a circle, a slanted cut that stays between the two ends gives an ellipse, and a cut parallel to the axis gives a rectangle.
Is the cross-sectional area the same as the base area?
For a straight cut, yes: every slice square to the axis of a right cylinder is the same circle as the base, πr². This is why the volume is that area times the height, πr²h.
How do you find the cross-sectional area of a pipe?
Use the inside diameter: A = π × (inside diameter)² ÷ 4. That is the open area the fluid flows through. For the area of the pipe wall itself, subtract the inside circle from the outside circle: π(R² − r²).
What is the area of a cylinder cut lengthwise?
A cut through the axis gives a rectangle 2r wide (the diameter) and h tall, so A = 2rh. For r = 3 in and h = 8 in, A = 48 in². A cut parallel to the axis but off-center gives a narrower rectangle.
Related
- Surface area of a cylinderThe surface area of a cylinder is A = 2πr² + 2πrh. Get the total, lateral and base areas from a radius or diameter, with worked examples in inches and cm.
- Pipe volumePipe volume calculator: V = π × r² × L from the inside diameter. Gallons of water per foot of pipe, liters per meter, a full chart and worked examples.
- Lateral areaThe lateral area of a cylinder is L = 2πrh, the curved side only. Calculator, why the formula works, and examples with can labels and painted poles.
Updated October 12, 2026