Oblique cylinder volume
An oblique cylinder is a cylinder whose side leans, and its volume is exactly the same as a straight one: V = πr²h, where h is the perpendicular height between the two bases. With r = 3 in and a perpendicular height of 8 in, an oblique cylinder holds π × 9 × 8 ≈ 226.19 in³, whatever the tilt. If you only know the slanted side s and its angle α with the base, use h = s × sin α.
Contents
V = π × r² × h, with h the perpendicular height. From a slant side: h = s × sin α. With r = 3 in, s = 10 in and α = 60°: h = 8.660 in and V = 244.86 in³.
The calculator above takes either the perpendicular height or the slant length with its angle, and shows the height it used.
What is an oblique cylinder?
A cylinder has two parallel, congruent circular bases. When the line joining their centers (the axis) is perpendicular to the bases, the cylinder is a right cylinder. When the axis is tilted, the cylinder is oblique: a stack of coins pushed sideways, a leaning tower, a can squashed at an angle while keeping its round ends level.
The bases stay circles and stay parallel. Only the side leans. That is why the volume does not change, and why you must not use the slanted length in the formula.
Why the volume is the same: Cavalieri's principle
Cavalieri's principle says: if two solids have the same height, and every plane parallel to the base cuts them in slices of equal area, then the solids have equal volumes.
Take a right cylinder and an oblique cylinder with the same radius r and the same perpendicular height h. Cut both at any level. Each slice is a circle of radius r, area πr². The slices of the oblique cylinder are only shifted sideways. Shifting coins in a stack does not change how much metal there is, so the volumes are equal:
V(oblique) = V(right) = πr²h.
The same principle explains the formula of the right cylinder itself, built from thin disks; the derivation of the cylinder volume formula shows both the stacking argument and the integral.
How to find the volume of a slanted cylinder
- Get the radiusMeasure across a base and halve the diameter.
- Get the perpendicular heightMeasure square to the bases, or compute it from the slant side.
- Apply V = πr²hSquare the radius, multiply by π and by the perpendicular height.
The perpendicular height from the slant
The slant side s, the perpendicular height h and the sideways offset x form a right triangle. Two ways to get h:
| You know | Height | Example |
|---|---|---|
| Slant s and angle α with the base | h = s × sin α | s = 10, α = 60° → h = 8.660 |
| Slant s and angle β with the vertical | h = s × cos β | s = 10, β = 30° → h = 8.660 |
| Slant s and sideways offset x | h = √(s² − x²) | s = 10, x = 6 → h = 8 |
Example 1: inches, slant and angle
An oblique cylinder has r = 3 in, a slant side of 10 in and leans at 60° to its base. h = 10 × sin 60° ≈ 8.660 in. V = π × 3² × 8.660 ≈ 244.86 in³, about 1.06 US gallons (÷ 231).
Using the slant length directly, π × 9 × 10 ≈ 282.74 in³, overstates the volume by about 15 %.
Example 2: centimeters
r = 5 cm, slant side 20 cm, angle 75° with the base. h = 20 × sin 75° ≈ 19.319 cm. V = π × 25 × 19.319 ≈ 1,517.3 cm³, or about 1.517 L.
Example 3: from the offset
The top of a 10 in slanted cylinder sits 6 in to the side of the bottom, and the radius is 3 in. h = √(100 − 36) = 8 in, so V = π × 9 × 8 ≈ 226.19 in³, the same as a right cylinder 8 in tall.
Example 4: a leaning column in feet
A column of radius 4 ft has an axis 30 ft long leaning at 80° to the ground. h = 30 × sin 80° ≈ 29.544 ft. V = π × 16 × 29.544 ≈ 1,485.1 ft³.
A check by the tilted cross-section
You can also slice square to the tilted axis. That slice is an ellipse with semi-axes r and r × sin α, area πr² sin α. Multiply by the axis length s: V = πr² sin α × s = πr² × (s sin α) = πr²h. For example 1, π × 9 × sin 60° × 10 ≈ 244.86 in³ again. Both routes agree, which is a good test of your numbers.
A cylinder cut at a slant
A different shape is often called a slanted cylinder too: a right cylinder whose top has been cut by a sloping plane, like a pipe end cut at an angle. Its bases are not parallel. If the cut does not reach the bottom, the volume is the base area times the average height:
V = πr² × (h₁ + h₂) ÷ 2, where h₁ and h₂ are the shortest and longest sides.
A pipe of radius 1 in cut so that one side is 4 in long and the other 6 in long holds π × 1 × 5 ≈ 15.71 in³, exactly halfway between the 4 in cylinder (12.57 in³) and the 6 in cylinder (18.85 in³). In metric: r = 10 cm, sides of 30 cm and 40 cm, V = π × 100 × 35 ≈ 10,996 cm³, about 11.0 L.
Surface area of an oblique cylinder
The two bases still add 2πr². The curved side is different: its area equals the perimeter of the tilted cross-section (the ellipse above) times the slant length s. The perimeter of an ellipse has no exact formula in elementary terms, so this area is computed numerically or with an approximation. For example 1, the ellipse has semi-axes 3 in and 2.598 in, a perimeter of about 17.61 in, so the curved side is about 176.1 in², more than the 163.2 in² of the right cylinder with the same height. Do not use 2πrh or 2πrs for an oblique cylinder.
Common mistakes
- Using the slant length as h. The answer is too large by a factor 1 ÷ sin α.
- Using the cosine of the angle with the base. h = s × sin α when α is measured from the base; cos goes with the angle from the vertical.
- Calculator in radians. sin 60 in radian mode is −0.305. Switch to degrees.
- Confusing oblique and cut-at-a-slant. The first has parallel bases; the second uses the average height.
Same base, same perpendicular height, same volume. The tilt never matters for the volume of an oblique cylinder.
For the general method, read how to find the volume of a cylinder. To solve for the height when the volume is given, use the page on the height of a cylinder, and for a straight cylinder in one step, the cylinder volume calculator. Oval bases, rather than tilted circles, are covered on the elliptical cylinder volume page.
FAQ
What is the formula for the volume of an oblique cylinder?
V = πr²h, the same as for a right cylinder, where r is the radius of the circular base and h is the perpendicular height between the two bases, not the length of the slanted side.
Why does an oblique cylinder have the same volume as a right one?
By Cavalieri's principle: if two solids have the same height and every horizontal slice of one has the same area as the matching slice of the other, their volumes are equal. Every horizontal slice of an oblique cylinder is a circle of area πr², just like in the right cylinder.
How do you find the volume of a slanted cylinder from the slant length?
Find the perpendicular height first: h = s × sin α, where s is the slant side and α the angle it makes with the base. Then V = πr²h. With r = 3 in, s = 10 in and α = 60°, h ≈ 8.660 in and V ≈ 244.86 in³.
How many faces, edges and vertices does an oblique cylinder have?
The same as a right cylinder: two flat circular faces, one curved face, two circular edges and no vertices. Tilting the side does not add or remove any face.
What is the difference between an oblique cylinder and a right cylinder?
In a right cylinder the axis joining the centers of the bases is perpendicular to them. In an oblique cylinder the axis is tilted, so the side leans. Volume is unchanged for the same base and perpendicular height; the curved surface area is larger.
Related
- How to find the volumeHow to find the volume of a cylinder: square the radius, multiply by π and the height. Worked examples in inches, feet, cm and m, plus common mistakes.
- DerivationWhy the volume of a cylinder is πr²h: stacked disks, polygon prisms, Cavalieri, the integral ∫ πr² dz, disks and shells, and why dV/dr is the side area 2πrh.
- Height of a cylinderHow to find the height of a cylinder: h = V ÷ (πr²) from the volume, or from the surface area. Calculator, tank examples in gallons and liters, common traps.
Updated October 12, 2026