What is a cylinder? Definition, parts and types
A cylinder is a three-dimensional solid with two flat, parallel bases of the same size and shape, joined by one curved surface. In school geometry the bases are circles and the side stands straight up: this is the right circular cylinder, the shape of a can, a pipe or a drum. Its volume is the base area times the height, V = πr²h.
Contents
V = π × r² × h. A cylinder with a radius of 3 in and a height of 8 in holds π × 9 × 8 = 226.19 in³, about 0.979 US gal.
The calculator above takes the radius (or diameter) and height of any cylinder and returns its volume, surface area and capacity, so you can test each definition below on real numbers.
Cylinder definition: two levels
The word comes from the Greek kylindros, "roller". Mathematics uses it at two levels of precision, and most confusion about cylinders comes from mixing them up.
The school definition (geometry class)
In a middle school or high school geometry course, "cylinder" means a right circular cylinder: a solid bounded by two congruent circles in parallel planes and the curved surface that joins them, with the line through the two centers perpendicular to the circles. Every formula on this site, from V = πr²h to the surface area 2πr² + 2πrh, assumes this shape unless the page says otherwise.
The general definition (higher mathematics)
In general geometry, a cylindrical surface is the surface swept by a straight line, called a generatrix, that moves parallel to a fixed direction while touching a fixed closed curve, called the directrix. A solid cylinder is the region enclosed by such a surface and two parallel planes. Under this definition the base can be a circle, an ellipse or any other closed curve, and the side does not have to be perpendicular to the base. An ellipse gives an elliptical cylinder. A polygon gives a prism, which is why some higher-level texts treat prisms as a special kind of cylinder, while school geometry keeps the two apart; the page is a cylinder a prism? settles the question for each context.
In analytic geometry, the surface x² + y² = r² in three dimensions is an infinite cylinder: x and y are tied to a circle while z is free. The page on the equation of a cylinder covers this form, shifted axes, elliptical cylinders and cylindrical coordinates.
Parts of a cylinder: the vocabulary
These terms appear in every problem about cylinders. Learn them once and the formulas read themselves.
| Term | Meaning | Symbol or formula |
|---|---|---|
| Base | Each of the two flat, congruent ends; circles in a circular cylinder | area πr² |
| Radius | Distance from the center of a base to its edge | r |
| Diameter | Distance across a base through its center | d = 2r |
| Axis | The line joining the centers of the two bases | — |
| Height | Perpendicular distance between the planes of the bases | h |
| Lateral (curved) surface | The curved side that joins the two bases | area 2πrh |
| Generatrix (element) | Any straight segment on the curved side joining the two base edges, parallel to the axis | length h in a right cylinder |
| Edge | Each circle where a base meets the curved side | circumference 2πr |
A cylinder has no corners. The usual school count is 3 faces (2 flat, 1 curved), 2 curved edges and 0 vertices, but the count depends on how a textbook defines "face" and "edge". The page on the faces, edges and vertices of a cylinder explains both answers and why Euler's formula does not apply as is.
Types of cylinders
Two questions classify any cylinder: what shape is the base, and is the axis perpendicular to it? A third covers solids that are not "full".
| Type | What makes it different | Volume |
|---|---|---|
| Right circular cylinder | Circular bases, axis perpendicular to them; the standard school cylinder | πr²h |
| Oblique cylinder | Axis tilted; the side leans, bases stay parallel | πr²h with h measured vertically |
| Elliptical cylinder | Bases are ellipses with semi-axes a and b | πabh |
| Hollow cylinder (tube) | A smaller cylinder removed along the axis; outer radius R, inner radius r | π(R² − r²)h |
| Half cylinder | A cylinder cut in two along a plane through its axis | πr²h ÷ 2 |
Right and oblique cylinders
In a right cylinder the axis meets the bases at 90°, so the side is vertical and the height equals the length of the side. In an oblique cylinder the axis is tilted: picture a stack of coins pushed sideways. Each coin keeps its size, so by Cavalieri's principle the volume does not change, provided you use the perpendicular height and not the slant length. The page on oblique cylinder volume shows how to get that height from the slant length and the angle.
Circular and elliptical cylinders
A circular cylinder has circular bases. An elliptical cylinder has oval bases, with a long semi-axis a and a short semi-axis b; the base area is πab instead of πr². With a = 4 cm, b = 3 cm and a height of 10 cm, the volume is π × 4 × 3 × 10 = 376.99 cm³. The elliptical cylinder volume page covers oval tanks and how to measure the axes.
Hollow and half cylinders
A hollow cylinder is a tube: pipe walls, rings, bushings. Its volume is the outer cylinder minus the inner one. With an outer radius of 4 in, an inner radius of 3 in and a length of 10 in, the material fills π × (16 − 9) × 10 = 219.91 in³; the hollow cylinder volume guide also works from diameters and wall thickness. A half cylinder is cut lengthwise through the axis, like a trough; with r = 3 in and h = 8 in it holds half of 226.19 in³, that is 113.10 in³ (details on the half cylinder volume page).
Properties of a cylinder
- Constant cross-section. Every slice cut parallel to the bases is a copy of the base. This is the property that makes the volume "base area × height".
- Three kinds of slices. A cut square to the axis gives a circle; a cut along the axis gives a rectangle 2r wide and h tall; a slanted cut through the side gives an ellipse. The page on the cross-sectional area of a cylinder computes each one.
- It unrolls flat. The curved side unrolls into a rectangle whose width is the circumference 2πr, which is why the lateral area is 2πrh. The two bases plus that rectangle form the net of a cylinder.
- Volume grows with the square of the radius. Doubling the radius multiplies the volume by 4; doubling the height only by 2.
- Symmetry. A right circular cylinder is symmetric about its axis (it looks the same after any rotation around it) and about the plane halfway between the bases.
- It can roll and stack. A cylinder rolls in a straight line on its curved side and stands stable on a flat base.
- Sphere inside a cylinder. A sphere that just fits inside a cylinder (h = 2r) has exactly ⅔ of the cylinder's volume, a result known since Archimedes. With r = 3 in: cylinder 169.65 in³, sphere 113.10 in³.
- Least material for a given volume. Among closed right circular cylinders holding the same volume, the one with the smallest total surface area has a height equal to its diameter (h = 2r).
Cylinder formulas at a glance
All formulas below are for a right circular cylinder with radius r and height h. The examples use r = 3 in and h = 8 in, then r = 5 cm and h = 12 cm, computed with full-precision π and rounded.
| Quantity | Formula | r = 3 in, h = 8 in | r = 5 cm, h = 12 cm |
|---|---|---|---|
| Base area | πr² | 28.27 in² | 78.54 cm² |
| Lateral area | 2πrh | 150.80 in² | 376.99 cm² |
| Total surface area | 2πr² + 2πrh = 2πr(r + h) | 207.35 in² | 534.07 cm² |
| Volume | πr²h | 226.19 in³ | 942.48 cm³ |
| Capacity | in³ ÷ 231 · cm³ ÷ 1,000 | 0.979 US gal | 0.942 L |
Run your own numbers through the cylinder volume calculator on the home page. For the area side, the guide to the surface area of a cylinder covers the total area, the curved side alone and open-top cases. For the step-by-step volume method with diameter, units and mistakes to avoid, read how to find the volume of a cylinder.
Measure the radius, not the diameter. The formulas use r; a diameter put in by mistake makes the volume four times too large and the base area four times too large.
Cylinder shaped objects in everyday life
Real objects are rarely perfect cylinders: cans have rims, bottles have necks, tree trunks taper. Still, the cylindrical shape is close enough that V = πr²h gives useful estimates for all of these:
- Food cans, paint cans and drinking glasses with straight sides
- Pipes, hoses, drinking straws and the tube inside a paper towel roll (hollow cylinders)
- Water heaters, propane and air tanks, oil drums and rain barrels
- Round concrete columns, fence post holes and wells
- Batteries, candles, crayons, rolling pins and hockey pucks (a very short cylinder)
- Logs and tree trunks over a short length, coins and round cakes
In everyday speech a cylinder is often just called a can, a tube, a drum or a roll. "Cylinder" also has technical meanings outside geometry: the chamber where a piston moves in an engine, a pressurized gas container, or the rotating part of a revolver. All are named after the same round, straight-sided shape.
Cylinder vs cone, prism and sphere
A cylinder is easiest to understand next to its neighbors. The comparison below uses the same radius r and height h where it applies.
| Solid | Bases | Side | Volume | Vertices |
|---|---|---|---|---|
| Cylinder | 2 congruent circles | 1 curved surface | πr²h | 0 |
| Cone | 1 circle | 1 curved surface narrowing to a point | ⅓πr²h | 1 (apex) |
| Prism | 2 congruent polygons | flat rectangles or parallelograms | base area × h | yes, at every corner |
| Sphere | none | 1 curved surface | ⁴⁄₃πr³ | 0 |
A cone with the same base and height holds exactly ⅓ of the cylinder. A prism shares the cylinder's volume rule but has flat faces and corners; as the number of sides of its base grows, a regular prism looks more and more like a cylinder, which the page on the differences between a cylinder and a prism shows with a table. With h = 2r, a cone, a sphere and a cylinder of the same radius have volumes in the ratio 1 : 2 : 3; the cylinder, cone and sphere volume page works through it.
Common mistakes about cylinders
- Using the slant length as the height of an oblique cylinder. The height is always the perpendicular distance between the bases.
- Confusing lateral area and total area. The lateral area 2πrh is the curved side only; add 2πr² for the two bases.
- Mixing units. A radius in inches and a height in feet give a meaningless product. Convert first.
- Calling a cylinder a prism or a polyhedron in a school answer. In school geometry, prisms and polyhedra have flat polygonal faces only.
- Spelling. The word is spelled c-y-l-i-n-d-e-r; "cilinder", "cylander" and "cylindar" are frequent misspellings.
Explore the shape of a cylinder
FAQ
What is the definition of a cylinder in math?
In school geometry, a cylinder is a three-dimensional solid with two congruent, parallel circular bases joined by one curved surface. In general geometry, a cylinder is the surface traced by a straight line that moves parallel to itself along a closed curve, so the base can be any closed curve, not only a circle.
What is a right cylinder?
A right cylinder is a cylinder whose axis is perpendicular to its bases. Its side stands straight up, so the height equals the length of the side. When the bases are circles, it is a right circular cylinder, the shape used by default in most textbooks and formulas.
Is a can a cylinder?
A can is a common everyday example of a cylinder: two flat circular ends and one curved side. A real can has rims and rounded edges, so it is only close to a perfect geometric cylinder, but its volume is well estimated with V = πr²h using the inside measurements.
What does cylindrical mean?
Cylindrical means shaped like a cylinder: round in cross-section and the same width all along its length. Pipes, candles, batteries and tree trunks are described as cylindrical even when they are not perfect cylinders.
How many faces does a cylinder have?
The usual school answer is 3 faces (2 flat circular faces and 1 curved face), 2 curved edges and 0 vertices. Some textbooks count only flat surfaces as faces and say a cylinder has 2 faces and 1 curved surface.
Is a cylinder a polyhedron?
No. A polyhedron is made only of flat polygonal faces with straight edges. A cylinder has a curved surface and circular edges, so it is not a polyhedron and not a prism in the strict sense, even though its volume follows the same rule: base area times height.
Related
- Faces, edges, verticesA cylinder has 3 faces (2 flat, 1 curved), 2 curved edges and 0 vertices. See why some textbooks disagree and how it compares with cones and prisms.
- Cylinder vs prismIs a cylinder a prism? Not in school geometry: a prism has polygon bases. Both share V = base area × height. Compare volumes, areas and examples.
- 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.
Updated October 12, 2026