How many edges does a cylinder have? Faces, edges and vertices
How many edges does a cylinder have? Two, both curved: the circle around the top and the circle around the bottom. In the usual school count it also has 3 faces (2 flat circles and 1 curved surface) and 0 vertices, because it has no corners. Some textbooks count differently, and the reasons are below.
Contents
Net of a cylinder: two circles of radius r and a rectangle 2πr wide and h tall. With r = 3 in and h = 8 in, the rectangle is 18.85 in × 8 in and the paper area is 2πr² + 2πrh = 207.35 in².
The net plotter above unfolds a cylinder into its pieces: two circles and one rectangle. Counting those pieces is the quickest way to see the three faces.
Cylinder faces, edges and vertices: the school answer
| Part | How many | What they are |
|---|---|---|
| Faces | 3 | 2 flat circular faces (the bases) and 1 curved face (the side) |
| Edges | 2 | 2 curved edges, the circles where each base meets the side |
| Vertices | 0 | No corners: the edges never meet |
This is the answer expected in most elementary and middle school worksheets in the US and the UK. If your teacher or textbook uses a different count, it is not wrong: it is a different definition of "face" or "edge", explained in the next sections. For the shape itself, its parts and its types, see the guide what is a cylinder.
The two edges
An edge is a line where two faces meet. On a cylinder, the top base meets the curved side along a circle, and the bottom base meets it along another circle. That gives 2 edges. Each one is as long as the circumference of the base, 2πr: with a radius of 3 in, each edge measures 18.85 in; with a radius of 4 cm, 25.13 cm.
The zero vertices
A vertex is a point where edges meet, a corner. The two circles of a cylinder are parallel and never touch, and a circle has no end points. So a cylinder has no vertex. Students often confuse a cylinder with a cone here: a cone does have 1 vertex, its tip (apex).
The three faces
Cut the curved side along a straight line from top to bottom and unroll it: you get a rectangle. Together with the two circles, that makes three pieces, which is the net of a cylinder. With r = 4 cm and h = 10 cm, the two flat faces cover 2 × π × 4² = 100.53 cm², the curved face covers 2 × π × 4 × 10 = 251.33 cm², and the whole surface is 351.85 cm². These pieces are exactly the terms of the surface area of a cylinder formula, 2πr² + 2πrh.
Why textbooks give different answers
Faces, edges and vertices were first defined for polyhedra: solids made only of flat polygons, like cubes, prisms and pyramids. In that setting a face is a flat polygon, an edge is a straight segment where two faces meet, and a vertex is a point where edges meet. A cylinder has a curved surface and round edges, so it is not a polyhedron, and the words have to be stretched to apply to it. Curricula stretch them in different ways:
| Convention | Faces | Edges | Vertices |
|---|---|---|---|
| Common school count (curved faces and edges allowed) | 3 (2 flat + 1 curved) | 2 curved | 0 |
| Faces must be flat | 2 flat faces + 1 curved surface | 2 curved | 0 |
| Strict polyhedron definitions (flat faces, straight edges) | 2 | 0 | 0 |
All three describe the same object. The safe move in a test is to answer with the count your course uses and, when a short explanation is allowed, say which parts are flat and which are curved: "2 flat circular faces, 1 curved surface, 2 curved edges, no vertices" is correct under every convention.
Quick check with a real object: hold a can. Count the flat ends you could stand it on (2), the curved side you can roll it on (1), the rims you can run a finger around (2) and the sharp corners (none).
Euler's formula and the cylinder
For every convex polyhedron, Euler's formula says F − E + V = 2. A cube gives 6 − 12 + 8 = 2. Apply it to the school count of a cylinder and you get 3 − 2 + 0 = 1. The formula does not "fail": it simply was never meant for a solid whose faces are curved and whose edges are closed loops with no vertex.
Mathematicians get 2 back by cutting the surface into pieces that behave like polygons. Put one point on the top circle and one on the bottom circle (2 vertices), and draw a vertical seam joining them. The edges are now the top circle, the bottom circle and the seam (3 edges). The faces are the two bases and the side, which the seam opens into a rectangle (3 faces). Then F − E + V = 3 − 3 + 2 = 2. This is the same idea as the net: the seam is where you cut the side to unroll it.
Cylinder vs cone, sphere, prism and cube
The table compares the cylinder with the solids it is most often confused with. The cylinder, cone and sphere rows use the common school convention; the polyhedra rows have no ambiguity.
| Solid | Faces | Edges | Vertices | F − E + V |
|---|---|---|---|---|
| Cylinder | 3 (2 flat, 1 curved) | 2 (curved) | 0 | 1 |
| Cone | 2 (1 flat, 1 curved) | 1 (curved) | 1 (apex) | 2 |
| Sphere | 1 (curved) | 0 | 0 | 1 |
| Triangular prism | 5 | 9 | 6 | 2 |
| Cube or rectangular prism | 6 | 12 | 8 | 2 |
| Hexagonal prism | 8 | 18 | 12 | 2 |
| Square pyramid | 5 | 8 | 5 | 2 |
The cone's result of 2 is a coincidence of counting, not a proof that Euler's formula applies to it. The prism rows show the pattern that links prisms to cylinders: a prism whose base has n sides has n + 2 faces, 3n edges and 2n vertices. As n grows, the side faces get thinner and the prism looks more and more like a cylinder, but it always keeps its corners. The page cylinder vs prism explains why a cylinder is not a prism in school geometry, even though both follow the volume rule "base area × height".
Common mistakes
- Saying a cylinder has 1 vertex. That is the cone. A cylinder has no point where edges meet.
- Saying a cylinder has 0 edges in a school worksheet. Most worksheets expect 2 curved edges; 0 is only the answer under the strict polyhedron definition.
- Counting the curved side as two faces. The side is one continuous surface, even though it unrolls into a rectangle with two long sides.
- Calling a cylinder a polyhedron. Polyhedra have only flat polygonal faces; a cylinder has a curved one.
From counting to measuring
Once you know the parts, each one has a size. The flat faces each have an area of πr², the curved face 2πrh, the edges a length of 2πr each, and the whole solid a volume of πr²h. A cylinder with a radius of 3 in and a height of 8 in has two faces of 28.27 in², a curved face of 150.80 in², and holds 226.19 in³. The cylinder volume calculator on the home page gives the volume, surface area and lateral area from any radius or diameter, in inches, feet, centimeters or meters.
FAQ
How many edges does a cylinder have?
A cylinder has 2 edges: the circle around the top and the circle around the bottom, where each flat base meets the curved side. Both edges are curved, not straight. Under the strict polyhedron definition, where an edge must be a straight segment, a cylinder has 0 edges.
How many vertices does a cylinder have?
A cylinder has 0 vertices. A vertex is a corner point where edges meet, and the two circular edges of a cylinder never meet each other or end at a point.
How many faces does a cylinder have?
The common answer is 3 faces: 2 flat circular faces and 1 curved face. Textbooks that count only flat surfaces as faces say a cylinder has 2 faces and 1 curved surface. Follow the definition used in your course.
Do cylinders have edges?
Yes, in most school curricula: the two circles where the curved side meets the bases are counted as curved edges. Only when edges are defined as straight segments between two vertices, as for polyhedra, does a cylinder have no edges.
Does Euler's formula work for a cylinder?
Not as is. With 3 faces, 2 edges and 0 vertices, F − E + V = 1, not 2. Euler's formula is stated for polyhedra, and a cylinder is not a polyhedron. If you add one point on each circle and a vertical seam, you get 3 faces, 3 edges and 2 vertices, and 3 − 3 + 2 = 2.
Related
- What is a cylinderA cylinder is a solid with two parallel, congruent circular bases joined by a curved surface. Learn its parts, types, properties and formulas.
- Net of a cylinderA cylinder net is a rectangle 2πr wide and h tall plus two circles of radius r. Draw one to size, print it, and use it to find the surface area.
- 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.
Updated October 12, 2026