When we look at any shape—whether it’s a flat 2D shape like a triangle, square, or hexagon, or a solid 3D shape like a cube, pyramid, or cylinder—we can describe it using three key features: vertices, edges, and faces.
We’ll take a closer look at what each of these means in math and how they help us understand shapes better.
Read on for simple definitions, examples with visuals, answers to common questions, and a quick test to check what you’ve learned.
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A vertex (plural: vertices) is a point where two or more line segments, rays, or edges meet.
You can think of a vertex as a corner—like the corner of a book, a picture frame, or the pointy tip of a triangle.
We remember that 2D shapes—also called flat shapes—are shapes that only have length and width. They don’t have any depth or thickness, and they’re the kinds of shapes we usually draw on paper—like triangles, squares, rectangles, and hexagons.
Now, in the case of 2D shapes, vertices are the points where the sides of the shape meet. These are the corners we can count.
When we look at polygons—flat shapes with straight sides—we always find that the number of vertices is equal to the number of sides.
Let’s look at some of the common polygons and the number of vertices they have:
A triangle has 3 vertices.
A square has 4 vertices.
A rectangle has 4 vertices.
A pentagon has 5 vertices.
A hexagon has 6 vertices.
An octagon has 8 vertices.
But what about shapes like circles and ovals?
These shapes have curved edges, not straight sides—so we never see sides meeting at a sharp point. That means they do not have any vertices.
If a shape is too smooth to have a corner, then it’s too smooth to have a vertex.
3D shapes—also called solid shapes—have length, width, and depth. We can hold them, turn them, and see different sides from different angles.
In 3D shapes, vertices are the points where three or more edges meet. Just like in flat shapes, we can think of these points as corners. If you’ve ever held a cube, the sharp corners where the edges come together are its vertices.
Now, let’s look at some common polyhedra (singular: polyhedron)—which are 3D shapes made up of flat, straight-sided faces—and see how many vertices each one has:
A cube has 8 vertices.
A rectangular prism has 8 vertices.
A pyramid with a square base has 5 vertices.
A triangular prism has 6 vertices.
Just like with flat shapes, some 3D shapes—like spheres and cylinders—don’t have any vertices. They have curved surfaces, not sharp corners.
An edge is a line segment that connects two vertices.
You can think of an edge as the side of a shape, like the edge of a picture frame or the straight part of a ruler.
When it comes to 2D shapes, we often use the words sides and edges to mean the same thing. That’s because in flat shapes, each edge is also one of the shape’s sides.
In polygons, the number of edges (or sides) always matches the number of vertices because each side connects exactly two vertices, and every vertex is formed where two sides meet.
Let’s look at some common polygons and their number of edges:
A triangle has 3 edges.
A square has 4 edges.
A rectangle has 4 edges.
A pentagon has 5 edges.
A hexagon has 6 edges.
An octagon has 8 edges.
But remember, circles and ovals don’t have straight sides or vertices—so they don’t have edges either.
In 3D shapes, an edge is a straight line where two surfaces meet.
Think about a box or a pyramid. The lines where the flat parts of the shape come together—those are the edges. Each edge connects two vertices and helps form the shape’s structure.
Let’s look at the number of edges in some common 3D shapes students often see in class or at home:
A cube, like a dice or a gift box, has 12 edges.
A rectangular prism, like a cereal box, also has 12 edges.
A square pyramid, like the Great Pyramid of Giza, has 8 edges.
A triangular prism, which looks like a tent, has 9 edges.
Some 3D shapes don’t have edges at all.
A sphere is completely smooth, so it has no vertices and no edges.
A cylinder might look like it has edges in a drawing—around the top and bottom circles—but those are curved, not straight. And in geometry, an edge must be a straight line segment, so we say a cylinder has no edges.
A sphere is completely smooth, so it has no vertices and no edges.
A cylinder might look like it has edges in a drawing—around the top and bottom circles—but those are curved, not straight. And in geometry, an edge must be a straight line segment, so we say a cylinder has no edges.
A face is a flat surface on a 3D shape, also called a solid shape.
Each face is shaped like a 2D figure—such as a square, rectangle, or triangle—and together, the faces form the outside of the shape. You can think of a face as the part of a shape you can see, touch, and lay your hand flat on.
2D shapes do not have faces because they are already flat. In fact, a 2D shape is just one flat face—it doesn’t have multiple surfaces like a 3D shape does.
Let’s look at how many faces some common 3D shapes have:
A cube has 6 faces (each one is a square).
A rectangular prism has 6 faces (rectangles).
A square pyramid has 5 faces (1 square and 4 triangles).
A triangular prism has 5 faces (2 triangles and 3 rectangles).
Shapes like spheres and cylinders don’t have flat surfaces, so they don’t have faces in the geometric sense.
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Vertices, edges, and faces are connected by a rule called Euler’s Formula (not to be confused with Euler’s number).
We use this rule with polyhedra—3D shapes made of flat surfaces, such as cubes, pyramids, and prisms.
Euler’s Formula is written as:
Vertices − Edges + Faces = 2
(or V − E + F = 2)
So, what does this formula actually help us do?
This formula shows that if you know any two of these values in a polyhedron, you can calculate the third.
Let’s confirm this with an example.
If we know that a cube has 8 vertices and 12 edges, we can use Euler’s Formula to figure out how many faces it has.
V − E + F = 2
Now, we substitute values:
8 − 12 + F = 2
-4 + F = 2
F = 2 - (-4) = 2 + 4
F = 6
So, the cube has 6 faces—just like we expected. Euler’s Formula works!
Ready to practice what you’ve learned? Try our flash test.
When you’re done, check your answers at the bottom of the guide.
Look at the cube below. Can you count how many vertices, edges, and faces it has?
Number of vertices: ____
Number of edges: ____
Number of faces: ____
Look at the shape below. Which of the options describes it correctly?
Choose the correct set of features:
A) 6 vertices, 9 edges, 5 faces
B) 8 vertices, 12 edges, 6 faces
C) 5 vertices, 8 edges, 5 faces
A shape has 6 faces and 8 vertices. Use Euler’s Formula to find how many edges it has.
I am a 3D shape. I have 5 faces, 5 vertices, and 8 edges. One of my faces is a square, and the others are triangles. What shape am I?
A) Triangular prism
B) Square pyramid
C) Cube
Vertices, edges, and faces are an important part of geometry. Naturally, learning about them often leads to questions, especially as students start to recognize these features in both real objects and math problems.
Here are some of the most common questions we get at Mathnasium of West Chester, along with clear answers to help solve any dilemmas and build confidence.
Most students are introduced to vertices, edges, and faces in grades 2–3, as part of early geometry lessons. By grades 4 and 5, they’re expected to use this vocabulary to describe and compare 2D and 3D shapes and apply it when solving problems or identifying properties in diagrams.
Only if the shape is a polygon—a flat shape made of straight sides.
For example, a hexagon has 6 sides and 6 vertices. But shapes like circles and ovals don’t have vertices or straight edges, so this rule doesn’t apply.
That’s why it's important to check if a shape is made of straight lines before applying that rule.
Yes, some shapes can! For example, a square pyramid has 5 faces and 5 vertices. But this isn’t a general rule—it depends on the structure of the shape.
A cone has 1 flat face (the circular base) and 1 vertex (the point at the tip). However, it does not have any edges in the geometric sense, because the curved surface doesn’t meet the base along a straight line.
The boundary between the base and the curved surface looks like an edge in drawings, but it’s actually a curved line, not a straight one—so it’s not counted as an edge in geometry.
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Using a proprietary teaching approach called the Mathnasium Method™, our specially trained math tutors provide face-to-face instruction in an engaging and fun group environment to guide students in mastering any math class and topic, including vertices, edges, and faces.
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If you’ve given our flash test a try, check your answers below.
Task 1: Count the Features (Cube)
Number of vertices: 8
Number of edges: 12
Number of faces: 6
Task 2: Which One Matches the Shape? (Triangular Prism)
Correct Answer: A) 6 vertices, 9 edges, 5 faces
Task 3: Use Euler’s Formula
Correct Answer: 12 edges
(Using Euler’s Formula: 8 − E + 6 = 2 → E = 12)
Task 4: Which Shape Am I?
Correct Answer: B) Square pyramid
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