Explore 3D shapes — vertices, edges, faces, nets and cross-sections!
3D shapes have length, width and height. Unlike 2D shapes, they take up space.
| Shape | Faces | Vertices | Edges |
|---|---|---|---|
| Cube | 6 | 8 | 12 |
| Cuboid | 6 | 8 | 12 |
| Triangular prism | 5 | 6 | 9 |
| Square pyramid | 5 | 5 | 8 |
| Triangular pyramid (tetrahedron) | 4 | 4 | 6 |
| Cylinder | 3 (2 flat, 1 curved) | 0 | 2 curved |
| Cone | 2 (1 flat, 1 curved) | 1 (apex) | 1 curved |
| Sphere | 1 (curved) | 0 | 0 |
Faces + Vertices − Edges = 2
This works for all polyhedra (shapes with flat faces)!
Example — Cube: 6 + 8 − 12 = 2 ✅
Example — Triangular prism: 5 + 6 − 9 = 2 ✅
Prism: has two identical parallel bases (ends) connected by rectangles. Named after the shape of its base (triangular prism, square prism/cuboid, hexagonal prism).
Pyramid: has one polygon base with triangular faces meeting at an apex. Named after the shape of its base (square pyramid, triangular pyramid).
A net is a flat shape that folds up to make a 3D shape.
To check if a net is valid, imagine folding each face up. Make sure:
A cube has 11 possible nets. A cuboid has the same number of faces but different dimensions.
A cross-section is the 2D shape you see when you slice through a 3D shape.
| 3D Shape | Horizontal Cross-Section |
|---|---|
| Cube/Cuboid | Rectangle (or square) |
| Triangular prism (cut along length) | Triangle |
| Cylinder | Circle |
| Cone (cut parallel to base) | Circle |
| Sphere | Circle |
| Square pyramid (cut parallel to base) | Square |
Question: A triangular prism — how many faces, vertices and edges does it have? Verify with Euler's formula.
Step 1: Count the faces. A triangular prism has 2 triangular ends + 3 rectangular sides = 5 faces.
Step 2: Count the vertices (corners). Each triangle has 3 corners × 2 = 6 vertices.
Step 3: Count the edges. 3 along each triangle (×2 = 6) + 3 connecting edges = 9 edges.
Step 4: Check: 5 + 6 − 9 = 2 ✅
Question: A net has 1 square base, 4 triangles attached to each side. What 3D shape does it make?
Think: One square base + 4 triangular faces meeting at a point = a square pyramid.
A square pyramid has 5 faces (1 square + 4 triangles), 5 vertices, 8 edges.
Check Euler: 5 + 5 − 8 = 2 ✅
Question: What shape is the cross-section when you slice a cylinder parallel to its base?
A cylinder has circular bases. Slicing parallel to the base gives a circle.
If you slice a cylinder vertically through its axis, you get a rectangle.
Question: A shape has 8 faces and 12 vertices. How many edges does it have?
F + V − E = 2
8 + 12 − E = 2
20 − E = 2
E = 18 edges
Select a 3D shape to explore its properties!
Match each description to the correct 3D shape name. Drag the name to the correct description box.
1. 6 square faces, 8 vertices, 12 edges
2. 2 triangular faces + 3 rectangular faces, 6 vertices, 9 edges
3. 5 faces (1 square base + 4 triangles), apex at top
4. 2 circular faces + 1 curved surface, 0 vertices
5. 1 circular base + 1 curved face, 1 apex vertex
Each shape is shown. Drag the correct number of vertices to each shape.
1. Cube → number of vertices:
2. Triangular pyramid (tetrahedron) → number of vertices:
3. Square pyramid → number of vertices:
4. Triangular prism → number of vertices:
5. Sphere → number of vertices:
Use Euler's formula (F + V − E = 2) to find the missing value. Drag the answer.
1. Cube: F=6, V=8, E=?
2. Tetrahedron: F=4, V=4, E=?
3. Square pyramid: F=5, V=5, E=?
4. Triangular prism: F=5, V=6, E=?
5. Shape with F=8, V=12, E=? (hexagonal prism)
Is each shape a prism or a pyramid? Drag the label to sort them.
1. Triangular prism (two triangular ends connected by rectangles)
2. Square pyramid (square base, 4 triangular faces, apex)
3. Hexagonal prism (two hexagonal ends connected by rectangles)
4. Triangular pyramid / tetrahedron (4 triangular faces)
5. Cuboid (two rectangular ends connected by rectangles)
What 2D shape do you see when you slice each 3D shape parallel to its base? Drag the answer.
1. Cylinder (slice parallel to base)
2. Cube (slice parallel to base)
3. Triangular prism (slice through the length)
4. Cuboid (slice parallel to base)
5. Hexagonal prism (slice parallel to base)
Match each net description to the 3D shape it makes.
1. Net with 6 squares of equal size
2. Net with 1 square and 4 triangles
3. Net with 2 circles and 1 rectangle
4. Net with 2 triangles and 3 rectangles
5. Net with 6 rectangles (some of different sizes)
1. How many faces does a triangular prism have?
2. How many vertices does a square pyramid have?
3. A shape has 6 faces and 8 vertices. Use Euler's formula to find the number of edges.
4. What is the cross-section of a cylinder when sliced parallel to its base?
5. What is the difference between a prism and a pyramid?
6. How many edges does a tetrahedron (triangular pyramid) have?
7. What net does a cube have? How many faces must it have?
8. What is a cross-section?
9. How many faces does a hexagonal prism have?
10. A shape has 5 faces and 5 vertices. What shape is it?
11. What cross-section do you get when you slice a square pyramid parallel to its base?
12. How many vertices does a sphere have?
13. Name the 3D shape with exactly 4 faces.
14. A pentagonal prism has 5-sided bases. How many faces, vertices and edges does it have?
15. Is a cuboid a prism? Explain.
16. What shape is the net of a cone made of?
17. A shape has 8 faces, 6 vertices, and 12 edges. Verify Euler's formula.
18. What is the cross-section of a cone when sliced at an angle (not parallel to base)?
19. How many edges does a square pyramid have?
20. Which 3D shape has the same cross-section no matter where you slice it (parallel to any axis)?
1. A builder is making a dog kennel in the shape of a triangular prism. How many pieces of wood does she need for the faces (assume each face is one piece)?
2. Jamie says "My 3D shape has 12 edges and 8 vertices. What is it?" Use Euler's formula to find the number of faces and name the shape.
3. An architect designs a glass building that is a hexagonal prism. How many panes of glass are needed (one per face)?
4. A pyramid has a pentagon as its base. How many faces, vertices and edges does it have?
5. An ice cream factory makes 3D shape ice creams. A "double scoop" is a cylinder topped with a cone. Together, how many faces, vertices, and edges do these two shapes contribute?
6. A chocolate box is a triangular prism. The net is cut from a single sheet of card. List all the shapes in the net and how many of each.
7. A 3D shape has Faces = Vertices. Name two possible shapes this could be.
8. A gemstone is cut as an octahedron (8 faces, all equilateral triangles). How many vertices and edges does it have?
9. A child stacks a cube on top of another cube. How many faces, edges and vertices does the combined shape have? (The joined faces are internal and not counted.)
10. Why does Euler's formula NOT work for a sphere or cylinder? What's special about these shapes?