Grade by Grade
E. Spatial Sense · Spatial Sense: 3D Objects, Transformations, Circles, Surface Area and Volume

Classifying Prisms, Pyramids and Cylinders

E1.1 A1About 45 minutes
Learning goals
  • I can describe prisms, pyramids and cylinders by their bases, faces, edges and vertices.
  • I can tell a right prism or cylinder from an oblique one.
  • I can use patterns to predict the number of faces, edges and vertices of any prism or pyramid.
  • I can classify a 3D object and explain my reasoning.

Words to know

base
The face (or faces) an object is named after. A prism and a cylinder have 2 congruent, parallel bases. A pyramid has 1 base.
lateral face
A face that is not a base. In a prism the lateral faces are rectangles or parallelograms; in a pyramid they are triangles.
apex
The point where all the triangular faces of a pyramid meet.
edge
A line segment where two faces meet.
vertex
A corner point where edges meet. The plural is vertices.
right prism
A prism whose lateral faces are rectangles, so it stands straight up from its base.
oblique prism
A prism that leans: its lateral faces are parallelograms that are not rectangles.
cylinder
A 3D object with 2 congruent, parallel circular bases joined by one curved surface.
congruent
Same size and same shape.

Remember from earlier grades

In Grade 3 you named cubes, prisms, pyramids, cylinders and cones and counted their faces, edges and vertices. In Grade 6 you drew nets of prisms and pyramids and found their surface area.

In Grade 7 you will classify these objects more carefully, by the properties that make each family special. Next lesson you will add symmetry to the list.

Let's start in the kitchen

Open a kitchen cupboard. A cereal box, a can of soup, a box of crackers, maybe a box of chocolates shaped like a long tent. Each of these objects belongs to a family of 3D objects. If you know the family, you already know a lot about the object: how many faces it has, what shape they are, even how to find its volume (you'll learn that at the end of this unit).

Coloured boxes, tin cans and a triangular box on two shelves of an open kitchen cupboard.
Find the 3D families: rectangular prisms (boxes), cylinders (cans) and a triangular prism (the long red box at the bottom right). AI-generated image

Three families

Prisms

  • have 2 bases that are congruent polygons, parallel to each other
  • the other faces (the lateral faces) join matching edges of the two bases
  • are named after the shape of the base: triangular prism, rectangular prism, pentagonal prism ...

Pyramids

  • have 1 base, which is a polygon
  • all the lateral faces are triangles that meet at one point, the apex
  • are named after the base: triangular pyramid, square pyramid ...

Cylinders

  • have 2 bases that are congruent circles, parallel to each other
  • the bases are joined by one curved surface
  • have no vertices
basetriangular prismapexsquare pyramidbasecylinder
Three families: a triangular prism (its 2 triangle bases are the front and back), a square pyramid (1 square base, triangles meeting at the apex) and a cylinder (2 circular bases and a curved surface). Dashed lines are hidden edges.

Right or oblique?

In a right prism the lateral faces are rectangles, so the prism stands straight up. If you push the top base sideways, the prism leans. Now the lateral faces are slanted parallelograms. That is an oblique prism. It still has 2 congruent, parallel bases, so it is still a prism.

Cylinders work the same way. A neat stack of coins is a right cylinder. Push the stack so it leans and it becomes an oblique cylinder.

right prismoblique prismright cylinderoblique cylinderrectangle sidesparallelogram sidesstands straightleans
Push the top base sideways and a right prism becomes oblique: the bases are still congruent and parallel, but the side faces become parallelograms. Cylinders can lean the same way.
ObjectSides on the base (n)FacesEdgesVertices
triangular prism3596
rectangular prism46128
pentagonal prism571510
hexagonal prism681812
triangular pyramid3464
square pyramid4585
pentagonal pyramid56106
hexagonal pyramid67127
Look for patterns in each column.

The patterns

Call the number of sides on the base n.

Prism: faces = n + 2 (the n lateral faces plus 2 bases), edges = 3 × n (n on each base plus n going between them), vertices = 2 × n (n on each base).

Pyramid: faces = n + 1 (n triangles plus 1 base), edges = 2 × n (n on the base plus n going up to the apex), vertices = n + 1 (n on the base plus the apex).

Bonus pattern: for every prism and pyramid, faces + vertices − edges = 2. This is called Euler's formula (say "OY-ler"). It's a quick way to check your counting.

Worked example

Example 1: Predict without a model

How many faces, edges and vertices does an octagonal prism have? (An octagon has 8 sides.)

  1. It's a prism, so use the prism patterns with n = 8.
  2. Faces = 8 + 2 = 10. Edges = 3 × 8 = 24. Vertices = 2 × 8 = 16.
  3. Check with Euler's formula: 10 + 16 − 24 = 2 ✓

10 faces, 24 edges, 16 vertices.

Worked example

Example 2: Name the object

An object has 7 faces: 1 hexagon and 6 triangles that meet at a point. Name it and find its edges and vertices.

  1. Triangles meeting at one point means there is an apex, so it's a pyramid.
  2. There is only 1 polygon that isn't a triangle, the hexagon, so that is the base. It is a hexagonal pyramid, n = 6.
  3. Edges = 2 × 6 = 12. Vertices = 6 + 1 = 7.

A hexagonal pyramid with 12 edges and 7 vertices.

Worked example

Example 3: Work backwards

A prism has 21 edges. What shape is its base?

  1. For a prism, edges = 3 × n.
  2. 3 × n = 21, so n = 21 ÷ 3 = 7.
  3. A polygon with 7 sides is a heptagon.

A heptagonal prism (its bases are heptagons).

Classify carefully

  • A cube is a prism: a rectangular (square) prism whose faces are all squares. Any pair of opposite faces can be its bases.
  • A cylinder is not a prism, because its bases are circles (not polygons) and its side is curved. But it is like a prism in an important way: two congruent, parallel bases.
  • When you sort, say why. "It's a pyramid because it has one base and triangles that meet at an apex" is a full answer.
Hands-on activity

Kitchen classification hunt

You need: 5 or 6 objects from home (boxes, cans, a tube, a tent-shaped box, a pyramid-shaped toy or decoration), paper and a pencil.

  1. Make a table with columns: object, family (prism, pyramid or cylinder), base shape, right or oblique, faces, edges, vertices.
  2. Fill in a row for each object. Count edges by running a finger along each one and putting a small sticky note on it as you count, so you don't count any twice.
  3. Use the patterns to check your counts, and check prisms and pyramids with faces + vertices − edges = 2.
  4. Sort your objects into a Venn diagram with two circles: "has 2 parallel congruent bases" and "has a vertex". Where does each object go?

What to look for: cans go only in the "2 bases" circle; pyramids go only in the "vertex" circle; boxes and other prisms go in the overlap.

Just for fun

Why didn't the cylinder get invited to the vertex party?

It didn't have a single point to make.

Your turn

1

An object has 2 congruent pentagon bases and 5 rectangular faces. Name it.

2

How many faces, edges and vertices does a pentagonal pyramid have?

3

A box of tissues is a rectangular prism. How many faces, edges and vertices does it have?

4

Which object has a curved surface and 2 circular bases?

5

How many faces, edges and vertices does a heptagonal prism (7-sided base) have?

6

An object has 4 triangular faces and 1 square face. Name it.

7

True or false: the lateral faces of a right prism are rectangles.

More practice: warm-up

1

How many bases does a prism have?

2

How many bases does a pyramid have?

3

What is the point where the triangular faces of a pyramid meet called?

4

How many vertices does a cylinder have?

5

What shape are the bases of a triangular prism?

More practice: on your own

1

How many faces, edges and vertices does a decagonal pyramid (10-sided base) have?

2

A prism has 24 edges. What shape is its base?

3

A pyramid has 9 vertices. Name it and give its number of edges.

4

A stack of identical rectangular books is pushed so it leans to one side. What 3D object does it look like? Explain.

5

Check Euler's formula for a square pyramid.

Challenge

1

Can a prism have exactly 25 edges? Explain.

2

An object has 12 vertices and 18 edges. Is it a prism or a pyramid? Name it and give its number of faces.

3

Mia says, "A cylinder is just a prism with circle bases." Give one way she is right and one reason mathematicians put cylinders in their own family.

Check your understanding

Quiz

1.

How many faces does a hexagonal prism have?

2.

Which object has exactly one base?

3.

What shape are the lateral faces of a pyramid?

4.

A prism has 15 edges. What shape is its base?

5.

What makes a prism oblique?

6.

How many vertices does a square pyramid have?

For parents and teachers

Big idea: Ontario's Grade 7 expectation E1.1 asks students to describe and classify cylinders, pyramids and prisms by their geometric properties (this lesson), including plane and rotational symmetry (next lesson).

How to help at home: Collect real objects: boxes, cans, tubes, a tent-shaped chocolate box. Ask "What family is it in? How do you know?" Push for a reason that uses properties ("two congruent parallel bases"), not just "it looks like one."

Common difficulties:

  • Miscounting edges and vertices. Mark each edge with a small sticky note as you count, then check with faces + vertices − edges = 2.
  • Thinking a prism must be standing on its base. A triangular prism lying on a rectangle is still named for its triangle bases.
  • Calling a cylinder a prism. It shares some properties, but its bases are circles and its side is curved.

This lesson also builds A1 skills: looking for patterns, generalizing (n + 2, 3n, 2n) and explaining reasoning.