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E1.1 · Classify cylinders, pyramids, and prisms by properties and symmetry

Learn to classify cylinders, pyramids, and prisms by properties and symmetry through clear examples and targeted practice.

Ontario Grade 7 Mathematics

Spatial Sense

Use faces, bases, edges, vertices, and symmetry to describe three-dimensional figures

Imagine holding a cereal box, a party hat, and a can. The box has flat rectangular faces. The hat has one flat base and a surface that comes to a point. The can has two matching circular ends and a curved surface. These objects suggest three families of solids: prisms, pyramids, and cylinders. To classify a solid, look carefully at its parts and how they fit together. Symmetry is another useful clue. A solid is symmetrical when a reflection or turn can make it match its original position. In this lesson, you will use these ideas to describe and compare familiar solids.

What you will learn

Identify and describe the faces, bases, edges, and vertices of cylinders, pyramids, and prisms.
Classify a solid by comparing its properties with the defining properties of these families.
Describe symmetry in a solid using simple examples of reflection and turning.

Start with the parts of a solid

A solid is a three-dimensional shape. It has length, width, and height. Before classifying solids, review a few parts. A face is a flat surface. An edge is where two flat faces meet. A vertex is a corner where edges meet. The plural of vertex is vertices.

A base is a face used to describe the main shape of a solid. Some solids have two bases; others have one. A cylinder has two circular bases. A prism has two matching, parallel polygon bases. A polygon is a closed, flat shape with straight sides, such as a triangle or rectangle. A pyramid has one polygon base and triangular faces that meet at one point, called the apex.

A cylinder also has one curved surface joining its circular bases. Since that surface is not flat, it is not a face under the definition used here. A cylinder has no vertices. Its circular rims are not straight edges where two flat faces meet. This way of counting helps keep the three families distinct.

Faces are flat; edges meet flat faces; vertices are corners.
A cylinder has two circular bases and one curved surface.
A prism has two matching polygon bases; a pyramid has one polygon base and an apex.

Classify by the bases and side surfaces

A prism is named for the shape of its bases. For example, a triangular prism has triangular bases. The other faces join matching sides of the two bases. In a right prism, these joining faces are rectangles. A rectangular box is a familiar example of a rectangular prism. The key clue is the pair of matching, parallel bases—not the direction the solid is sitting.

A pyramid is also named for its base. A square pyramid has a square base. Each side of the base meets a triangular face, and all those triangular faces meet at one apex. A triangular pyramid has a triangular base and three triangular side faces. Its four vertices are all corners; one is the apex.

A cylinder is not a prism. Its bases are circles, not polygons, and its joining surface is curved rather than made of flat side faces. A cylinder can stand upright or lie on its side and still be a cylinder. Position does not change its defining properties.

When a solid is unfamiliar, check its bases first. Ask whether it has two matching polygon bases, one polygon base with triangular sides meeting at an apex, or two circular bases joined by a curved surface. Then check the faces and corners. This gives you more than a name: it gives a reason for the classification.

A triangular prism has two triangular bases.
A square pyramid has one square base, triangular side faces, and one apex.
A cylinder has circular bases and a curved joining surface.

Notice symmetry

A shape has symmetry when a change in position leaves it matching its original form. For a flat figure, a line of symmetry divides it into matching reflected halves. For a solid, a plane of symmetry is an imaginary flat slice that divides it into matching reflected halves. The two halves match as mirror images.

Some solids also match after a turn. Rotational symmetry means that turning a solid around an axis can bring it to a position that looks the same as its starting position. An axis is an imaginary line through or beside a solid that describes the turn. You do not need to count every possible turn to compare these families; first notice whether the solid has a clear matching structure.

A right circular cylinder has a plane of symmetry halfway between its circular bases. It also matches after many turns around the line through the centres of its bases. A square pyramid with its apex above the centre of its base has matching symmetry planes through the apex and suitable lines in the square base. A right prism with a regular base can also have symmetry planes. The exact symmetries depend on the base shape and how the solid is arranged.

Do not assume that every solid in a family has all the same symmetries. A slanted prism can still be a prism because it has two matching, parallel polygon bases, even if it does not have the mirror symmetry of a right prism. Describe the symmetry you can support from the solid’s shape, rather than guessing from its name.

A plane of symmetry divides a solid into matching mirror-image halves.
Rotational symmetry means a turn can make a solid match its starting position.
Symmetry depends on the particular solid, not only on its family name.

A reliable classification routine

Use a short routine when you classify a solid. First, look for its bases. Count them and identify their shapes. Next, inspect the surfaces joining the bases, or the side faces joining a base to an apex. Then note any vertices and edges. Finally, describe an obvious plane or turn symmetry, if one is shown or can be established from the stated shape.

A labelled picture can help, but remember that a drawing is only one view. A hidden edge may be dotted, and the solid may look different when turned. Use the stated features and the relationships between its parts. For example, two matching polygon bases joined by side faces still identify a prism, even when one base is partly hidden.

In the examples, the diagrams are labelled by family and base shape. They are schematic figures, not measurements. The written properties give the evidence for each classification.

Check bases, joining surfaces, faces, and corners in that order.
Use the solid’s properties, not just its outline in one drawing.
State symmetry only when the solid’s structure supports it.

Compare the defining properties

Solid	Bases	Joining surfaces	A useful symmetry clue
Prism	Two matching, parallel polygon bases	Flat side faces join matching sides	Depends on the base and arrangement
Pyramid	One polygon base	Triangular faces meet at one apex	A centred, regular arrangement may have mirror symmetry
Cylinder	Two circular bases	One curved surface	An upright circular cylinder has a middle plane of symmetry

Worked example

A solid with two triangular ends

A solid has two matching, parallel triangular bases. Three rectangular faces join corresponding sides of the bases. Classify it and describe one symmetry feature if it is a right triangular prism with matching ends.

Triangular prism

Schematic view of a prism with two matching triangular bases joined by side faces.

Check the bases
There are two bases, and both are triangles. They match and are parallel. That is the defining base pattern of a prism.
Check the joining faces
The three rectangular faces join the matching sides of the triangular bases. This supports the classification as a triangular prism.
Describe symmetry
For the stated right prism with matching ends, a plane halfway between the triangular bases divides it into matching halves. The matching condition matters; not every prism must have this symmetry.

Answer: It is a triangular prism. In the stated right-prism arrangement, a plane halfway between its triangular bases is a plane of symmetry.

Check: The two matching, parallel polygon bases identify the prism. The joining faces confirm its triangular-prism name.

Worked example

A square base and one apex

A solid has one square base. Four triangular faces meet at one point above the centre of the base. Classify the solid and describe a symmetry plane.

Square pyramid

Schematic square pyramid with four triangular side faces meeting at an apex.

Find the base and meeting point
There is one polygon base, and it is a square. The side faces meet at one point, the apex. These are the defining features of a pyramid.
Name the pyramid
Pyramids are named for their base shape. Since this base is a square, the solid is a square pyramid.
Use the stated arrangement
The apex is above the centre of the square. A vertical plane through the apex and a line that divides the square into matching halves gives matching mirror-image sides. This is a plane of symmetry for the stated solid.

Answer: It is a square pyramid. Its centred apex and square base give it planes of symmetry through the apex and matching dividing lines of the base.

Check: One square base plus triangular faces meeting at one apex identifies a square pyramid.

Worked example

A can-shaped solid

A closed solid has two equal circular bases, one above the other, joined by a curved surface. It is upright and has the same circular shape from top to bottom. Classify it and give one symmetry feature.

Cylinder

Schematic upright cylinder showing its two circular bases and curved joining surface.

Look at the bases
The solid has two equal circular bases. They are joined by a curved surface, not by flat polygon faces.
Classify it
Two circular bases joined by a curved surface are the defining properties of a cylinder. Circular bases also show why this is not a prism in this classification.
Describe symmetry
A plane halfway between the two bases divides this upright cylinder into matching halves. The line through the centres of the bases also gives an axis for turns that leave its appearance unchanged.

Answer: It is a cylinder. It has a plane of symmetry halfway between its circular bases and rotational symmetry around the line through the centres of those bases.

Check: The two circular bases and curved joining surface distinguish the cylinder from a prism or pyramid.

Common mistakes and how to avoid them

Calling every solid with a flat base a prism.

Correction: A prism has two matching, parallel polygon bases. A pyramid has one polygon base and side faces meeting at an apex.

Calling a cylinder a prism because it has two bases.

Correction: A cylinder’s bases are circular and its joining surface is curved. A prism has polygon bases and flat side faces.

Assuming every member of a family has the same symmetry.

Correction: Check the exact base shape and arrangement. Some prisms or pyramids may be slanted or arranged differently.

Counting a cylinder’s circular rim as a vertex or as an edge between flat faces.

Correction: A cylinder has no vertices. Its curved surface is not a flat face, so its circular rims are not edges where two flat faces meet.

Lesson summary

A prism has two matching, parallel polygon bases joined by side faces.
A pyramid has one polygon base and triangular faces that meet at one apex.
A cylinder has two circular bases joined by a curved surface.
A plane of symmetry divides a solid into matching mirror-image halves. Rotational symmetry describes a turn that leaves the solid matching its starting position.
Classify by checking bases and joining surfaces. Describe symmetry only when the solid’s particular shape supports it.

Check your understanding

Question 1

A solid has one pentagonal base, and five triangular faces meet at one apex. What is it?

A pentagonal pyramid
A pentagonal prism
A cylinder
A rectangular prism

Show answer and explanation

A pentagonal pyramid

One pentagonal base and triangular faces meeting at an apex identify a pentagonal pyramid.

Question 2

Which feature best distinguishes a cylinder from a prism?

A cylinder has a curved joining surface and circular bases.
A cylinder has two matching polygon bases.
A cylinder has triangular faces meeting at an apex.
A cylinder must have eight vertices.

Show answer and explanation

A cylinder has a curved joining surface and circular bases.

A cylinder has circular bases joined by a curved surface. A prism has polygon bases and flat side faces.

Question 3

A learner says, “Every prism has a plane of symmetry halfway between its bases.” Which response is best?

Check the particular prism; symmetry depends on its arrangement.
That is true because every prism has a circular base.
That is true because every prism has an apex.
No solid can have a plane of symmetry.

Show answer and explanation

Check the particular prism; symmetry depends on its arrangement.

A prism is identified by its two matching, parallel polygon bases. Its symmetry depends on the exact solid and arrangement.

Key terms

Apex: The point where the triangular side faces of a pyramid meet.
Base: A face used to describe the main shape of a solid.
Edge: A line segment where two flat faces meet.
Face: A flat surface of a solid.
Plane of symmetry: An imaginary flat slice that divides a solid into matching mirror-image halves.
Rotational symmetry: A property in which a turn can bring a solid to a position that matches its starting position.
Vertex: A corner where edges meet. The plural is vertices.

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Published by DoAssignment. This reviewed lesson follows Ontario Grade 7 Mathematics, expectation E1.1. It is a study resource, not an official curriculum publication.

Before publication, content is checked for structure, mathematical or chemical notation, calculations, course boundaries, and readability. Errors can still occur, so corrections are welcomed.

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