Minds On
Examining polygons
Examine the polygons and identify some of their key properties.
Think about the number of parallel sides, equal sides, types of angles, diagonals, rotational symmetry, and line symmetry.
Record your ideas using the Properties of Shapes fillable and printable document, or using a method of your choice.
Press the ‘Activity’ button to access Properties of Shapes.
Student Success
Check your understanding
Can you name the shape that matches this description?
This shape has:
- Four equal sides
- Opposite sides are parallel
- Four equal angles
- Diagonal lines bisect at a 90-degree angle
- Four lines of symmetry
- Rotational symmetry of 4
Note to teachers: See your teacher guide for collaboration tools, ideas and suggestions.
Press ‘Answer’ to to access the shape that is described.
This shape is a square.
Cylinders, pyramids, and prisms represent three broad categories of three-dimensional objects. In the next section we will explore these objects and their characteristics.
Action
Task 1: Three-dimensional objects
We can use our knowledge of two-dimensional shapes and apply it to three-dimensional objects such as cylinders, pyramids, and prisms.
Investigate a variety of three-dimensional objects.
- How many faces does each object have?
- Are there any pairs of faces the same size?
As you investigate each object, record your findings. You may complete the Three-Dimensional Objects graphic organizer in your notebook or using the following fillable and printable document.
Press the ‘Activity’ button to access Three-Dimensional Objects.
Consider a triangular prism.
Consider a square-based prism.
Consider a cylinder.
Consider a triangle-based pyramid.
Consider a square-based pyramid.
Press ‘Suggested Answers’ to reveal the number of faces for each object.
| Object | Triangular Faces | Rectangular Faces | Square Faces | Circular Faces | Curved Faces |
|---|---|---|---|---|---|
| Triangular Prism |
2 |
3 |
0 |
0 |
0 |
| Square-based Prism |
0 |
4 |
2 |
0 |
0 |
| Cylinder |
0 |
0 |
0 |
2 |
1 |
| Triangle-based Pyramid |
4 |
0 |
0 |
0 |
0 |
| Square-based Pyramid |
4 |
0 |
1 |
0 |
0 |
Task 2: Planes of symmetry
Symmetry for an object refers to whether it can be cut into 2 equal parts or rotated to look the same.
For two-dimensional shapes such as rectangles, triangles, circles, parallelograms, etc. there may be “line symmetry.”
Line symmetry means there’s a line (or more than one) that can be drawn through the shape, so it is cut into two equal parts.
Consider an equilateral triangle which has three lines of symmetry. The line of symmetry can be created from each vertex.
This is an image of three equilateral Triangles. Each of the equilateral Triangles shows a different line of symmetry.
Description
An equilateral triangle has 3 lines of symmetry. The first line of symmetry starts at the top corner and intersects perpendicularly with the bottom side. The second line of symmetry starts at the left corner and intersects perpendicularly with the right side. The third line of symmetry starts at the right corner and intersects perpendicularly with the left side.
Three-dimensional objects don’t have line symmetry but may have plane symmetry.
A plane is a flat surface that stretches in all directions without ending. It is two-dimensional, having length and width but no thickness.
A three-dimensional object has plane symmetry if the object can be divided into two parts that are identical reflections of one another by a flat surface known as a plane.
Consider a triangular prism which has four planes of symmetry.
The following is an image of four triangular prisms, each displaying a different plane of symmetry.
Description
Four triangular prisms represent the four planes of symmetry. The first prism has a plane of symmetry in the middle of the long sides. The second, third and, fouth planes are located between the edge where two rectangles meet and the middle line of the opposite rectangle.
Now let's consider the following five 3D shapes:
Are there any planes of symmetry?
When you are finished, click on the link below to see how many planes of symmetry these objects have. Some have a large number of planes of symmetry.
Press each title to reveal more information about planes of symmetry.
| Object | Number Planes of Symmetry |
|---|---|
|
Rectangular Prism |
3 |
|
Cylinder |
infinite |
|
Cube |
9 |
|
Square-based Pyramid |
4 |
|
Triangular-based Pyramid |
6 |
A rectangular prism has 3 planes of symmetry. Each plane cuts the prism into mirrored halves. All three planes lie parallel to the faces and go through the centre of the edges.
A cylinder has an infinite number of planes of symmetry. One plane lies parallel with the circular bases. It then has an infinite number of planes that cut the cylinder into mirrored halves by bisecting the circular bases.
The cube has nine symmetry planes. Each plane cuts the cube into two mirrored halves. Three planes lie parallel to the faces and go through the centre of the edges. Six planes go from edge to edge on a diagonal. They divide the cube into triangular-based prisms.
The cube has nine symmetry planes. Each plane cuts the cube into two mirrored halves. Three planes lie parallel to the faces and go through the centre of the edges. Six planes go from edge to edge on a diagonal. They divide the cube into triangular-based prisms.
A triangular-based pyramid has six planes of symmetry. Each plane of symmetry goes from one of the edges of the triangular-based pyramid, cutting the object into two mirrored halves.
Task 3: Rotational symmetry
Some three-dimensional objects have rotational symmetry. In these cases, the object looks the same when they are rotated around a line. Even though we use a line, this isn’t called line symmetry, but instead rotational symmetry.
Brainstorm
What do you think?
Imagine a prism with a rod inserted through the middle so that the shape can rotate or turn.
What happens as you rotate the shape less than a full turn and stop at various points? At each turn, does the shape perfectly match its original position?
This is an image of a cylinder, an equilateral triangular-based prism, and a square-based pyramid. Each shape shows an imaginary rod inserted through the mid-point so that it can rotate or turn.
Consider the same objects as in the planes of symmetry task. What rotational symmetries can you identify?
Click on the name of each shape to access their lines of symmetry.
| Object | Number of Rotational Symmetries |
|---|---|
|
Equilateral Triangular-based Prism |
2 |
|
Rectangular Prism |
3 (1 for each pair of opposite sides) |
|
Cylinder |
infinite |
|
Cube |
24 (there are 13 lines to rotate about and there are multiple rotations for each one – see image) |
|
Square-based Pyramid |
3 |
|
Triangular-based Pyramid |
7 |
In the rectangular prism, a line extends from the middle of one square end to the opposite square end. An arrow indicates the rotation from one corner of the square end to the opposite corner of the same square end.
In a cylinder, a line extends from the centre of the circular end to the centre of the opposite circular end. Three arrows indicate the rotation of the cylinder around this line. The cylinder can be rotated any angle around this line, and it will look the same.
A cube has 13 rotational symmetries. The first three cubes depict the rotational symmetry that occurs when you connect the middle of opposite faces with a line. The next four cubes depict the rotational symmetry that occurs when you connect any vertex with its opposite vertex. The last six images depict the rotational symmetry that occurs when you connect the middle of any edge to the opposite middle of an edge.
The line of symmetry extends from the top vertex of the pyramid to the center of the square base. Three arrows indicate the rotation from one corner of the square to the next corner, one corner to the second next corner and one corner to the third next corner.
A triangular-based pyramid (tetrahedron) has seven rotational symmetries. A tetrahedron is a four-sided object in which all sides are equilateral triangles. The tetrahedron is drawn so that the base has a right and left side and a back side on the triangle. This means the tetrahedron has left and right faces that are triangles and a back triangle as well. The first image depicts a line of symmetry passing through the middle of the back side of the bottom face and the middle of the line joining the front two faces. The second image depicts a line of symmetry passing through the middle of the left side of the left face and the middle of the bottom side of the right face. The third image depicts a line of symmetry passing through the middle of the bottom side of the left face and the middle of the right side of the right face. The fourth image depicts a line of symmetry passing through the middle of the bottom face and peak at the top of the tetrahedron. The fifth image depicts a line of symmetry passing through the middle left face and bottom right corner of the right face. The sixth image depicts a line of symmetry passing through the middle of the back face and the bottom corner of the left side of the right face. Seventh image depicts a line of symmetry passing through the middle of the right face and the bottom corner of the left side of the left face.
Consolidation
Task 1: A prism of a different kind
You have learned about classifying objects by listing the types of faces and number of vertices, planes of symmetry and rotational symmetry.
Create a description for this hexagonal-based prism shown below using what you have learned about classifying objects.
The ends are “regular” hexagons, meaning all six sides are equal in length.
Task 2: Classifying objects
Complete the Classify Objects in your notebook or using the following fillable and printable document.
Press the ‘Activity’ button to access Classify Objects.
Reflection
As you read the following descriptions, select the one that best describes your current understanding of the learning in this activity. Press the corresponding button once you have made your choice.
I feel...
Now, expand on your ideas by recording your thoughts using a voice recorder, speech-to-text, or writing tool.
When you review your notes on this learning activity later, reflect on whether you would select a different description based on your further review of the material in this learning activity.
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