Mathematics notes, unit 7
Shapes
Textbook Mathematics Year 3 (DLP), Part 1 and Part 2, pages Part 2, 85–96
What this chapter covers
- You can name square prisms (cubes), rectangular prisms (cuboids) and triangular prisms (7.1.1)
- You can count the flat surfaces, vertices and edges of a prism and find its base (7.1.2)
- You can tell prisms from non-prisms such as the sphere, cone, cylinder and pyramid (7.2.1)
- You can name regular polygons (pentagon, hexagon, heptagon, octagon) by their sides and corners (7.3.1)
- You can make patterns using polygons (7.3.2)
- You can find and draw axes of symmetry (7.4.1)
- You can solve problems about shapes with a checklist table, trial and error or drawing (7.5.1)
7.1Prisms
Look around your house. A box of tissue, a Rubik's cube and a slice of kek lapis all have the same kind of shape. They are prisms.
A prism is a 3-D (three-dimensional) shape that has 2 opposite surfaces of the same shape and size, called the bases. It has 5 or more flat surfaces and no curved surfaces.
A prism has three kinds of parts. Learn their names:
| Word | What it means |
|---|---|
| Flat surface | A flat side of the shape. You can put a book flat on it. |
| Edge | A line where two flat surfaces meet. |
| Vertex (more than one: vertices) | A corner where edges meet. It is a pointy part. |
| Base | One of the 2 opposite surfaces that have the same shape. A prism is named after its base. |
Three prisms you must know
The dashed lines show edges at the back that you cannot see.
| Prism | Shape of base | Flat surfaces | Vertices | Edges | Example |
|---|---|---|---|---|---|
| Cube (square prism) | square | 6 (all the same size) | 8 | 12 | dice, Rubik's cube |
| Cuboid (rectangular prism) | rectangle | 6 | 8 | 12 | tissue box, brick |
| Triangular prism | triangle | 5 | 6 | 9 | tent, Toblerone box |
Did you know? A cube is also called a square prism because its bases are squares. All 6 flat surfaces of a cube are squares of the same size.
Example: count the parts of a triangular prism
- Count the flat surfaces: 2 triangles (the bases) + 3 rectangles = 5 flat surfaces.
- Count the vertices: 3 on the front triangle + 3 on the back triangle = 6 vertices.
- Count the edges: 3 on the front + 3 on the back + 3 joining them = 9 edges.
If you cut a prism straight across (like slicing a loaf of bread), every slice has the same shape as the base. Cut a kek lapis and every slice is a rectangle!
Don't forget the hidden parts at the back. A cube has 8 vertices, not 7. Use the dashed lines to help you count.
A box of tissue is a cuboid. How many edges does it have?
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12 edges (a cuboid has 6 flat surfaces, 8 vertices and 12 edges).
7.2Prisms and non-prisms
Some 3-D shapes are not prisms. We call them non-prisms. Here are four of them.
A curved surface is a surface that is round, not flat. A ball rolls because it has a curved surface.
To check if a shape is a prism, ask three questions. A prism must get three ticks.
| Shape | 5 or more flat surfaces? | 2 opposite surfaces of the same shape? | No curved surfaces? | Prism? |
|---|---|---|---|---|
| Cuboid | ✓ | ✓ | ✓ | Yes |
| Sphere | ✗ | ✗ | ✗ | No |
| Cone | ✗ | ✗ | ✗ | No |
| Pyramid | ✓ | ✗ | ✓ | No |
| Cylinder | ✗ | ✓ | ✗ | No |
A cylinder has 2 opposite circles of the same size, so it looks like a prism. But it has a curved surface and only 2 flat surfaces. So a cylinder is a non-prism.
A pyramid has 5 flat surfaces and no curved surface, but it does not have 2 opposite surfaces of the same shape. Its top is a point (a vertex). So a pyramid is a non-prism.
Nets
A net is a flat shape that you can fold to make a 3-D shape. Cut it out, fold along the lines and stick the edges together.
Look for circles in a net. Circles mean a curved shape, so the shape is a non-prism (a cylinder or a cone).
Sort these shapes into prisms and non-prisms: Rubik's cube, football, tent, ice cream cone.
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Prisms: Rubik's cube (cube), tent (triangular prism). Non-prisms: football (sphere), ice cream cone (cone).
7.3Regular polygons and patterns
A polygon is a flat (2-D, two-dimensional) shape with straight sides. A regular polygon has 3 or more straight sides of equal length, 1 flat surface and no curved sides.
A polygon has sides (the straight lines) and corners (where two sides meet). A polygon always has the same number of corners as sides.
Four regular polygons to learn
| Name | Straight sides | Corners | Where you may see it |
|---|---|---|---|
| Pentagon | 5 | 5 | the patches on some footballs |
| Hexagon | 6 | 6 | honeycomb, the head of a nut (bolt) |
| Heptagon | 7 | 7 | some badges and tiles |
| Octagon | 8 | 8 | the "Berhenti" (stop) road sign |
Memory trick: an octopus has 8 legs, so an octagon has 8 sides. Hexagon and six both have an "x".
Regular or not?
Both pentagons have 5 sides. Only the first one is regular, because all its sides are the same length. The second one has sides of different lengths, so it is irregular. A circle has a curved side, so it is not a polygon at all.
A shape with 5 sides is always a pentagon, but it is only a regular pentagon when all 5 sides are equal.
Name the polygon that has 7 corners and 7 straight sides.
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Heptagon.
Making patterns
We can join polygons together to make beautiful patterns, like the tiles on a floor or a honeycomb made by bees.
Try this: cut out paper pentagons or hexagons and paste them to make a flower, a fish or a butterfly. Use bright colours, like the patterns on a songket or a Hari Raya card.
7.4Axis of symmetry
Fold a picture along a line. If the two parts overlap each other exactly, the fold line is an axis of symmetry.
An axis of symmetry is a straight line that divides a shape into two equal parts of the same shape and size. More than one axis are called axes of symmetry.
Did you know? A regular polygon has the same number of axes of symmetry as sides. A regular pentagon has 5, a regular hexagon has 6, a regular heptagon has 7 and a regular octagon has 8.
Example: is the diagonal of a rectangle an axis of symmetry?
- Cut a paper rectangle along the diagonal. You get two triangles of the same size.
- Now fold the rectangle along the diagonal instead. The corners stick out. The two parts do not overlap exactly.
- So the diagonal is not an axis of symmetry. A rectangle has only 2 axes: one across the middle and one down the middle.
Two parts of the same size are not enough. They must fit on top of each other exactly when you fold. Always test with an imaginary fold.
How many axes of symmetry does a regular octagon (like the "Berhenti" sign) have?
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8 axes of symmetry (a regular polygon has as many axes as sides).
7.5Solving problems
Here are three ways to solve problems about shapes.
Method 1: a checklist table
Puan Salina sticks pictures of a cone, a cuboid and a sphere on the whiteboard. Azian must choose the prism. Which one does she choose?
| Shape | 5 or more flat surfaces | 2 opposite surfaces the same | No curved surfaces |
|---|---|---|---|
| Cone | ✗ | ✗ | ✗ |
| Cuboid | ✓ | ✓ | ✓ |
| Sphere | ✗ | ✗ | ✗ |
Only the cuboid gets three ticks. Azian chooses the cuboid, because it has 6 flat surfaces, its opposite surfaces are the same shape and it has no curved surfaces.
Method 2: trial and error
Wong cuts a shape. It has 2 corners more than a pentagon. What is the shape?
- A pentagon has 5 corners.
- Try a hexagon: 6 corners. That is only 1 more. No.
- Try a heptagon: 7 corners. 7 − 5 = 2 more. Yes!
The shape is a heptagon.
Method 3: draw the axes of symmetry
Chin has three picture cards: a triangle road sign with a traffic light drawn inside it, a handkerchief (rectangle) and a kite. Zaki must take the card with more than one axis of symmetry.
- Traffic light road sign: 1 axis (down the middle). The lights are red at the top and green at the bottom, so no other fold works.
- Handkerchief: 2 axes (across and down).
- Kite: 1 axis (down the middle).
Zaki takes the handkerchief.
Jaslina has three cards: a circle, a regular pentagon and a shape with 4 sides of different lengths. She chooses the regular polygon. Which card does she choose? Why?
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The regular pentagon. Its 5 straight sides are all the same length. The circle has a curved side, and the other shape has sides of different lengths.
Summary
- A prism has 5 or more flat surfaces, 2 opposite bases of the same shape and size, and no curved surfaces. It is named after its base.
- Cube (square prism) and cuboid (rectangular prism): 6 flat surfaces, 8 vertices, 12 edges. Triangular prism: 5 flat surfaces, 6 vertices, 9 edges.
- Non-prisms: sphere, cone, cylinder (curved surfaces) and pyramid (no 2 opposite bases).
- A regular polygon is a flat shape with 3 or more straight sides of equal length. Pentagon 5, hexagon 6, heptagon 7, octagon 8 sides (and the same number of corners).
- Polygons can be joined to make patterns.
- An axis of symmetry divides a shape into two parts that overlap exactly when folded. A regular polygon has as many axes as sides. A rectangle has 2; its diagonal is not an axis.
- Solve problems with a checklist table, trial and error, or by drawing axes of symmetry.