E. Spatial Sense · Spatial Sense: 3D Objects, Transformations, Circles, Surface Area and VolumePlane Symmetry and Rotational Symmetry of 3D Objects
E1.1About 45 minutes
Learning goals- I can explain what a plane of symmetry is and find the planes of symmetry of prisms, pyramids and cylinders.
- I can find an axis of rotational symmetry and the order of rotational symmetry of a 3D object.
- I can use symmetry, along with faces, edges and vertices, to describe and classify 3D objects.
Words to know
plane
A perfectly flat surface that goes on forever in every direction, like an endless sheet of glass.
plane of symmetry
A plane that cuts a 3D object into two halves that are mirror images of each other.
axis of rotation
An imaginary straight line (like a skewer) that an object turns around.
rotational symmetry
An object has rotational symmetry if it looks exactly the same after turning it less than a full turn around an axis.
order of rotational symmetry
The number of times an object matches itself during one full turn around an axis.
regular polygon
A polygon with all sides equal and all angles equal, like an equilateral triangle or a square.
Remember from Grade 6
In Grade 6 you found lines of symmetry of 2D shapes (fold lines that make two matching halves) and the order of rotational symmetry (how many times a shape matches itself in one full turn). A square has 4 lines of symmetry and rotational symmetry of order 4.
In 3D, a line of symmetry becomes a plane of symmetry, and you turn objects around an axis instead of a point.
Let's start with a sheet of glass
Imagine a giant, perfectly thin sheet of glass slicing a cereal box straight down the middle. If the two halves are mirror images of each other, the glass is on a plane of symmetry.
Now imagine pushing a skewer straight down through the centre of a square pyramid and spinning it. After a quarter turn (90°), it looks exactly the same as before. The skewer is an axis of rotation, and the pyramid has rotational symmetry.
How to count
Planes of symmetry: look for every flat cut that makes two mirror-image halves. For a prism, think about the base: every line of symmetry of the base gives a vertical plane. Then check for a horizontal plane halfway up.
Rotational symmetry: pick an axis, usually through the centres of the two bases of a prism, or through the apex and the centre of the base of a pyramid. Count how many times the object matches itself in one full turn. That count is the order. The smallest angle of turn is 360° ÷ order.
| Object | Planes of symmetry | Order of rotational symmetry (axis through the bases, or apex and base) |
|---|
| rectangular prism (length, width, height all different) | 3 | 2 |
| square prism that isn't a cube | 5 | 4 |
| cube | 9 | 4 |
| right triangular prism, equilateral triangle base | 4 | 3 |
| right hexagonal prism, regular hexagon base | 7 | 6 |
| right square pyramid (apex above the centre) | 4 | 4 |
| right cylinder | infinitely many | infinite: any angle works |
Spot the pattern
For a right prism with a regular n-sided base (that isn't a cube): n vertical planes + 1 horizontal plane = n + 1 planes, and rotational symmetry of order n around the axis through the centres of the bases.
For a right pyramid with a regular n-sided base: n planes (all vertical) and order n. A pyramid has no horizontal plane of symmetry, because the top half has the apex and the bottom half has the base.
A right cylinder is special: any vertical cut through the middle is a plane of symmetry, so there are infinitely many, plus one horizontal plane. It matches itself after a turn of any angle.
Worked exampleExample 1: A fence post
A wooden fence post is a square prism: its base is a square 10 cm by 10 cm and it is 120 cm tall. How many planes of symmetry does it have?
- The base is a square. A square has 4 lines of symmetry (2 through the middles of opposite sides and 2 along the diagonals). Each gives a vertical plane: 4 planes.
- A horizontal cut halfway up (at 60 cm) makes a top half and a bottom half that are mirror images: 1 more plane.
- It is not a cube (120 cm ≠ 10 cm), so there are no extra planes.
Worked exampleExample 2: A hexagonal pencil
An unsharpened pencil is a hexagonal prism with a regular hexagon base. Find its order of rotational symmetry around the axis through the centres of its two ends, and the smallest angle of turn.
- Turn the pencil around its long axis. Each time a new flat side comes to the top, it looks the same.
- A hexagon has 6 sides, so this happens 6 times in one full turn.
- Smallest angle: 360° ÷ 6 = 60°.
Order 6; smallest angle of turn 60°.
Looking straight at the end of the pencil, the axis is the centre dot. Each 60° turn brings a new flat side to the top, so the order of rotational symmetry is 6.Worked exampleExample 3: Classify using symmetry
An object has an apex, exactly 4 planes of symmetry, and rotational symmetry of order 4 around one axis. What could it be?
- An apex means it is a pyramid.
- For a right pyramid with a regular base, the number of planes and the order both equal the number of sides on the base. Both are 4.
- A regular 4-sided polygon is a square.
The mirror test
Hold a small mirror straight against an object, right where you think a plane of symmetry is. If the object plus its reflection looks like the whole object, you found a plane of symmetry. The mirror acts like the sheet of glass.
Hands-on activitySymmetry with a mirror and a turn
You need: a small plastic (shatterproof) mirror, a few objects (a cereal box, a can, a square tissue box, a hexagonal pencil), a sticky note.
- Planes: Hold the mirror against each object in different places: straight down the middle, corner to corner, across the middle. Count the planes of symmetry you find.
- Rotation: Put a sticky note on the table in front of the object. Stand the object on its base and slowly turn it one full turn. Count how many times it looks exactly as it did at the start. That is the order.
- Record your results in a table and compare them with the table in this lesson.
What to look for: a square tissue box (if its height differs from its width) should give 5 planes and order 4 around its vertical axis; a can should match itself at every angle.
Just for funA cube has 9 planes of symmetry. That's 9 different ways to cut a cube-shaped cake into two perfectly equal halves, so nobody can complain that their piece is smaller!
Your turn
1How many planes of symmetry does a cube have?
9. 3 planes are parallel to the faces (cutting it into two equal boxes), and 6 go diagonally through pairs of opposite edges.
2What is the order of rotational symmetry of a right square pyramid around the axis through its apex and the centre of its base?
4. It matches itself every 90°.
3How many planes of symmetry does a right cylinder have?
Infinitely many. Every vertical plane through the middle works, plus the horizontal plane halfway up.
4How many planes of symmetry does a right triangular prism with an equilateral triangle base have?
4. 3 vertical planes (one for each line of symmetry of the triangle) + 1 horizontal plane.
5What is the smallest angle of turn for a right hexagonal prism (regular base) around the axis through its bases?
60°. Order 6, and 360° ÷ 6 = 60°.
6Why doesn't a pyramid have a horizontal plane of symmetry?
A horizontal cut makes a small pointed top piece and a bigger bottom piece with the base. They are not mirror images.
More practice: warm-up
1A plane of symmetry cuts an object into two ______ halves.
2How many planes of symmetry does a rectangular prism with length, width and height all different have?
3What is the order of rotational symmetry of a right pentagonal prism (regular base) around the axis through its bases?
4An object has rotational symmetry of order 4. What is the smallest angle of turn?
5True or false: a right cylinder looks the same after turning it any angle around its central axis.
More practice: on your own
1A right octagonal prism has a regular octagon base and is not especially tall or short. How many planes of symmetry does it have? What is its order of rotational symmetry around the axis through its bases, and its smallest angle of turn?
9 planes, order 8, smallest angle 45°. A regular octagon has 8 lines of symmetry, giving 8 vertical planes, plus 1 horizontal plane. 360° ÷ 8 = 45°.
2A right hexagonal pyramid has a regular hexagon base. How many planes of symmetry? What order of rotational symmetry around the axis through the apex?
6 planes, order 6. One vertical plane for each line of symmetry of the hexagon; no horizontal plane, because of the apex.
3A square prism has a base 5 cm by 5 cm and a height of 5 cm. How many planes of symmetry does it have?
9. All its edges are 5 cm, so it's a cube, which has 9 planes (not 5).
4A tent is shaped like a right triangular prism whose triangle is isosceles but not equilateral. The tent stands on its rectangular floor. How many planes of symmetry does it have? Does it have rotational symmetry around the axis through the centres of its two triangle ends?
2 planes: the vertical plane down the middle of the tent (through the ridge line), because an isosceles triangle has 1 line of symmetry, and the plane that cuts the tent into a front half and a back half. No rotational symmetry around that axis: the triangle ends only match themselves after a full turn.
5Which has more planes of symmetry: a right square pyramid or a square prism that isn't a cube? Why?
The square prism (5 planes vs 4). Both have the 4 vertical planes from the square, but only the prism has a horizontal plane, because its top and bottom match.
Challenge
1A regular tetrahedron is a triangular pyramid whose 4 faces are all equilateral triangles. Each plane of symmetry contains one edge and passes through the middle of the opposite edge. It has 6 edges. How many planes of symmetry does it have?
6. There is one plane for each of the 6 edges. (That's more than the 3 planes of a triangular pyramid whose side faces aren't equilateral, because now every face could be the base.)
2An object has an apex, exactly 3 planes of symmetry, and rotational symmetry of order 3 around one axis. Describe it.
A right triangular pyramid with an equilateral triangle base, with the apex directly above the centre of the base. Apex → pyramid; order 3 and 3 planes → a regular 3-sided base.
3A box is 8 cm long, 4 cm wide and 4 cm high. Ana says it has 3 planes of symmetry, like any box. Is she right? Explain.
No, it has 5. Its two ends are 4 cm by 4 cm squares, so it's really a square prism lying on its side. The square gives 4 planes along the length, and there is 1 plane cutting it into two 4 cm halves. Around the long axis through the centres of the square ends it has rotational symmetry of order 4.
Quiz
1. What does a plane of symmetry do?
2. How many planes of symmetry does a cube have?
3. What is the order of rotational symmetry of a right square pyramid around the axis through its apex?
4. What is the smallest angle of turn for a right hexagonal prism (regular base) around the axis through its bases?
5. How many planes of symmetry does a rectangular prism with length, width and height all different have?
6. Which object has infinitely many planes of symmetry?
Big idea: E1.1 includes classifying prisms, pyramids and cylinders by plane symmetry and rotational symmetry. Students extend their Grade 6 work on lines and rotational symmetry of 2D shapes into 3D.
How to help at home: Use real objects and a small mirror. For rotational symmetry, stand the object on a table and turn it slowly, counting how many times it looks the same. A pencil, a can and a tissue box make a good set.
Common difficulties:
- Forgetting the horizontal plane in prisms and cylinders, or wrongly adding one to pyramids.
- Assuming all boxes have 3 planes. If two dimensions are equal, a box has more symmetry (a cube has 9).
- Confusing lines with planes. In 3D, the mirror is a flat sheet, not a line.
These questions are hard to picture. It's fine (and smart) to use real objects or modelling clay to check.