Symmetry planes and axes in solids
Extend reflection and rotation to specified prisms, cylinders, pyramids and cones.
Extended only.
Before you begin
- Describe 2D line and rotational symmetry.
- Recognise right prisms and regular pyramids.
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
What does a line of symmetry become when extended along a right prism?
What you will learn
- Identify a plane of symmetry in three dimensions.
- Describe a rotational symmetry axis and its order.
- Use a solid's precise shape rather than generalising from its name.
A plane reflects the whole solid
A plane of symmetry is a flat slice such that reflection in it sends the entire solid onto itself. A cuboid with three different edge lengths has three symmetry planes, each through its centre parallel to a pair of opposite faces. A diagonal slice is not generally a symmetry plane; extra equal dimensions can create extra symmetry, as in a cube.
For a right prism, each mirror line of its cross-section extends along the prism to give a longitudinal symmetry plane. A plane halfway between the two ends also swaps the matching ends. A right equilateral triangular prism therefore has three longitudinal planes plus one halfway plane. These statements assume a right prism with unchanged cross-section; a slanted solid may have different symmetry.
Specify the axis and the matching turn
A right regular square-based pyramid has four vertical symmetry planes, each through its apex and a symmetry line of the square base. Its vertical axis through apex and base centre has rotational order four. There is no horizontal symmetry plane: reflection would send the apex to a point below the base that is not part of the pyramid.
A right circular cylinder has infinitely many symmetry planes through its central axis, plus the plane halfway between its ends. It matches under any rotation about that axis. A right circular cone also has infinitely many planes containing its axis and matches under any axial rotation, but has no halfway horizontal plane. For a right equilateral triangular prism the longitudinal axis has order three. Always name the axis: a non-cubic cuboid has order two about each central axis parallel to an edge, while these are different axes from a cylinder's continuous one.
Pause and explain
Which extra plane belongs to a right cylinder but not a right cone?
Worked example
Describe the symmetry planes and vertical rotational symmetry of a right regular square-based pyramid.
Show the worked solution
- Extend each of the square base's four mirror lines upward through the apex to give four vertical planes.
- Quarter-turns around the apex-to-base-centre axis preserve the solid, giving order four.
- A horizontal reflection would not preserve the apex and base, so there is no horizontal symmetry plane.
Answer 4 vertical symmetry planes; vertical axis of order 4; no horizontal plane.
Try a topic test
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Test this topic →From guided practice to a new situation
Write your answer and reasoning first. Use a hint only if you are stuck. The model solution is for self-checking; your written work is not automatically marked.
How many symmetry planes does a cuboid with dimensions 2 cm, 3 cm and 5 cm have? Describe them.
Give me a hint
Unequal dimensions exclude the cube's extra diagonal planes.
Compare my reasoning
- A central plane parallel to each pair of opposite faces reflects the solid onto itself.
- There are three such orientations, one for each dimension.
- No diagonal exchange of unequal dimensions gives an additional plane.
3 central planes, each parallel to a pair of opposite faces.
Look for these in your work
- Specified three different dimensions.
- Described planes rather than only drawing lines.
A right equilateral triangular prism has how many symmetry planes, and what is the order about its longitudinal axis?
Give me a hint
Use the triangle's three mirror lines and its rotational order.
Compare my reasoning
- Each of the three mirror lines gives one longitudinal plane.
- The central plane between the two ends adds a fourth plane.
- Rotation by 120° about the longitudinal axis matches, so that axis has order three.
4 planes; longitudinal rotational order 3.
Look for these in your work
- Included the plane swapping the two ends.
- Named the rotational axis.
A right circular cone-shaped ornament has a mark on one side. Would the undecorated cone's continuous rotational symmetry necessarily remain?
Give me a hint
Test the whole object including the decoration.
Compare my reasoning
- An undecorated right circular cone matches after any rotation about its axis.
- Rotating moves the single side mark to a different place.
- The decorated object no longer necessarily has that continuous symmetry; the mark must also match.
No; the decoration must be preserved as well as the solid's surface.
Look for these in your work
- Stated the undecorated cone's symmetry.
- Tested all features of the actual object.
Common mistakes
- Calling a drawn line a 3D symmetry plane.
- Giving all prisms the same symmetries regardless of their bases.
- Assuming a cone has the cylinder's halfway plane.
What can you explain now?
Does a right regular square-based pyramid have a horizontal symmetry plane halfway up?
Compare with the explanation
No.
Such reflection would exchange unlike cross-sections and cannot map the apex and square base onto matching parts.
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Make it stick
Tomorrow, describe one symmetry plane and one axis for a specified prism or pyramid. Explain which parts must swap or stay fixed.
If you needed the solution, close it and solve the problem again from a blank page. A correct answer is stronger when you can explain the reason for each step.