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.
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.
After trying it yourself, choose your next review. This is your self-assessment.
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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.
Original StudyForA teaching content · AI-assisted checks · Human teacher review pending.
Number contains 19 sequenced lessons; Algebra and graphs 21; Coordinate geometry 10; Geometry 15; Mensuration 13; Trigonometry 13; Transformations and vectors 11; Probability 7; Statistics 13. All 72 syllabus section entries now link to teaching and staged practice. Seven mixed checkpoints sample skills and suggest review lessons. Nine earlier overviews remain available. Full cumulative assessment and human teacher review remain pending. Written lesson practice, charts and constructions are self-checked, not automatically graded. Checkpoints do not save an exam grade or certify mastery.
Check the official syllabus 2025–2027 Open related practice and resources