Reflect a whole solid through a plane rather than reflecting one face.
Extended · Syllabus E4.1, E4.5
Notes and key methods
A plane of symmetry divides a solid into two mirror-image halves. A line drawn on one visible face is not itself a plane. Count distinct planes of the whole solid, including planes not parallel to a face.
A cube has nine symmetry planes: three through opposite face centres and six diagonal planes. A cuboid with all three dimensions distinct has only three. An equilateral triangular prism has three longitudinal planes and one midplane; a right regular square pyramid has four vertical planes.
Check the conditions
Use the stated units and precision.
Show the method as well as the final answer.
Three worked examples
Example 1 · Solid shapes and planes of symmetry
Consider a right prism with equilateral triangular ends. Count the faces, edges and vertices of the complete solid, including hidden ones.
How many planes of symmetry does it have?
How many faces does it have?
How many edges does it have?
How many vertices does it have?
Show the reasoning
(a)
Use symmetry of the entire named solid and its stated regularity.
The complete solid has 4 distinct mirror planes.
Answer: \(4\).
(b)
Include both visible and hidden faces of the closed solid.
Counting every polygonal face gives 5.
Answer: \(5\).
(c)
Do not double-count an edge because it belongs to two faces.
Count each shared face boundary once to obtain 9.
Answer: \(9\).
(d)
Count distinct vertices, including hidden ones.
The complete solid has 6 distinct corners.
Answer: \(6\).
Example 2 · Solid shapes and planes of symmetry
Consider a right regular square-based pyramid. Count the faces, edges and vertices of the complete solid, including hidden ones.
How many planes of symmetry does it have?
How many faces does it have?
How many edges does it have?
How many vertices does it have?
Show the reasoning
(a)
Use symmetry of the entire named solid and its stated regularity.
The complete solid has 4 distinct mirror planes.
Answer: \(4\).
(b)
Include both visible and hidden faces of the closed solid.
Counting every polygonal face gives 5.
Answer: \(5\).
(c)
Do not double-count an edge because it belongs to two faces.
Count each shared face boundary once to obtain 8.
Answer: \(8\).
(d)
Count distinct vertices, including hidden ones.
The complete solid has 5 distinct corners.
Answer: \(5\).
Example 3 · Solid shapes and planes of symmetry
Consider a cube. Count the faces, edges and vertices of the complete solid, including hidden ones.
How many planes of symmetry does it have?
How many faces does it have?
How many edges does it have?
How many vertices does it have?
Show the reasoning
(a)
Use symmetry of the entire named solid and its stated regularity.
The complete solid has 9 distinct mirror planes.
Answer: \(9\).
(b)
Include both visible and hidden faces of the closed solid.
Counting every polygonal face gives 6.
Answer: \(6\).
(c)
Do not double-count an edge because it belongs to two faces.
Count each shared face boundary once to obtain 12.
Answer: \(12\).
(d)
Count distinct vertices, including hidden ones.
The complete solid has 8 distinct corners.
Answer: \(8\).
Common mistakes
Keep intermediate values exact until the final step.