Distinguish reflection axes from rotational order.
Core & Extended · Syllabus C4.1, C4.5, E4.1, E4.5
Notes and key methods
A line of symmetry reflects a shape onto itself. Rotational symmetry counts how many times a shape matches itself during a full 360° turn, including the final turn. A shape with no nontrivial rotational symmetry has order 1.
The smallest positive rotation is 360° divided by the rotational order. A generic parallelogram has half-turn symmetry but no line of symmetry; a rectangle has two reflection axes. Extra equal lengths or angles can change the answer, so use every qualifier in the named shape.
Check the conditions
Use the stated units and precision.
Show the method as well as the final answer.
Three worked examples
Example 1 · Shapes and their symmetries
Consider an equilateral triangle.
How many lines of symmetry does it have?
Find its order of rotational symmetry.
Find its interior-angle sum in degrees.
Find its smallest positive angle of rotational symmetry in degrees.
Show the reasoning
(a)
Use the full shape description, including exclusions of special cases.
The stated shape has 3 reflection axes.
Answer: \(3\).
(b)
Count matching positions during one full revolution.
It matches itself 3 times in a complete turn, including 360°.
Answer: \(3\).
(c)
Use the polygon angle sum, not an assumption that all angles are equal.
For 3 vertices the sum is \(\displaystyle 180(3-2)=180\)°.
Answer: \(180\).
(d)
Convert rotational order into a rotation angle.
Divide a full turn by order 3 to obtain \(120\)°.
Answer: \(120\).
Example 2 · Shapes and their symmetries
Consider a regular pentagon.
How many lines of symmetry does it have?
Find its order of rotational symmetry.
Find its interior-angle sum in degrees.
Find its smallest positive angle of rotational symmetry in degrees.
Show the reasoning
(a)
Use the full shape description, including exclusions of special cases.
The stated shape has 5 reflection axes.
Answer: \(5\).
(b)
Count matching positions during one full revolution.
It matches itself 5 times in a complete turn, including 360°.
Answer: \(5\).
(c)
Use the polygon angle sum, not an assumption that all angles are equal.
For 5 vertices the sum is \(\displaystyle 180(5-2)=540\)°.
Answer: \(540\).
(d)
Convert rotational order into a rotation angle.
Divide a full turn by order 5 to obtain \(72\)°.
Answer: \(72\).
Example 3 · Shapes and their symmetries
Consider a nonequilateral isosceles triangle.
How many lines of symmetry does it have?
Find its order of rotational symmetry.
Find its interior-angle sum in degrees.
Find its smallest positive angle of rotational symmetry in degrees.
Show the reasoning
(a)
Use the full shape description, including exclusions of special cases.
The stated shape has 1 reflection axes.
Answer: \(1\).
(b)
Count matching positions during one full revolution.
It matches itself 1 times in a complete turn, including 360°.
Answer: \(1\).
(c)
Use the polygon angle sum, not an assumption that all angles are equal.
For 3 vertices the sum is \(\displaystyle 180(3-2)=180\)°.
Answer: \(180\).
(d)
Convert rotational order into a rotation angle.
Divide a full turn by order 1 to obtain \(360\)°.
Answer: \(360\).
Common mistakes
Keep intermediate values exact until the final step.