DWG NO. 09.5 — Lesson 5 of 5

Symmetry

Unit 9: Transformations · ~15–30 min

Objective

Identify all lines of symmetry in a figure, and find the order and angle of rotational symmetry when a figure has it.

Line symmetry

A figure has line symmetry (also called reflectional symmetry) if a line can be drawn through it so that reflecting the figure over that line maps it exactly onto itself. That line is called a line of symmetry. A figure can have zero, one, or many lines of symmetry — an isosceles triangle has exactly one, a rectangle has two, and a regular polygon with n sides has exactly n.

Rotational symmetry

A figure has rotational symmetry if it can be rotated less than a full 360° about its center and land exactly on itself. The order of rotational symmetry is the number of positions (including the starting position) in which the figure looks the same during one full turn. The angle between those positions is found by dividing 360° by the order.

angle of rotational symmetry = 360° ÷ order

A regular polygon with n sides has rotational symmetry of order n, so its angle of rotational symmetry is 360° ÷ n — the same n that gives it n lines of symmetry.

A regular pentagon: 5 lines of symmetry and rotational symmetry of order 5.

Worked Example 1 · Lines of symmetry in a regular hexagon

ProblemHow many lines of symmetry does a regular hexagon have?

1.A regular polygon with n sides has n lines of symmetry.

6 lines of symmetry

Worked Example 2 · Order and angle of rotational symmetry

ProblemFind the order and angle of rotational symmetry for a square.

1.A square lands on itself 4 times in a full turn, so the order is 4.

2.Angle = 360° ÷ 4 = 90°.

Order 4, angle 90°

Worked Example 3 · A letter with rotational symmetry

ProblemDoes the letter S have rotational symmetry? Does it have line symmetry?

1.Rotating S by 180° about its center maps it onto itself, so it has rotational symmetry of order 2.

2.No line through S produces a mirror match — there is no line of symmetry.

Rotational symmetry (order 2, 180°), no line symmetry.

Guided practice

1.How many lines of symmetry does a regular octagon have?
Show answer
A regular polygon with n sides has n lines of symmetry, so an octagon has 8.
2.Find the order and angle of rotational symmetry for a regular pentagon.
Show answer
Order 5 (one for each side); angle = 360° ÷ 5 = 72°.
3.An equilateral triangle is rotated about its center. What is the smallest angle of rotation, greater than 0°, that maps it onto itself?
Show answer
Its rotational order is 3, so the smallest such angle is 360° ÷ 3 = 120°.
4.How many lines of symmetry does a non-square rectangle have, and where are they?
Show answer
Two: the vertical line and the horizontal line through the center. The two diagonals are not lines of symmetry, since a non-square rectangle's diagonal reflection does not map it onto itself.
5.A figure has rotational symmetry of order 6. What angle of rotation maps it onto itself, and how many total lines of symmetry would a regular version of that figure (a regular hexagon) have?
Show answer
Angle = 360° ÷ 6 = 60°. A regular hexagon (n = 6) has 6 lines of symmetry.

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