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Math · Spatial sense
Lines of Symmetry: Fold, Match and Count
Find lines of symmetry by folding, count them in squares, triangles and letters, reflect points across a mirror line, and spot the diagonal trap in rectangles.
The idea
Fold a paper heart down the middle and the two halves land exactly on top of each other. The fold is called a line of symmetry. A line of symmetry splits a shape into two halves that are mirror images: if you folded along the line, every part of one half would cover the matching part of the other half.
Some shapes have several lines of symmetry. A square has 4: one across the middle, one up and down, and its two diagonals. A rectangle that is not a square has only 2. Its diagonals cut it into two equal triangles, but folding along a diagonal does not make the corners meet, so the diagonals are not lines of symmetry.
Some shapes have none. A scalene triangle, with three different side lengths, has 0 lines of symmetry. A regular polygon, with all sides and all angles equal, has as many lines of symmetry as it has sides: an equilateral triangle has 3, a regular pentagon 5 and a regular hexagon 6. A circle has infinitely many, because every line through its centre works.
You can also use a line of symmetry like a mirror. To reflect a point across a mirror line, keep it at the same height and put it the same distance away on the other side. A point on the line stays where it is.
Capital letters are good practice, but be careful: the answer depends on how the letter is drawn. In this lesson, letters are simple block capitals, like the ones on a classroom alphabet strip.
Worked examples
Example 1 · Test a fold
Is the line from the top-left corner to the bottom-right corner of a long rectangle a line of symmetry?
- Imagine folding the rectangle along that diagonal.
- The top-right corner would swing over, but it would not land on the bottom-left corner; it would stick out past the edge.
- The two triangles have the same area, but they do not cover each other when folded.
No. A rectangle's diagonal is not a line of symmetry (unless the rectangle is a square).
Example 2 · Count the lines in a regular hexagon
How many lines of symmetry does a regular hexagon have?
- Draw lines joining opposite corners: there are 3.
- Draw lines joining the middles of opposite sides: there are 3 more.
- Check that you did not count any line twice: each line runs all the way across.
- 3 + 3 = 6, which matches the rule for regular polygons: one line per side.
6 lines of symmetry.
Example 3 · Reflect a point
On a grid, a vertical mirror line runs down the middle. A dot is 2 squares to the left of the line and 3 squares up from the bottom. Where is its reflection?
- A reflection stays at the same height: still 3 squares up.
- It is the same distance from the line on the other side: 2 squares to the right.
- Check: the dot and its reflection are both 2 squares from the line, so folding along the line would put one on the other.
2 squares to the right of the line, 3 squares up.
Example 4 · Letters
Sort these block capitals: A, B, H, F.
- A: the left and right halves match, so it has a vertical line.
- B: the top and bottom halves match, so it has a horizontal line.
- H: it has both a vertical and a horizontal line.
- F: no fold matches the halves, so it has none.
A vertical only; B horizontal only; H both; F none.
Example 5 · Complete a symmetrical shape
The left half of a shape has corners 1, 3 and 2 squares from a vertical mirror line. Complete the right half.
- For each corner, count its distance from the line.
- Mark a new corner the same distance on the right, at the same height.
- Join the new corners in the same order as the left half.
The right-half corners are 1, 3 and 2 squares from the line, at matching heights.
Common mistakes
These are tempting because they use familiar operations. Check what the question is actually measuring.
Why this fails: The diagonal splits a rectangle into equal triangles, but folding along it does not make the halves match. Only a square's diagonals work.
Why this fails: A line runs all the way across the shape. If you count from both ends, a square seems to have 8 lines instead of 4.
Why this fails: Scalene triangles and many parallelograms have 0. Zero is a correct answer.
Why this fails: The reflection is the same distance from the mirror line, not twice as far. Measure from the line, not from the original point.
Why this fails: A rhombus that is not a square has four equal sides but only 2 lines of symmetry, because its angles are not all equal.
Foundation
Foundation practice
Fold in your head, then count. Each line must match the two halves exactly.
1. You fold a paper square along the line joining the middle of the top side to the middle of the bottom side. The two halves match exactly. What is that fold line?
- a diagonal
- a side
- a line of symmetry
- a right angle
2. How many lines of symmetry does a square have?
- 4
- 2
- 8
- 1
3. How many lines of symmetry does a rectangle that is not a square have?
- 4
- 1
- 0
- 2
4. How many lines of symmetry does an equilateral triangle have?
- 1
- 3
- 6
- 0
5. A scalene triangle has three sides of different lengths. How many lines of symmetry does it have?
- 1
- 3
- 0
- 2
6. Which capital letter, drawn in simple block style, has a vertical line of symmetry?
- A
- F
- L
- R
7. Which capital letter, drawn in simple block style, has a horizontal line of symmetry but not a vertical one?
- H
- T
- N
- B
Practice
Practice practice
Count lines in more shapes and reflect points across a mirror line.
8. How many lines of symmetry does a regular hexagon (6 equal sides and 6 equal angles) have?
- 3
- 6
- 12
- 4
9. How many lines of symmetry does a regular pentagon have?
- 2
- 10
- 5
- 1
10. An isosceles triangle has exactly two equal sides. How many lines of symmetry does it have?
- 1
- 3
- 2
- 0
11. A parallelogram has no right angles and its sides are not all equal. How many lines of symmetry does it have?
- 2
- 4
- 1
- 0
12. How many lines of symmetry does a rhombus that is not a square have?
- 4
- 2
- 1
- 0
13. A dot is 3 squares to the left of a vertical mirror line. Where is its reflection?
- 6 squares to the right of the line
- 3 squares above the dot
- 3 squares to the right of the line, at the same height
- On the mirror line
14. A dot sits right on the mirror line. Where is its reflection?
- In the same place, on the line
- One square to the right
- At the edge of the grid
- It has no reflection
15. How many lines of symmetry does the block capital H have?
- 1
- 4
- 0
- 2
Think
Think practice
These questions target common mistakes. Explain why the tempting answer fails.
16. Ali says a rectangle's diagonal is a line of symmetry because it cuts the rectangle into two equal triangles. Why is he wrong?
- The triangles have different areas.
- The two triangles are the same size, but folding along the diagonal does not make them land on each other.
- Diagonals are never lines of symmetry in any shape.
- A rectangle has no lines of symmetry.
17. Mei says, "Every shape with four equal sides has 4 lines of symmetry." Which shape proves her wrong?
- a square
- a rectangle
- a rhombus that is not a square
- an equilateral triangle
18. Why is folding a paper cut-out a good way to check a line of symmetry?
- If the two halves cover each other exactly, the fold line is a line of symmetry.
- Folding makes every shape symmetrical.
- Folding measures the angles.
- Only squares can be folded.
19. True or false: a circle has exactly 4 lines of symmetry.
- True
- False
20. Leo counts the lines on a square by drawing them and gets 8. What mistake did he probably make?
- He forgot the diagonals.
- He used a rectangle by mistake.
- Squares really have 8 lines.
- He counted each line twice, once from each end.
21. True or false: if a shape is turned upside down, it gains a new line of symmetry.
- True
- False
22. An isosceles trapezoid (two parallel sides and two equal slanted sides) has how many lines of symmetry?
- 0
- 1
- 2
- 4
Challenge
Challenge practice
Combine shape rules, reflections and careful counting. Show your reasoning.
23. Half of a shape is drawn on a grid to the left of a vertical mirror line. Its corners are 1, 2 and 4 squares from the line. When you complete the shape, how far from the line are the new corners on the right?
- 2, 4 and 8 squares
- 4, 2 and 1 squares, all moved up
- 1, 2 and 4 squares
- All 4 squares
24. A regular octagon (like a stop sign) has how many lines of symmetry?
- 8
- 4
- 16
- 6
25. You want a four-sided shape with exactly 1 line of symmetry. Which shape works?
- a square
- a rectangle
- a parallelogram
- a kite
26. Which word, written in block capitals, has a vertical line of symmetry down its middle?
- CAT
- TOT
- BIB
- BOB
27. A shape has exactly 3 lines of symmetry and 3 equal sides. What is it, and how do you know?
28. How many lines of symmetry do these shapes have in total: one square, one rectangle (not a square) and one scalene triangle?
- 8
- 7
- 6
- 4
29. A paper heart is folded down its middle and the halves match. If you cut it along the fold, what is true about the two pieces?
- They are the same size and shape, mirror images of each other.
- One piece is bigger.
- They are different shapes.
- They cannot be compared.
30. Why does a regular hexagon have more lines of symmetry than a regular pentagon?
Answer key
- 1. a line of symmetry
A line of symmetry splits a shape into two halves that match exactly when folded. A diagonal is a different line (corner to corner), and a side is on the edge, not through the middle. - 2. 4
A square has 2 lines through the middles of opposite sides and 2 diagonals: 4 in all. 2 forgets the diagonals; 8 counts each line twice (once from each end). - 3. 2
One line goes across the middle and one goes up and down the middle. The diagonals are not lines of symmetry: fold along one and the corners do not land on each other. 4 is true only for a square. - 4. 3
Each line goes from a corner to the middle of the opposite side, and there are 3 corners. 1 is the answer for an isosceles triangle that is not equilateral; 6 counts each line twice. - 5. 0
No fold can match the halves, because every side and angle is different. A shape can have zero lines of symmetry. 1 would need at least two equal sides. - 6. A
The left half of A mirrors the right half across an up-and-down line. In block style, F, L and R have no line of symmetry: their halves do not match either way. - 7. B
B's top half mirrors its bottom half (if the bumps are drawn the same size), but its left and right halves differ. H has both lines, T has only a vertical line, and N has none. - 8. 6
3 lines join opposite corners and 3 lines join the middles of opposite sides: 6 in all. For any regular polygon, the number of lines equals the number of sides. 3 counts only one kind of line; 12 counts each line twice. - 9. 5
A regular polygon with 5 sides has 5 lines, each from a corner to the middle of the opposite side. 10 counts each line from both ends. - 10. 1
The only line runs from the corner between the equal sides to the middle of the opposite side. 3 is the equilateral answer; 2 would need a second matching fold that does not exist. - 11. 0
Folding along a diagonal or through the middle never matches the slanted corners. A parallelogram can be turned to fit onto itself, but turning is not folding. 2 is the rectangle answer. - 12. 2
Both diagonals of a rhombus are lines of symmetry. Lines through the middles of opposite sides do not work unless the rhombus is a square, which would give 4. - 13. 3 squares to the right of the line, at the same height
A reflection is the same distance from the mirror line on the other side, at the same height. 6 squares right adds the distance twice, and moving up changes the height. - 14. In the same place, on the line
Its distance from the line is 0, so its reflection is 0 squares away on the other side: the same spot. It does not jump to the edge of the grid. - 15. 2
H matches itself folded up and down (vertical line) and folded top to bottom (horizontal line), so 2. 1 misses one of them; 4 would need matching diagonal folds, which H does not have. - 16. The two triangles are the same size, but folding along the diagonal does not make them land on each other.
Equal halves are not enough: the halves must match when folded. Fold a paper rectangle corner to corner and the corners stick out. Cutting a shape in half by area is not the same as reflecting it. - 17. a rhombus that is not a square
A rhombus has four equal sides but only 2 lines of symmetry (its diagonals). A square does have 4, and a rectangle does not have four equal sides, so it cannot test her claim. - 18. If the two halves cover each other exactly, the fold line is a line of symmetry.
Folding tests the definition directly: matching halves. Measuring only the sides can miss slanted parts, and how pretty a shape looks is not evidence. - 19. False
Any line through the centre of a circle is a line of symmetry, so it has infinitely many, not just 4. 4 is the square's answer. - 20. He counted each line twice, once from each end.
Each line of symmetry runs all the way across the shape. Counting both ends of the same line doubles the answer. A square has 4. - 21. False
Turning a shape does not change its shape, so it keeps the same lines of symmetry. They just point in a new direction. - 22. 1
The only line goes up and down through the middles of the two parallel sides. Its diagonals do not work, and a side-to-side fold puts the long base on top of the short one. - 23. 1, 2 and 4 squares
Each reflected corner is the same distance from the line on the other side. Doubling to 2, 4 and 8 squares would measure from the wrong place. - 24. 8
A regular polygon has as many lines of symmetry as sides. An octagon has 8 sides, so 8 lines: 4 through opposite corners and 4 through the middles of opposite sides. - 25. a kite
A kite has one line of symmetry along its long diagonal. A square has 4, a rectangle has 2, and a general parallelogram has 0. - 26. TOT
For a whole word to mirror across its middle, every letter needs its own vertical line and the word must read the same backward. T and O both have vertical lines, and TOT is the same backward. CAT is not the same backward, and B in BIB and BOB has only a horizontal line, so its left and right halves differ. - 27. An equilateral triangle, because 3 equal sides and 3 matching folds from each corner to the opposite side's middle fit only that triangle.
Regular polygons have as many lines as sides, and the only regular 3-sided shape is the equilateral triangle. - 28. 6
Square 4 + rectangle 2 + scalene triangle 0 = 6. Getting 8 means counting the rectangle's diagonals; getting 7 means giving the scalene triangle a line. - 29. They are the same size and shape, mirror images of each other.
Matching halves across a line of symmetry are mirror images: same size and shape, flipped. They are not different sizes, and one is not a turned copy unless the shape also has turn symmetry. - 30. Because a regular polygon has one line of symmetry for each side, and a hexagon has 6 sides while a pentagon has 5.
More equal sides and angles give more ways to fold the shape so the halves match.