MathIt BlogLines of symmetry explained: what they are and how many each shape has
What is a line of symmetry, and how many lines of symmetry does a shape have?
A line of symmetry is a line that splits a shape into two halves that match exactly when you fold along it, like a reflection in a mirror. The number of lines of symmetry depends on the shape: a square has 4, a rectangle has 2, an equilateral triangle has 3, a regular hexagon has 6, a parallelogram has 0 and a circle has infinitely many.
Key takeaways
- A line of symmetry is a line that divides a shape into two halves that match exactly when the shape is folded along it.
- A square has 4 lines of symmetry, a rectangle has 2, an equilateral triangle has 3 and a regular hexagon has 6.
- The diagonals of a rectangle are not lines of symmetry, because folding a rectangle along a diagonal leaves two corners sticking out.
- A regular polygon with n sides has exactly n lines of symmetry, and a circle has infinitely many.
- Line symmetry and rotational symmetry are different properties: a parallelogram has no line of symmetry but looks the same after a half turn.
Your third grader folds a sheet of paper in half, cuts half a heart along the fold and opens a perfect heart. That crease answers "what is a line of symmetry?", and the same idea is behind the worksheet that asks how many lines of symmetry a shape has. Finding lines of symmetry takes a fold, a mirror and one careful habit: check that the two halves match point for point.
What is a line of symmetry?
A line of symmetry is a line that splits a shape into two halves that are mirror images of each other. Common Core standard 4.G.A.3 asks fourth graders to "recognize a line of symmetry for a two-dimensional figure as a line across the figure such that the figure can be folded along the line into matching parts", and to "identify line-symmetric figures and draw lines of symmetry".
The paper heart has one line of symmetry, down the middle. A square has four, and a scalene triangle has none. In England, the national curriculum for year 4 asks pupils to "identify lines of symmetry in 2-D shapes presented in different orientations" and to "complete a simple symmetric figure with respect to a specific line of symmetry".
Wikipedia's article on reflection symmetry lists "line symmetry" and "mirror-image symmetry" as names for the same idea; a line of symmetry is also called an axis of symmetry or a mirror line.
Line symmetry vs rotational symmetry
Line symmetry and rotational symmetry are two different properties, and worksheets in later grades ask about both. Line symmetry means the shape folds onto itself. Rotational symmetry means the shape looks the same after you turn it less than a full turn around its center.
A square has both: it folds onto itself 4 ways, and it looks the same after a quarter turn of 90°. A parallelogram with no right angles has rotational symmetry (it matches itself after a half turn of 180°) but no line of symmetry at all. The capital letters S, N and Z behave the same way.
How to find lines of symmetry, step by step
Lines of symmetry are found by testing every possible fold, one at a time, and counting only the folds where both halves match exactly. The method works for any flat shape on a grade 3 or grade 4 worksheet.
- Trace or cut out the shape. Draw the shape on thin paper or cut it out of scrap paper.
- Try the vertical fold. Fold the shape left to right and hold it up to a window. If every edge of one half lies on top of the other half, the crease is a line of symmetry.
- Try the horizontal fold. Open the paper and fold it top to bottom, then check the edges again.
- Try the diagonal folds. For a polygon, test folds from corner to opposite corner and from the middle of one side to the middle of the opposite side. Each end of a polygon's line of symmetry sits on a corner or a side's midpoint.
- Use the mirror test for drawings. Stand a small mirror upright on the line you want to test. If the half you see plus its reflection looks like the whole shape, the line is a line of symmetry.
- Draw and count. Draw each line that passed with a ruler as a dashed line, then count the lines once each.
Grade 4 geometry sits in the same school year as multi-digit multiplication and division (Common Core 4.NBT.B.5 and 4.NBT.B.6), and the math game for 4th graders covers that number work.
Free printable worksheetMultiplication — 4th GradersWorksheet · PDFThe part people get wrong: the rectangle's diagonals
The diagonals of a rectangle are not lines of symmetry, and this is the most common wrong answer on symmetry worksheets. A diagonal cuts the rectangle into two triangles of the same size, which looks like it should count. Folding along the diagonal tells the truth: the folded triangle lands at an angle, and two corners stick out past the other half.
The halves must be mirror images, so every point on one side has a partner at the same distance on the other side. A square passes the diagonal fold because all its sides are equal, which is why a square has 4 lines and a rectangle has 2.
How many lines of symmetry does each shape have?
The number of lines of symmetry for each common shape is fixed, so a child can check a worksheet answer against one list. The counts assume the most general version of each shape: a rectangle that is not a square, an isosceles triangle that is not equilateral and a rhombus that is not a square.
| Shape | Lines of symmetry | Where the lines are | Rotational symmetry? |
|---|---|---|---|
| Circle | Infinitely many | Every line through the center | Yes, any turn |
| Square | 4 | 2 through midpoints of opposite sides, 2 diagonals | Yes, every 90° |
| Rectangle | 2 | Through midpoints of opposite sides | Yes, 180° |
| Equilateral triangle | 3 | Each corner to the middle of the opposite side | Yes, every 120° |
| Isosceles triangle | 1 | From the corner between the equal sides to the middle of the base | No |
| Scalene triangle | 0 | None | No |
| Regular pentagon | 5 | Each corner to the middle of the opposite side | Yes, every 72° |
| Regular hexagon | 6 | 3 corner to corner, 3 midpoint to midpoint | Yes, every 60° |
| Parallelogram | 0 | None | Yes, 180° |
| Rhombus | 2 | Both diagonals | Yes, 180° |
| Kite | 1 | The diagonal through the corners where equal sides meet | No |
The triangle rows connect to the triangle names children learn in the same years, and the guide to types of triangles and angles explains equilateral, isosceles and scalene with side lengths.
Why a regular polygon with n sides has n lines
A regular polygon has n lines of symmetry when it has n equal sides and n equal angles. A polygon with an odd number of sides, like the pentagon, has one line from each corner to the middle of the opposite side, so 5 corners give 5 lines. A polygon with an even number of sides, like the hexagon, has n ÷ 2 lines from corner to corner and n ÷ 2 lines from midpoint to midpoint: 3 + 3 = 6.
The turning angle follows the same pattern: a regular polygon with n sides looks the same after a turn of 360° ÷ n. A hexagon matches itself every 360 ÷ 6 = 60°, and a pentagon every 360 ÷ 5 = 72°. Those angles are division facts.
Play free in your browserDivision FactsGrade 3+Symmetry activities at home
Symmetry activities at home turn the definition into something your child can see and touch. Each takes about ten minutes with things you already own.
- Letters of the alphabet: Write the capital letters in a plain font and test each one with a mirror. In a font like Arial, A, M, T, U, V, W and Y have a vertical line; B, C, D, E and K have a horizontal line; and H, I, O and X have both.
- Butterflies and leaves: Compare a butterfly picture with a garden leaf. Both are close to mirror images, which starts a good talk about "exact" in math.
- Ink-blot painting: Fold a sheet of paper, open it, drop paint on one side of the crease and press the halves together. The print is symmetric about the fold every time.
- Cut-out shapes: Fold paper once, cut a shape along the fold and predict what it will look like opened.
Folding a shape in half also shows halves in the fraction sense: each part is 1/2 of the whole, and the guide to fractions for parents builds on the same fold.
Mirror drawing
Mirror drawing is the activity that matches the year 4 goal of completing "a simple symmetric figure". Draw half of a house on squared paper beside a straight line, and ask your child to draw the other half with each corner the same number of squares from the line.
A hand mirror on the line lets your child check the drawing alone. Move from a vertical line to a horizontal one, then a diagonal: the "different orientations" part of the curriculum.
How to check progress with symmetry
Five short questions show whether a child understands lines of symmetry or is guessing from the look of a shape.
- The square and the rectangle: Ask for the number of lines in each. The answers are 4 and 2, and the child should explain why the rectangle's diagonals fail.
- The three triangles: Show an equilateral, an isosceles and a scalene triangle. The answers are 3, 1 and 0.
- The turned shape: Draw an isosceles triangle lying on its side. A child who still finds the 1 line has met the year 4 "different orientations" goal.
- The letters: Ask which of H, E, S and A have a line of symmetry. H, E and A do; S has none.
- The half picture: Give half a shape beside a line and ask for the rest.
A five-minute kitchen-table test
A five-minute symmetry test uses three cut-out shapes: a square, a rectangle and a parallelogram. Ask your child to predict the number of lines for each, then fold to check.
The predictions should read 4, 2 and 0, and a child who predicts 4 for the rectangle has met the diagonal trap. The math game for 3rd graders and the overview of what math each grade should know show where symmetry sits beside the rest of the year's work.
Common mistakes with lines of symmetry
Four mistakes cause most wrong answers on symmetry worksheets, and each one has a short fix.
- Counting the diagonals of a rectangle: A rectangle gets 4 lines instead of 2. Fold a paper rectangle along a diagonal and let the corners show the answer.
- Counting the diagonals of a parallelogram: A parallelogram gets 2 lines instead of 0. Each diagonal splits it into two equal triangles, but neither fold matches.
- Counting a line twice: A child counts the top half and the bottom half of the same line as two lines. Draw each line once from edge to edge, then count the drawn lines.
- Missing a tilted line: A shape printed at an angle hides its lines. Turn the worksheet and test again.
Edge case: the circle and the regular polygons
A circle has infinitely many lines of symmetry, because every line through its center folds it onto itself. Wikipedia's reflection symmetry article states that "a circle has infinitely many axes of symmetry", so "infinitely many" is the expected answer.
A regular polygon with n sides has exactly n lines, so a regular octagon has 8 and a regular 12-sided polygon has 12. The rule holds only for regular polygons.
In algebra the same idea returns as the axis of symmetry of a graph. OpenStax Elementary Algebra 2e gives the axis of symmetry of y = ax² + bx + c as the line x = −b ÷ 2a, so the graph of y = x² − 6x + 8 folds onto itself along x = 3. The guide to area and perimeter covers the other shape measurements of the same years.
Your lines of symmetry checklist and next steps
- Fold test: fold a paper copy and check that every edge lies on top of its partner.
- Every direction: test the vertical, the horizontal and both diagonal folds before you count.
- Rectangle trap: remember that a rectangle has 2 lines and its diagonals do not count.
- Regular shapes: a regular polygon with n sides has n lines, and a circle has infinitely many.
- Count once: draw each line edge to edge, then count the drawn lines.
- Two kinds: keep line symmetry (fold) and rotational symmetry (turn) apart.
Practise it in the MathIt app
About 90 math games with adaptive difficulty, daily practice and 2–5 player battles — free on iOS and Android.
MathIt is a free app for iOS and Android with about 90 game types, adaptive difficulty and a short daily practice for the number facts that sit beside geometry at school. Parents and children can also play 2 to 5 player real-time math battles, and the math game for parents and kids page shows how a family round works.
Frequently asked questions
How many lines of symmetry does a square have?
A square has 4 lines of symmetry. Two of them join the midpoints of opposite sides, one vertical and one horizontal, and the other two are the diagonals that join opposite corners. A paper square folded along any of the 4 lines gives two halves that match exactly.
How many lines of symmetry does a triangle have?
The number of lines of symmetry in a triangle depends on its type. An equilateral triangle has 3, one from each corner to the middle of the opposite side. An isosceles triangle has 1, running down the middle between its two equal sides. A scalene triangle, with three different side lengths, has 0.
Does a rectangle have 4 lines of symmetry?
A rectangle that is not a square has 2 lines of symmetry. The two lines join the midpoints of opposite sides. The diagonals look like they should work, but folding a rectangle along a diagonal leaves two corners sticking out past the other half, so the halves do not match.
What is the axis of symmetry?
The axis of symmetry is another name for a line of symmetry. In algebra the phrase usually means the vertical line that splits the graph of a parabola into two mirror halves. For a graph y = ax² + bx + c, the axis of symmetry is the line x = −b ÷ 2a, so y = x² − 6x + 8 has its axis at x = 3.
Which letters of the alphabet have a line of symmetry?
Sixteen capital letters have a line of symmetry in a plain capital font such as Arial. A, M, T, U, V, W and Y have a vertical line; B, C, D, E and K have a horizontal line; H, I, O and X have both. Decorative fonts change the answer, so check the letters on your child's worksheet.
