How Many Ways Can You Fold a Shape in Half?
Learn to find every line of symmetry in a figure, complete a half-drawn shape, and count the faces, edges and vertices of cubes, cones, prisms and pyramids.
What is a line of symmetry?
A line of symmetry is a line that divides a figure into two halves which fold exactly onto each other. If you could fold the shape along that line and every edge matched, the line is a line of symmetry — which is why it is sometimes called the fold line or the mirror line.
This page covers everything in the ICSE Class 6 Mathematics chapter on symmetry and solids: counting lines of symmetry, completing a symmetrical figure, naming solids and their parts, and matching a solid to its net.
This page covers everything in the ICSE Class 6 Mathematics chapter on symmetry and solids: counting lines of symmetry, completing a symmetrical figure, naming solids and their parts, and matching a solid to its net.
How many lines of symmetry does each shape have?
The count depends on the shape's own regularity, and these are worth memorising:
- square: 4 — two through opposite sides, two along the diagonals
- rectangle: 2 — through opposite sides only, never the diagonals
- rhombus: 2 — both along the diagonals
- isosceles triangle: 1 — through the vertex between the equal sides
- equilateral triangle: 3 — one from each vertex
- circle: infinitely many — every diameter is a line of symmetry
- parallelogram: 0
Letters behave the same way. A, H, M, T and U have one line of symmetry; H, I, O and X have two; F, G, J, L, P, Q, R, S and Z have none.
The result that surprises students most is the rectangle's diagonal. Fold a rectangle along its diagonal and the two halves do not match — the corners overshoot. So a rectangle has 2 lines of symmetry, not 4.
- square: 4 — two through opposite sides, two along the diagonals
- rectangle: 2 — through opposite sides only, never the diagonals
- rhombus: 2 — both along the diagonals
- isosceles triangle: 1 — through the vertex between the equal sides
- equilateral triangle: 3 — one from each vertex
- circle: infinitely many — every diameter is a line of symmetry
- parallelogram: 0
Letters behave the same way. A, H, M, T and U have one line of symmetry; H, I, O and X have two; F, G, J, L, P, Q, R, S and Z have none.
The result that surprises students most is the rectangle's diagonal. Fold a rectangle along its diagonal and the two halves do not match — the corners overshoot. So a rectangle has 2 lines of symmetry, not 4.
How do you complete a symmetrical figure?
Given one half and the line of symmetry, you build the other half point by point.
For each corner of the given half, measure its perpendicular distance from the line of symmetry. Mark a matching point on the other side at exactly the same distance, measured at right angles to the line. Then join your new points in the same order as the original.
So if a vertex sits 3 cm from the mirror line, its partner sits 3 cm on the other side, along the same perpendicular.
In real life this is how a rangoli or a paper-cut design is made: fold the paper, cut one half, and the fold guarantees the other half matches.
To construct a line of symmetry with ruler and compasses, use the perpendicular bisector construction on a pair of matching points — the bisector of the segment joining them is the mirror line.
Measuring along a slanted direction instead of perpendicular to the line is the usual error, and it distorts the whole figure.
For each corner of the given half, measure its perpendicular distance from the line of symmetry. Mark a matching point on the other side at exactly the same distance, measured at right angles to the line. Then join your new points in the same order as the original.
So if a vertex sits 3 cm from the mirror line, its partner sits 3 cm on the other side, along the same perpendicular.
In real life this is how a rangoli or a paper-cut design is made: fold the paper, cut one half, and the fold guarantees the other half matches.
To construct a line of symmetry with ruler and compasses, use the perpendicular bisector construction on a pair of matching points — the bisector of the segment joining them is the mirror line.
Measuring along a slanted direction instead of perpendicular to the line is the usual error, and it distorts the whole figure.
What are the faces, edges and vertices of a solid?
A three-dimensional solid has faces (flat or curved surfaces), edges (where two faces meet) and vertices (corner points where edges meet).
- cube: 6 faces (all squares), 12 edges, 8 vertices
- cuboid: 6 faces (rectangles), 12 edges, 8 vertices
- cylinder: 3 faces (2 flat circles, 1 curved), 2 edges, 0 vertices
- cone: 2 faces (1 flat circle, 1 curved), 1 edge, 1 vertex (the apex)
- sphere: 1 curved face, 0 edges, 0 vertices
- triangular prism: 5 faces, 9 edges, 6 vertices
- square pyramid: 5 faces, 8 edges, 5 vertices
A prism has two identical parallel ends joined by rectangles; a pyramid has one base and triangular faces meeting at an apex.
For example, a dice is a cube, a tin of food is a cylinder, an ice-cream cone is a cone, and a laddu is a sphere.
The sphere is the boundary case: with no flat surfaces meeting anywhere, it has no edges and no vertices at all.
- cube: 6 faces (all squares), 12 edges, 8 vertices
- cuboid: 6 faces (rectangles), 12 edges, 8 vertices
- cylinder: 3 faces (2 flat circles, 1 curved), 2 edges, 0 vertices
- cone: 2 faces (1 flat circle, 1 curved), 1 edge, 1 vertex (the apex)
- sphere: 1 curved face, 0 edges, 0 vertices
- triangular prism: 5 faces, 9 edges, 6 vertices
- square pyramid: 5 faces, 8 edges, 5 vertices
A prism has two identical parallel ends joined by rectangles; a pyramid has one base and triangular faces meeting at an apex.
For example, a dice is a cube, a tin of food is a cylinder, an ice-cream cone is a cone, and a laddu is a sphere.
The sphere is the boundary case: with no flat surfaces meeting anywhere, it has no edges and no vertices at all.
How do you match a solid to its net?
A net is the flat shape you get by unfolding a solid along its edges. Fold the net back up and you rebuild the solid, so the net must contain exactly the faces the solid has — no more, no fewer.
The net of a cube is six squares arranged so they fold into a closed box. The net of a cuboid is six rectangles, opposite pairs matching in size. The net of a cylinder is a rectangle with two circles attached, the rectangle's length equal to the circle's circumference. The net of a cone is a circle plus a sector. The net of a square pyramid is a square with four triangles on its sides.
So to identify a net, count its pieces and check their shapes. Five faces made of one square and four triangles can only fold into a square pyramid.
A sphere has no net, because a curved surface cannot be flattened without stretching or tearing it — which is also why every flat map of the world distorts something.
The net of a cube is six squares arranged so they fold into a closed box. The net of a cuboid is six rectangles, opposite pairs matching in size. The net of a cylinder is a rectangle with two circles attached, the rectangle's length equal to the circle's circumference. The net of a cone is a circle plus a sector. The net of a square pyramid is a square with four triangles on its sides.
So to identify a net, count its pieces and check their shapes. Five faces made of one square and four triangles can only fold into a square pyramid.
A sphere has no net, because a curved surface cannot be flattened without stretching or tearing it — which is also why every flat map of the world distorts something.
Exam tip
Exam tip: counting lines of symmetry by folding, not by eye
Students judge symmetry by whether a line "looks like it cuts the shape in half", which is why the rectangle's diagonal and the parallelogram both get miscounted.
Apply the fold test mentally, one line at a time: would the two halves land exactly on each other, corner onto corner? For a parallelogram, sliding one half across matches it, but folding does not — and symmetry requires the fold.
A quick self-check for regular polygons: a regular polygon with sides has exactly lines of symmetry. An equilateral triangle has 3, a square 4, a regular pentagon 5. If your count disagrees, recheck by folding.
Apply the fold test mentally, one line at a time: would the two halves land exactly on each other, corner onto corner? For a parallelogram, sliding one half across matches it, but folding does not — and symmetry requires the fold.
A quick self-check for regular polygons: a regular polygon with sides has exactly lines of symmetry. An equilateral triangle has 3, a square 4, a regular pentagon 5. If your count disagrees, recheck by folding.
Did you know
Why does a sphere have no edges or vertices?
An edge is defined as the line where two faces meet, and a vertex as the point where edges meet. A sphere has a single continuous curved surface, so there is nowhere for two faces to meet.
With no edges, there can be no vertices either. It is the reason a ball rolls smoothly in every direction while a dice tumbles from face to face.
With no edges, there can be no vertices either. It is the reason a ball rolls smoothly in every direction while a dice tumbles from face to face.
Key takeaways
Symmetry and solids in 30 seconds
- A line of symmetry folds a figure into two halves that match exactly; a square has 4, a rectangle 2, an equilateral triangle 3, a circle infinitely many and a parallelogram none.
- To complete a symmetrical figure, mark each new point at the same perpendicular distance from the mirror line.
- Faces, edges and vertices count a solid's surfaces, meeting lines and corners; a cube and cuboid both have 6, 12 and 8.
- A prism has two identical parallel ends, a pyramid one base and an apex; a sphere has one curved face, no edges and no vertices.
- A net is the unfolded solid and must contain exactly its faces — a sphere has no net, since curved surfaces cannot flatten.
You will remember all of this far better after answering five questions on it than after reading it twice.
- To complete a symmetrical figure, mark each new point at the same perpendicular distance from the mirror line.
- Faces, edges and vertices count a solid's surfaces, meeting lines and corners; a cube and cuboid both have 6, 12 and 8.
- A prism has two identical parallel ends, a pyramid one base and an apex; a sphere has one curved face, no edges and no vertices.
- A net is the unfolded solid and must contain exactly its faces — a sphere has no net, since curved surfaces cannot flatten.
You will remember all of this far better after answering five questions on it than after reading it twice.