ShapesTypes of triangles and quadrilaterals
Triangles and quadrilaterals each come in a small family of named types. Knowing the family tree matters, because a property proved for one type holds for every type inside it: whatever is true of all parallelograms is true of every rectangle, rhombus and square.
Triangles by their sides
- Scalene: all three sides different, and so all three angles different.
- Isosceles: two sides equal, and the two angles opposite them equal.
- Equilateral: three sides equal, and every angle 60°.
Triangles by their angles
- Acute-angled: all three angles less than 90°.
- Right-angled: one angle exactly 90°.
- Obtuse-angled: one angle more than 90°.
The two ways of sorting combine. A triangle can be right-angled and isosceles (angles 90°, 45°, 45°), but never right-angled and equilateral, since an equilateral triangle has only 60° angles. A triangle can have at most one right angle or one obtuse angle, because the three must total 180°.
The quadrilateral family
- Trapezium: one pair of opposite sides parallel.
- Parallelogram: both pairs of opposite sides parallel.
- Rectangle: a parallelogram with right angles.
- Rhombus: a parallelogram with all sides equal.
- Square: a rectangle that is also a rhombus.
- Kite: two pairs of equal sides next to each other.
The angles of every quadrilateral add up to 360°.
Properties at a glance
| Property | Parallelogram | Rectangle | Rhombus | Square | Kite |
|---|---|---|---|---|---|
| Opposite sides parallel | yes | yes | yes | yes | no |
| Opposite sides equal | yes | yes | yes | yes | no |
| All sides equal | no | no | yes | yes | no |
| All angles 90° | no | yes | no | yes | no |
| Diagonals bisect each other | yes | yes | yes | yes | one only |
| Diagonals equal | no | yes | no | yes | no |
| Diagonals at right angles | no | no | yes | yes | yes |
| Lines of symmetry | 0 | 2 | 2 | 4 | 1 |
"No" here means "not in general". A particular parallelogram may happen to have equal diagonals, and then it is a rectangle.
Questions that test the family tree
- Is every square a rectangle? Yes. It has four right angles.
- Is every rectangle a square? No. Its sides need not be equal.
- Is every rhombus a parallelogram? Yes.
- Is a square a rhombus? Yes. All its sides are equal.
- Is a parallelogram a trapezium? By the definition "at least one pair of parallel sides", yes. Many Indian school books define a trapezium as having exactly one pair, and then it is not. Follow your book.
How to identify a shape from its diagonals
Diagonals that bisect each other mean a parallelogram. If they are also equal, it is a rectangle. If they are also at right angles, it is a rhombus. If both, it is a square. This test is often quicker than measuring sides and angles.
Each shape has a page with its formulas: triangle, right-angled triangle, equilateral, isosceles, square, rectangle, parallelogram, rhombus, trapezium and kite.