A quadrilateral is any polygon with four sides. The named kinds are not separate boxes. They are nested, so a shape can honestly carry several names at once, and a square carries almost all of them.
The family
| Name | Requires | Also true of it |
|---|---|---|
| Trapezoid | at least one pair of parallel sides | nothing further |
| Parallelogram | both pairs of opposite sides parallel | opposite sides and opposite angles are equal |
| Rectangle | a parallelogram with four right angles | diagonals are equal |
| Rhombus | a parallelogram with four equal sides | diagonals cross at right angles |
| Square | a rectangle and a rhombus at once | everything above applies |
Why a square is also a rectangle
A rectangle is defined by its angles, not by having unequal sides. A square has four right angles, so it satisfies the definition, and it is a rectangle. It also has four equal sides, so it is a rhombus. And since both are parallelograms, it is a parallelogram too.
This is where classroom arguments start. The answer is that is a rectangle and is only a rectangle are different claims. Every square is a rectangle. Not every rectangle is a square.
A note on trapezoids
United States textbooks generally define a trapezoid as having at least one pair of parallel sides, which makes every parallelogram a trapezoid as well. Some other countries define it as having exactly one pair, which excludes parallelograms. Both definitions are in use, so it is worth knowing which one your book follows.
Angle sum
The four angles of any quadrilateral add to 360 degrees, because it splits into two triangles of 180 degrees each.
Finding a missing angle
three angles measure 100, 85 and 72 360 - 100 - 85 - 72 = 103 degrees
Common mistakes
Treating the names as exclusive
Answering that a shape is a square and therefore not a rectangle is the classic error. It is both.
Calling any tilted square a rhombus and nothing more
A square rotated is still a square. Tilting a shape does not change what it is.
Check yourself
Is every rectangle a parallelogram?
Yes. Its opposite sides are parallel, which is all a parallelogram requires.
Is every parallelogram a rectangle?
No. A parallelogram only becomes a rectangle when its angles are right angles.
Three angles of a quadrilateral are 90, 90 and 65. Find the fourth.
360 - 90 - 90 - 65 = 115 degrees.
Name every category a square belongs to.
Quadrilateral, trapezoid under the United States definition, parallelogram, rectangle, rhombus and square.
See also
Sources
- Quadrilateral classification as taught in United States elementary geometry, including the inclusive definition of a trapezoid used by most United States textbooks. Examples on this page are our own.