Rectangles, Rhombi & Squares
Rectangles, rhombi, and squares are the special parallelograms. Each one is a parallelogram first — so opposite sides are congruent and parallel, opposite angles are congruent, consecutive angles are supplementary, and the diagonals bisect each other — and then each adds its own extra requirement on top.
The definitions are short: a rectangle is a parallelogram with four right angles, a rhombus is a parallelogram with four congruent sides, and a square is both at once. What sets the three apart on most test questions is what their diagonals do.
The definitions and the hierarchy
A rectangle constrains the angles: all four are . A rhombus constrains the sides: all four are congruent. A square satisfies both definitions, so every square is a rectangle and every square is a rhombus.
The arrows only run one way. A rhombus can lean, so it is not always a square, and a rectangle can be longer than it is tall, so it is not always a square either. When a question says always, check whether the smaller category is fully contained in the bigger one.
What the diagonals do
In a rectangle, the diagonals are congruent: . In a rhombus, the diagonals are perpendicular, and each diagonal bisects the pair of vertex angles it connects. A square, being both, has diagonals that are congruent and perpendicular.
These diagonal facts stack on top of the parallelogram fact that the diagonals bisect each other. So in a rhombus, the diagonals cut each other in half and cross at — which splits the rhombus into four congruent right triangles. That is why so many rhombus problems are secretly Pythagorean theorem problems: half of one diagonal and half of the other are the legs, and the side of the rhombus is the hypotenuse.
The figure shows a rhombus with its four congruent sides and its perpendicular diagonals crossing at the center.
Choosing the right fact for the problem
Expressions on the two diagonals of a rectangle — set the whole diagonals equal, because they are congruent. Diagonal lengths of a rhombus with a side to find — halve both diagonals and use . An angle inside a rhombus — look for the right angle where the diagonals cross and use the triangle angle sum. A classification statement — test it against the definitions and the one-way hierarchy.
Worked examples
Example 1: congruent diagonals in a rectangle
In rectangle , and . Find .
Answer:
Example 2: side of a rhombus from its diagonals
The diagonals of a rhombus measure cm and cm. Find the side length.
Answer: cm
Example 3: an angle in a rhombus
In rhombus , the diagonals intersect at and . Find .
Answer:
Try one yourself
Common questions
Is every square a rhombus?
Yes. A square has four congruent sides, which is exactly the rhombus definition, so a square is a rhombus with the bonus of right angles. It is also a rectangle for the same reason: four right angles.
Are the diagonals of a rhombus congruent?
Not in general — they are perpendicular but usually different lengths. The diagonals are only congruent when the rhombus is also a rectangle, which makes it a square.
Do rectangles, rhombi, and squares keep the parallelogram properties?
Yes, all of them. Opposite sides congruent and parallel, opposite angles congruent, consecutive angles supplementary, diagonals bisecting each other — every one still holds. The special names only add properties; they never take any away.
How do I decide whether a parallelogram is a rectangle, a rhombus, or a square?
Check angles and sides separately. Four right angles (or congruent diagonals) makes it a rectangle. Four congruent sides (or perpendicular diagonals) makes it a rhombus. If it passes both checks, it is a square.
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