Parallelogram and Rhombus
A parallelogram is a quadrilateral with two pairs of parallel sides, and a rhombus is a parallelogram with four equal sides. In brief: in a parallelogram opposite sides and opposite angles are equal and the diagonals bisect each other. The area is base times height, A = bh, or ab sin θ. In a rhombus the diagonals are also perpendicular, and the area is (1/2)d1d2. For diagonals 6 and 8 the rhombus has area 24 and side 5.
These two shapes appear in tilings, linkages and crystal structures. They share many properties, and a handful of tests tell you when a quadrilateral is a parallelogram or a rhombus. This guide lists the properties, the tests (converses) with proofs of the key facts, area and perimeter formulas and worked problems, with a comparison table.
What are the properties of a parallelogram?
- Opposite sides are parallel and equal.
- Opposite angles are equal.
- Consecutive angles are supplementary (add up to 180°), because the sides are parallel.
- The diagonals bisect each other. Their intersection is the center of symmetry (point symmetry), but a general parallelogram has no axis of symmetry.
- Each diagonal splits the parallelogram into two congruent triangles.
Why the opposite sides are equal: the diagonal AC creates triangles ABC and CDA with AC common and equal alternate angles at A and C (parallel sides). By ASA, the triangles are congruent, so AB = CD and BC = DA.
How do you prove a quadrilateral is a parallelogram?
Any one of the following conditions is enough.
- Both pairs of opposite sides are parallel (the definition).
- Both pairs of opposite sides are equal.
- One pair of opposite sides is both parallel and equal.
- Both pairs of opposite angles are equal.
- The diagonals bisect each other.
Example: a quadrilateral with vertices A(0, 0), B(5, 1), C(7, 4), D(2, 3). Then AB = (5, 1) and DC = (5, 1) are equal vectors, so AB ∥ DC and AB = DC. A pair of opposite sides is parallel and equal: a parallelogram. Check the diagonals: the midpoints of AC and BD are both (3.5, 2) ✓.
Counter-example for the condition: one pair of equal opposite sides is not enough. An isosceles trapezoid has equal legs but it is not a parallelogram.
What are the properties of a rhombus?
A rhombus has all the properties of a parallelogram and in addition:
- All four sides are equal.
- The diagonals are perpendicular.
- The diagonals bisect the angles of the rhombus.
- It has two axes of symmetry (the diagonals) and point symmetry.
- It has an inscribed circle, and its center is the intersection of the diagonals.
Tests: a parallelogram with two adjacent equal sides is a rhombus. A parallelogram with perpendicular diagonals is a rhombus. A quadrilateral with four equal sides is a rhombus.
The diagonals cut a rhombus into four congruent right triangles with legs d1/2 and d2/2, so the side is a = (1/2)√(d12 + d22). For d1 = 6 and d2 = 8: a = (1/2) · 10 = 5, a 3-4-5 triangle.
How do you find the area and perimeter?
Parallelogram: A = b · h, where h is the height perpendicular to the base b, or A = ab sin θ, where θ is the angle between sides a and b. The perimeter is 2(a + b). Note that the height is not the slanted side.
Rhombus: A = (1/2)d1d2 = a2 sin θ = a · h. The perimeter is 4a.
Worked problems
Example 1: parallelogram area
A parallelogram has sides 10 and 6 with an angle of 30°. A = 10 · 6 · sin 30° = 30. The height onto the side 10 is 6 sin 30° = 3. The perimeter is 32.
Example 2: angles
One angle of a parallelogram is 65°. The opposite angle is 65° and the other two are 180° − 65° = 115°. The sum is 2 · 65 + 2 · 115 = 360° ✓.
Example 3: a rhombus from its diagonals
Diagonals 16 and 12. Area = 96, side = (1/2)√(256 + 144) = (1/2) · 20 = 10, perimeter 40. The height is A/a = 9.6. The angles: tan(θ/2) = 6/8 for the half-diagonals, so θ/2 ≈ 36.87° and the angles are about 73.7° and 106.3°.
Example 4: using the diagonals of a parallelogram
The sum of the squares of the diagonals equals the sum of the squares of the four sides: d12 + d22 = 2(a2 + b2). For sides 5 and 7 and one diagonal 8, the other diagonal satisfies 64 + d2 = 2(25 + 49) = 148, so d2 = 84 and d ≈ 9.17.
Parallelogram and rhombus compared
| Feature | Parallelogram | Rhombus |
|---|---|---|
| Sides | opposite equal | all four equal |
| Angles | opposite equal | opposite equal |
| Diagonals | bisect each other | bisect each other, perpendicular, bisect the angles |
| Axes of symmetry | 0 (general case) | 2 |
| Area | bh or ab sin θ | d1d2/2 or a2 sin θ |
| Perimeter | 2(a + b) | 4a |
| Inscribed circle | no (generally) | yes |
Where do these shapes come from?
Euclid defined the rhombus and the “rhomboid” (parallelogram with unequal adjacent sides) in Book I of the Elements and proved that opposite sides and angles of a parallelogram are equal. The area formula base times height was used by Egyptian and Babylonian surveyors, who treated a parallelogram as a rectangle rearranged by moving a triangle from one side to the other. The word parallelogram is from Greek, meaning “bounded by parallel lines”.
Where are they used today?
The parallelogram law is used in physics for adding two vectors: the diagonal of the parallelogram built on two vectors is their sum. Mechanical linkages such as pantographs and scissor lifts use parallelograms to keep parts parallel. Rhombus patterns appear in tilings, fences with diamond lattices and the shape of playing-card diamonds. In crystallography, unit cells are often parallelograms or rhombuses.
Which mistakes are common?
Using the slanted side as the height
The height is the perpendicular distance between the parallel sides. The slanted side is longer.
Assuming a parallelogram has equal diagonals
Equal diagonals occur only in a rectangle. In a rhombus, the diagonals are perpendicular, not equal (except in a square).
Assuming equal opposite sides are enough
Both pairs of opposite sides must be equal, or one pair equal and parallel.
Mixing up the area formulas
(1/2)d1d2 works for rhombuses (and kites), not for general parallelograms.