Parallelogram - Properties, Area, and Formulas
Parallelogram - Properties, Area, and Formulas
Definition
A parallelogram is a four-sided flat shape (a quadrilateral) whose opposite sides are both parallel and equal in length. Its area equals base times height: A=b×h, where b is the length of one side (the base) and h is the perpendicular distance between that base and the side parallel to it. Every rectangle, rhombus, and square is a parallelogram — but not every parallelogram is a rectangle.
Quick Reference
| Field | Value |
|---|---|
| Definition | Quadrilateral with both pairs of opposite sides parallel and equal |
| Area formula | A=b×h |
| Perimeter formula | P=2(a+b) |
| Diagonal lengths | d₁,d₂ — generally unequal; satisfy d₁² + d₂² = 2(a²+b²) |
| Special cases | Rectangle, Rhombus, Square |
| Used in | Geometry, vector addition, structural engineering, crystallography |
What Is A Parallelogram?
A parallelogram is the most general quadrilateral with parallel sides. Take any pair of parallel line segments, the same length, and connect their endpoints with two more segments of the same length to each other. What you have is a parallelogram. The internal angles can be anything — as long as opposite angles are equal — and the diagonals can have different lengths.
Three properties follow from the definition and define what makes a parallelogram a parallelogram:
- Opposite sides are equal — both pairs.
- Opposite angles are equal — both pairs.
- Diagonals bisect each other — they cross at their midpoints.
If a four-sided shape satisfies any one of these conditions plus parallel-opposite-sides, it is a parallelogram.
Types of Parallelograms
Three special parallelograms each add one extra constraint:
| Type | Extra constraint | What changes |
|---|---|---|
| Rectangle | All four angles are 90° | Diagonals are equal in length |
| Rhombus | All four sides are equal | Diagonals are perpendicular and bisect angles |
| Square | All four angles are 90° AND all four sides equal | Both above hold; the most special parallelogram |
A general parallelogram has none of these extra constraints. It just has parallel opposite sides.
Area, Perimeter, and Diagonals
The area of a parallelogram is base times perpendicular height: A=b×h.
The perimeter sums all four sides. Since opposite sides are equal: P=2(a+b), where a and b are the two side lengths.
The diagonals of a parallelogram are generally not equal. They satisfy the parallelogram law: d₁² + d₂² = 2(a² + b²).
Worked Examples
Example 1: Find the area
A parallelogram has base 12 cm and height 5 cm. Find its area.
A = b × h = 12 × 5 = 60 cm².
Final answer: 60 cm².
Example 2: Use of the slant side (the wrong path first)
A parallelogram has base 10 cm and slant side 8 cm, with the slant side at 30° to the base. Find the area.
The instinct is to multiply base times slant: A = 10 × 8 = 80. That is wrong — the slant side is not the perpendicular height.
The perpendicular height is the slant times the sine of the angle: h = 8 × sin 30° = 8 × 0.5 = 4 cm.
Then:
A = 10 × 4 = 40 cm².
Final answer: 40 cm².
Example 3: Find a diagonal using the parallelogram law
A parallelogram has sides a=5 and b=7, and one diagonal d₁=6. Find the other diagonal.
d₁² + d₂² = 2(a² + b²)
36 + d₂² = 2(25 + 49) = 148.
d₂² = 112.
d₂ ≈ 10.58.
Final answer: The other diagonal is approximately 10.58.
The Mathematicians Who Shaped The Parallelogram
The parallelogram is so old that no single person is credited with it. Euclid (c. 300 BCE, Alexandria) gave the first systematic treatment in Elements Book I, where the basic parallelogram theorems appear as Propositions 33–45. The parallelogram law of vector addition is a much later development, codified in the work of Stevin (1548–1620, Flemish) for force vectors and later given rigorous mathematical form by Hamilton and Grassmann in the 1800s.
Common Mistakes of Parallelogram
Mistake 1: Using the slant side instead of the perpendicular height for area.
Where it slips in: The problem gives the side length and an angle, and the student multiplies side by side without finding the perpendicular distance.
Don't do this: A = b × slant.
The correct way: Drop a perpendicular from one parallel side to the other, find that distance, and use it as h. If only the slant and angle are given, h = slant × sin(angle).
Mistake 2: Assuming all parallelograms have equal diagonals.
Where it slips in: The student carries the rectangle property over to all parallelograms.
Don't do this: Assume d₁ = d₂ for any parallelogram.
The correct way: Diagonals are equal only when the parallelogram is a rectangle.
Mistake 3: Confusing the parallelogram with the trapezium.
Where it slips in: A four-sided shape with one pair of parallel sides is a trapezium, not a parallelogram.
Don't do this: Treat any four-sided shape with at least one pair of parallel sides as a parallelogram.
The correct way: Both pairs of opposite sides must be parallel for a parallelogram.
Mistake 4: Using the wrong formula for the rhombus.
Where it slips in: The rhombus has the parallelogram area formula A=b×h but also a special diagonal formula A=1/2d₁d₂ — the second uses both diagonals at right angles.
Don't do this: Apply A=1/2d₁d₂ to a general parallelogram.
The correct way: The diagonal-product formula applies only when the diagonals are perpendicular — which happens for the rhombus and the square, not for the general parallelogram.
Frequently Asked Questions
Is a square a parallelogram? Yes. Every square is also a rectangle, every rectangle is a parallelogram, so every square is a parallelogram.
What is the difference between a rhombus and a parallelogram? A rhombus is a parallelogram with the extra rule that all four sides are equal. A general parallelogram has only opposite sides equal.
Why does the parallelogram area formula use perpendicular height? Because shearing a rectangle into a parallelogram preserves area. The perpendicular height stays the same during the shear; the slant side changes.
What are the diagonals of a parallelogram used for? Beyond the parallelogram law for finding lengths, diagonals appear in vector addition and in proofs about congruence between the two triangles a diagonal cuts the parallelogram into.
Are the angles in a parallelogram always equal? Opposite angles are equal; adjacent angles sum to 180°. So a parallelogram with one 70° angle has angles 70°, 110°.