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: $A = b \times h$, 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 | $A = b \times h$ |
Perimeter formula | $P = 2(a + b)$, where $a$ and $b$ are the two side lengths |
Diagonal lengths | $d_1, d_2$ — generally unequal; satisfy $d_1^2 + d_2^2 = 2(a^2 + b^2)$ |
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:
$$A = b \times h$$
The slant side is not used in the area calculation — only the perpendicular distance between the parallel sides matters. This is the same formula as the rectangle's area, which is no coincidence: a parallelogram is just a rectangle that has been sheared sideways. The area is preserved by the shear.
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:
$$d_1^2 + d_2^2 = 2(a^2 + b^2)$$
This identity, sometimes attributed to Pappus, is the key reason parallelograms appear in vector addition — adding two vectors using the parallelogram rule produces a diagonal whose length is computable from this law.
Worked Examples
Example 1: Find the area
A parallelogram has base $12,\text{cm}$ and height $5,\text{cm}$. Find its area.
$$A = b \times h = 12 \times 5 = 60,\text{cm}^2$$
Final answer: $60,\text{cm}^2$.
Example 2: Use of the slant side (the wrong path first)
A parallelogram has base $10,\text{cm}$ and slant side $8,\text{cm}$, with the slant side at $30°$ to the base. Find the area.
The instinct is to multiply base times slant: $A = 10 \times 8 = 80$. That is wrong — the slant side is not the perpendicular height. Stop and re-check.
The perpendicular height is the slant times the sine of the angle:
$$h = 8 \times \sin 30° = 8 \times 0.5 = 4,\text{cm}$$
Then:
$$A = 10 \times 4 = 40,\text{cm}^2$$
Final answer: $40,\text{cm}^2$.
Example 3: Find a diagonal using the parallelogram law
A parallelogram has sides $a = 5$ and $b = 7$, and one diagonal $d_1 = 6$. Find the other diagonal.
$$d_1^2 + d_2^2 = 2(a^2 + b^2)$$
$$36 + d_2^2 = 2(25 + 49) = 148$$
$$d_2^2 = 112$$
$$d_2 = \sqrt{112} \approx 10.58$$
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 shape's importance was already clear: Euclid uses it to prove area-equivalence results that became foundational for geometry.
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 $1800$s. The same shape that Euclid drew in the dust now describes how forces, velocities, and electromagnetic fields combine.
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: $A = b \times \text{slant}$, treating the slant as the height.
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, $h = \text{slant} \times \sin(\text{angle})$.
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 $d_1 = d_2$ for any parallelogram.
The correct way: Diagonals are equal only when the parallelogram is a rectangle. In a general parallelogram, the diagonals have different lengths and are computed by the parallelogram law $d_1^2 + d_2^2 = 2(a^2 + b^2)$.
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 (or trapezoid in US English), not a parallelogram. The student sees parallel sides and assumes 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. If only one pair is parallel, it is a trapezium and obeys different formulas.
Mistake 4: Using the wrong formula for the rhombus.
Where it slips in: The rhombus has the parallelogram area formula $A = b \times h$ but also a special diagonal formula $A = \frac{1}{2} d_1 d_2$ — the second uses both diagonals at right angles.
Don't do this: Apply $A = \frac{1}{2} d_1 d_2$ 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.
The real-world version of Mistake 1 is one of the most common errors in carpentry and tile-laying. A craftsman cutting parallelogram tiles for a herringbone floor must measure the perpendicular height of the tile, not its slant side, when calculating how many tiles fit in a given length. Tradesmen who use slant length end up short by the cosine factor — and a $30°$ tile pattern measured this way undercuts the floor by about $14%$. See vector parallelogram law for the formal mathematical version.
Try These Next
Now try this: a parallelogram has base $9,\text{cm}$, slant side $6,\text{cm}$, and an angle of $60°$ between them. Find the area. If you get stuck on identifying the perpendicular height, return to Example $2$.
If your child is comfortable with parallelogram area, the natural next step is the trapezium area formula, then the vector parallelogram law for adding vectors. At Bhanzu, trainers connect the area formula to vector addition so the same diagram does double duty across geometry and physics.
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