What is a parallelogram?
A parallelogram is a quadrilateral with two pairs of opposite sides parallel. Opposite sides have equal lengths, opposite angles are equal, and each pair of adjacent angles adds to 180°. These relationships make it possible to derive area, height, perimeter and diagonals from a relatively small set of measurements.
For an education-focused calculation, the key idea is to separate length from perpendicular height. The side b is not generally the height. Instead, its perpendicular component is h = b sin(A). Once that height is known, the familiar area rule K = a h follows.
How to use the parallelogram calculator
Choose the shape
Use the general parallelogram mode, or switch to rectangle, rhombus or square for their special constraints.
Enter known values
For the standard case, enter a, b and the included angle A. Other fields can be used to reverse-solve.
Read the geometry
The calculator fills area, height, perimeter, diagonals and the adjacent angle B.
Rectangle, rhombus and square: special cases
A rectangle is a parallelogram with four right angles. A rhombus is a parallelogram with four equal sides. A square satisfies both conditions. The general formulas still apply, but the special constraints simplify them.
| Shape | Defining condition | Area | Perimeter |
|---|---|---|---|
| Parallelogram | Opposite sides parallel | a·b·sin(A) | 2(a+b) |
| Rectangle | A = 90° | a·b | 2(a+b) |
| Rhombus | a = b | a²·sin(A) | 4a |
| Square | a = b and A = 90° | a² | 4a |
Parallelogram formulas
The formulas below cover the main quantities used by this calculator. They are closely connected: once one geometric relationship is known, the others follow through trigonometry and the parallelogram law.
Here h is the perpendicular distance between the pair of sides of length a.
The sine extracts the perpendicular component of b.
For the chosen base a, height is the perpendicular component of side b.
Opposite sides are equal, so each side length occurs twice.
This is the cosine-law form for the diagonal joining the endpoints of the two side vectors.
The other diagonal uses the opposite vector direction and therefore the plus sign.
Diagonals and the parallelogram law
The diagonals are not generally equal and not generally perpendicular. What is always true is that they bisect one another and satisfy the parallelogram law:
This identity is useful as a consistency check. If you know a, b, p and q, the equation can also be rearranged to recover a missing side.
Angles and height
Adjacent angles are supplementary, so B = 180° − A. Because sin(A) = sin(180° − A), either adjacent angle produces the same area when used with the same two side lengths.
| Quantity | Relationship | Meaning |
|---|---|---|
| Opposite angles | A = C, B = D | Opposite corners match. |
| Adjacent angles | A + B = 180° | Neighboring corners are supplementary. |
| Height | h = b sin(A) | Perpendicular component of b. |
| Maximum area | A = 90° | For fixed a and b, sin(A) is largest. |
Worked examples
Example 1: sides 10 cm and 6 cm with a 60° included angle
Start with K = a·b·sin(A). Substituting a = 10, b = 6 and A = 60° gives K = 10 × 6 × sin(60°) ≈ 51.96 cm². The height is h = 6 sin(60°) ≈ 5.20 cm, so K = 10 × 5.20 gives the same result to rounding.
Example 2: area and side b
If K and a are known, height is h = K/a. If b is also required and the angle is known, rearrange K = a b sin(A) to b = K/(a sin(A)). This is useful when a geometry problem gives area rather than height.
Example 3: using a diagonal
If a, b and diagonal p are known, the included angle can be recovered with the cosine rule:
Once A is known, the calculator can recover the height, area, other diagonal and perimeter.
Common checks and mistakes
| Check | What to remember |
|---|---|
| Side vs height | Do not use b as the height unless A = 90°. |
| Degrees | The calculator uses degrees; trigonometric functions internally use radians. |
| Diagonal signs | p and q use opposite signs on the 2ab cos(A) term. |
| Angle range | A valid non-degenerate parallelogram has 0° < A < 180°. |
| Units | Length results are in cm; area is in cm². |
Parallelogram calculator FAQ
The basic formula is K = a·h, where a is a base and h is the perpendicular height. If the two sides and included angle are known, the equivalent formula is K = a·b·sin(A).
For base a and adjacent side b, use h = b·sin(A). The sine extracts the perpendicular component of b.
Not in general. They are equal for a rectangle, but a general parallelogram usually has two different diagonal lengths. Both diagonals bisect each other.
For fixed a and b, K = ab sin(A), so the maximum occurs at A = 90°, where sin(A) = 1. The parallelogram is then a rectangle.
Yes, when enough additional information is known. For example, with a, b and p, the included angle can be recovered with the cosine rule, after which the remaining quantities follow.