What is the area formula for a parallelogram?

The area of a parallelogram equals the base times the height, where the height is the perpendicular distance between the two parallel bases, not the slant side length. Alternatively, if you know two adjacent sides a and b and the included angle theta, the area equals a times b times sin(theta). These two formulas are equivalent because the height h equals b times sin(theta).

How do you find the height of a parallelogram?

The height of a parallelogram is the perpendicular distance between the base and the opposite side. If you know the slant side s and the included angle theta, the height equals s times sin(theta). This height is different from the side length: a parallelogram with the same base and side can have different areas depending on how tilted it is.

How are the diagonals of a parallelogram calculated?

A parallelogram has two diagonals. Using the law of cosines: the shorter diagonal d1 squared equals a squared plus b squared minus 2ab times cos(theta), and the longer diagonal d2 squared equals a squared plus b squared plus 2ab times cos(theta), where a and b are the adjacent sides and theta is the included angle.

Parallelogram Calculator

A parallelogram is a four-sided flat shape where the opposite sides are parallel and equal in length. Because the two pairs of sides are parallel, opposite angles are always equal and adjacent angles always add up to 180 degrees. Rectangles and rhombuses are special cases of parallelograms: a rectangle is a parallelogram with four right angles, and a rhombus is a parallelogram with all four sides equal. Parallelograms appear throughout engineering drawing, structural design, and vector mathematics, where the area of a parallelogram formed by two vectors gives the magnitude of their cross product. This calculator takes the base length, the side length, and the included angle between them to compute all key measurements in one step. It returns the area, the perpendicular height (the distance between the two parallel bases), the perimeter, and both diagonal lengths computed using the law of cosines. The angle must be between 0 and 180 degrees exclusive. At 90 degrees the shape becomes a rectangle. Enter your measurements below and all results update instantly. The default example uses base = 8, side = 5, angle = 60 degrees, giving area = 34.64, height = 4.33, and perimeter = 26.

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Base (a)

units

Side (b)

units

Included angle (θ)

deg

34.64

area (square units)

Height4.33

Perimeter26.00

Diagonal d17.00

Diagonal d211.36

Angle must be between 0 and 180 degrees exclusive. At 90 degrees the shape is a rectangle.

How it works

Given base a, side b, and included angle theta in degrees: the height h = b * sin(theta). Area = a * h = a * b * sin(theta). Perimeter = 2 * (a + b). The diagonals are found by the law of cosines: d1^2 = a^2 + b^2 - 2ab * cos(theta), giving the shorter diagonal; d2^2 = a^2 + b^2 + 2ab * cos(theta), giving the longer diagonal (using the supplementary angle 180 - theta). All angles are converted to radians for the trigonometric functions.

Worked example

For base = 8, side = 5, angle = 60 degrees: height = 5 * sin(60) = 5 * 0.8660 = 4.33. Area = 8 * 4.33 = 34.64. Perimeter = 2 * (8 + 5) = 26. Diagonal d1 = sqrt(64 + 25 - 80 * cos(60)) = sqrt(89 - 40) = sqrt(49) = 7.00. Diagonal d2 = sqrt(64 + 25 + 80 * cos(60)) = sqrt(89 + 40) = sqrt(129) = 11.36. These match the defaults above.

Related calculators

Rhombus Calculator: a parallelogram with all sides equal.
Pentagon Calculator: area, perimeter, diagonal, and apothem of a regular pentagon.
Circle Segment Calculator: area and arc length of a circular segment.
Law of Sines and Cosines: solve any triangle from sides and angles.

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