How is the area of a parallelogram calculated?

Area = base × side × sin(angle), which equals base × height. The height is the side length times the sine of the included angle.

How are the two diagonals found?

They come from the law of cosines: d₁ = √(a² + b² − 2ab·cos(angle)) and d₂ = √(a² + b² + 2ab·cos(angle)), where a is the side and b is the base.

What is the perimeter of a parallelogram?

The perimeter is 2 × (base + side), because opposite sides are equal in length. The angle does not affect the perimeter, only the area and the diagonals.

Why does the height depend on the angle?

The height is the perpendicular distance between the base and its opposite side. It equals the side length times sin(angle), so a flatter parallelogram (smaller angle) has a shorter height and less area.

What happens at a 90-degree angle?

At 90° the parallelogram becomes a rectangle: the height equals the side, the area equals base × side, and both diagonals are equal at √(a² + b²).

Which angle should I enter?

Enter the interior angle between the base and the adjacent side, in degrees. The opposite angle is the same, and the other two interior angles are its supplement (180° minus your value).

Is any data uploaded?

No — all calculations happen in your browser and nothing leaves your device.

What is the Parallelogram Calculator?

Free parallelogram calculator — enter the base, side and included angle to get area, height, perimeter and both diagonals. Runs entirely in your browser, nothing uploaded. It runs free in your browser on Gera Tools, with nothing uploaded.

Parallelogram Calculator

Name: Parallelogram Calculator
Creator: Gera Tools
License: https://creativecommons.org/licenses/by/4.0/

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Parallelogram calculator

A parallelogram is a four-sided shape with two pairs of parallel, equal-length sides and two pairs of equal angles. It is the general case of which rectangles and rhombuses are special forms. Enter the base, the adjacent side and the included angle between them, and this tool returns the area, the perpendicular height, the perimeter and both diagonals.

How all five values are calculated

With base b, side a and included angle θ in degrees (converted to radians for the trig functions), the tool computes:

height    = a × sin(θ)             perpendicular distance between the two base lines
area      = b × a × sin(θ)         = base × height
perimeter = 2 × (b + a)            opposite sides are equal
d1        = √(a² + b² − 2ab·cos(θ))   longer diagonal when θ < 90°
d2        = √(a² + b² + 2ab·cos(θ))   shorter diagonal (uses supplementary angle)

The diagonal formulas come from the law of cosines applied to the two triangles formed by each diagonal. The d1 formula uses the included angle θ; d2 uses the supplementary angle (180° − θ), which is why the cosine term changes sign.

Worked example

Base 10, side 6, angle 60°:

Height: 6 × sin(60°) = 6 × 0.8660 = 5.196
Area: 10 × 5.196 = 51.96 square units
Perimeter: 2 × (10 + 6) = 32 units
d1: √(36 + 100 − 120 × cos60°) = √(136 − 60) = √76 ≈ 8.718
d2: √(36 + 100 + 120 × cos60°) = √(136 + 60) = √196 = 14.0

Special cases at specific angles

The angle you enter controls how “slanted” the shape is, and three angle values produce well-known special shapes:

Angle	Shape produced	Area formula
90°	Rectangle	base × side (height = side)
Any angle, a = b	Rhombus	side² × sin(θ)
90°, a = b	Square	side²

At 90° both diagonals become equal (they both equal √(a² + b²) from Pythagoras), which is the property distinguishing rectangles from other parallelograms.

Common mistakes

Entering the slant height instead of the perpendicular height: the “height” of a parallelogram is the perpendicular distance between the parallel sides, not the length of the slanted side. The tool computes this from the side and angle, so always enter the side length and angle rather than the height directly.

Using the wrong angle: enter the interior angle at the vertex between base and side (the “included angle”). The opposite angle at the other pair of vertices is the same value; the other two angles are the supplement (180° − θ). If you know one interior angle, you know all four.

Everything is computed locally in your browser — nothing is uploaded.