Parallelogram Solver

Solve a parallelogram from base, side, and either height or included angle. Get area, perimeter, both interior angles, both diagonals, and the derived height in one clear geometry check.

How It Works

Formula

A = bh

A = bs\sin(\theta)

P = 2(b+s)

h = s\sin(\theta)

d_{AC} = \sqrt{b^2 + s^2 + 2bs\cos(\theta)}

d_{BD} = \sqrt{b^2 + s^2 - 2bs\cos(\theta)}

\theta_{acute} = \arcsin\left(\frac{h}{s}\right)

Variables, symbols and units

$b$: Base length(linear unit)
$s$: Side length(linear unit)
$h$: Perpendicular height(linear unit)
$\theta$: Included interior angle(deg or rad)
$A$: Area(square unit)
$P$: Perimeter(linear unit)
$d_{AC}, d_{BD}$: The two diagonals(linear unit)

Calculation method explained

The calculator keeps the geometry to one clean choice: either you know the perpendicular height or you know the included interior angle. From there it derives the full parallelogram set: area first, then perimeter, the acute/obtuse angle pair, and both diagonals.

In base + side + height mode, the calculator uses $A = bh$ directly, then recovers the acute angle with $\arcsin(h/s)$ . In base + side + included angle mode, it derives height with $h = s\sin(\theta)$ and then uses $A = bh = bs\sin(\theta)$ . Diagonals come from the two law-of-cosines style expressions shown above, and the result surface sorts them into short and long for you.

Frequently Asked Questions

When should I use height mode versus angle mode?

Use height mode when your sketch gives the perpendicular rise from the base. Use angle mode when the drawing labels the interior opening angle between the base and the side instead.

Why can’t the height be longer than the side?

The height is the perpendicular component of the side. A component cannot exceed the full side length, so a height larger than the side would describe impossible geometry.

Why do you show both an acute and an obtuse angle?

Every parallelogram has two angle values: one acute and one obtuse. They are supplementary, so the larger one is always 180° minus the smaller one, or pi minus the smaller one in radians.

How do I know which diagonal is longer?

The result list sorts the two diagonals for you. The diagram also distinguishes them so you can see the shorter crossing line and the longer corner-to-corner line at a glance.

Is this enough for fabrication approval?

No. This tool is for geometry planning, homework checks, and layout double-checking. Real cuts, fit-up, and tolerances still need your project-specific review.

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Parallelogram Solver

How It Works

Frequently Asked Questions

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