Parallelogram calculator
Area from a base and height, or from two sides and the angle between them.
Fill in the fields and the result will appear here automatically.
Solves a parallelogram two ways: from a base with its height, and from two sides with the angle between them. The second mode also gives the perimeter, the height and both diagonals; the first gives only the area, because the second side does not follow from a base and a height, and a dash is shown instead of a plausible perimeter. At 0 or 180 degrees the figure collapses into a line: that input is rejected rather than returning an area of zero.
How it works
Formula and logic
From base a and perpendicular height h: S = ah. From adjacent sides a, b and included angle θ: S = ab sin θ, h = b sin θ, P = 2(a+b). Enter θ in degrees. The sorted diagonals are Dmax/min = √[a²+b² ± 2ab|cos θ|]. Equivalent half-angle identities avoid losing the smaller diagonal when nearly equal squared terms are subtracted.
Example
Sides of 10 and 8 cm at an angle of 30° give an area of 40 cm² and a perimeter of 36 cm.
Fields and units
- Length unit — list option
- What is known — list option
- Side a — cm
- Height to side a — cm
- Side b — cm
- Angle between sides — °
How to use
- — Choose base and height, or two adjacent sides and their included angle.
- — Measure the height perpendicular to the base.
- — Enter the angle in degrees strictly between 0° and 180°.
- — Lengths must be positive and share one unit; base and height alone do not determine the perimeter or diagonals.
Method and limitations
- Calculation method
- Formula and logic
- Data or methodology source
- OpenStax: height b sin θ and triangle area ab sin θ/2 Euclid I.34: a diagonal bisects a parallelogram’s area
- Limitation
- Assumes two pairs of parallel sides in one plane. Angles of 0° and 180° are excluded; a genuinely positive small angle is not treated as zero. Height and area alone do not uniquely determine inclination or the other side; results are rounded.
FAQ
Why is no perimeter shown in the height mode?
Because the second side does not follow from a base and a height: infinitely many parallelograms of different slant share the same area. Reporting a perimeter would be an invention.
What happens at 90 degrees?
The sine is one and the parallelogram becomes a rectangle: the area is the product of the sides.
Why is 180 degrees rejected?
At that angle the figure collapses into a line and stops being a parallelogram. An area of zero would be formally right but meaningless, so the calculator reports the problem instead.
How does a parallelogram differ from a rhombus?
A rhombus has all sides equal. Enter the same a and b and the calculation still holds for it.
Does replacing θ with 180°−θ change the parallelogram’s area?
No: sin(180°−θ) = sin θ. Height, perimeter and the sorted pair of diagonals also agree; the slant changes. At θ = 90° both diagonals are equal and the figure is a rectangle.