Parallelogram Calculator

What does this parallelogram calculator compute?

A parallelogram of sides a>0, b>0 and included angle θ∈(0°,180°) has area ab sin θ, perimeter 2(a+b), and diagonals √(a²+b²±2ab cos θ). θ=90° recovers a rectangle.

The published formula or method is A = ab sin θ, P = 2(a+b), d² = a²+b²±2ab cos θ. The following references support the stated mathematics or method:

• Euclid’s Elements — I.35: Parallelograms on the Same Base: Equal-base, equal-height parallelograms have equal area. Comparing with a rectangle gives A = base × perpendicular height; side length is not generally the height. Mathematical reference only; not validation or endorsement of this implementation.
• Gilbert Strang, Calculus — Chapter 1: Introduction to Calculus (MIT OpenCourseWare): Right-triangle definitions of sine, cosine and tangent, the sine/cosine identities, and the law-of-cosines and parallelogram exercises 31–32. Chord length follows by bisecting the radii triangle. Regular-polygon apothems and areas follow by splitting the polygon into equal triangles. The law of sines follows by equating a triangle’s altitude computed from either adjacent side; SSA can give two triangles. These applications are derived from the stated relations. Mathematical reference only; not implementation validation or an endorsement of CalculatorX.

Which inputs and limits should you check?

Use positive dimensions in one consistent length unit and the angle convention shown by the selected shape. Perimeter uses length units and area uses squared length units.

The worked example uses a = 2, b = 3, θ = 30° with A = 6 sin 30°. Compare it with the Example calculations and the complete result for the same inputs.

How can you verify and cite this result?

CalculatorX is responsible for this page. The mathematical references above explain the model; implementation checks and their scope are documented separately in the math.geometry.plane v1.6.0 Evidence and capability contract. Read the displayed verification and production state together with the specification's numerical limits.

Use the clean calculator URL to cite the tool. A shared input link identifies a particular calculation. Content updates, numerical verification, and domain reviews have separate dates and scopes.

Numerical domain, polygon order and reproducibility

Use one consistent length unit; area uses its square. Side/base lengths, radii and heights must be positive normal binary64 numbers. Angles are degrees strictly between0 and180; degree-to-radian conversion must remain normal. Regular n is a safe integer3 through9007199254740991. The annulus requires R>r>0; its perimeter includes both circles. The trapezoid model is isosceles, including the equal-base rectangle case.

For irregular polygons provide3–8 ordered distinct vertices as x,y; x,y, a flat even coordinate list, [x,y] arrays or {x,y} objects in machine calls. An optional repeated first vertex closes the ring. Clockwise and concave polygons are supported. Exact predicates on normalized binary64 reject self-intersections, nonadjacent touching/overlap, duplicate vertices, adjacent backtracking and zero area; straight intermediate vertices without overlap are allowed.

Strict decimal text is limited to1024 characters per scalar (32768 for a vertex list); unknown fields and conflicting active aliases are rejected. Nonzero text converted to zero returns NUMERIC_UNDERFLOW. Nonzero inputs and all positive output measures must be finite normal binary64 (at least2^-1022); otherwise NUMERIC_RANGE requires changing units or translating coordinates. Exact zero coordinates are supported.

Polynomial measures use exact dyadic intermediates, scaled norms and stable half-angle diagonals. Annulus area uses the factored difference; polygon area and topology use exact binary64 predicates. Transcendental functions remain binary64. No absolute small-value clamp or display rounding replaces a nonzero result with zero. Thin rings, nearly flat shapes and tiny offsets relative to large coordinates are sensitive to input rounding: predicates concern normalized values, not the original arbitrary-precision decimal geometry. Relative1e-11 verification compares arithmetic with independent high precision; it does not bound input uncertainty.

Figures use equal coordinate scales. A thin, nearly collapsed or unresolvable figure, or a regular polygon with more than128 sides, is explicitly omitted while complete numerical results remain readable and exportable. This bound keeps rendering independent of a very large n.

Interactive calculation runs locally. A shared URL sends parameters to the server for replay/SSR and may enter browser history, access logs or analytics. REST/MCP send inputs for server execution. Site analytics may receive page URLs and interaction metadata. No zero-log, fixed retention, deletion or processing-region guarantee is claimed.