BYTETOOLS

Parallelogram Calculator

Find the area, perimeter, both heights, both diagonals and the angle between them from two sides and an angle, a base and height, or a side and a diagonal.

20.7846
Area (cm²)
20
Perimeter (cm)
60°
Acute angle
120°
Obtuse angle

Where each number came from

  • Area = a·b·sin θ = 20.7846 cm²
  • Perimeter = 2(a + b) = 20 cm
  • Height onto a: ha = area / a = 3.4641 cm
  • Height onto b: hb = area / b = 5.1962 cm
  • Short diagonal = √(a² + b² − 2ab·cos θ) = 5.2915 cm
  • Long diagonal = √(a² + b² + 2ab·cos θ) = 8.7178 cm
  • Angle between the diagonals = 115.693°

Solved from two sides and the included angle.

Shape and the parallelogram law

Parallelogram law check

d₁² + d₂² = 104  ·  2(a² + b²) = 104

The two figures agree, which confirms the diagonals are consistent with the sides.

Opposite sides are parallel and equal, and the diagonals bisect each other.

What is the Parallelogram Calculator?

The ByteTools Parallelogram Calculator solves the shape from whichever three measurements you happen to have.

  • Three input modes: sides and angle, base and height, or sides and a diagonal
  • Area, perimeter, both heights h(a) and h(b), and both interior angles
  • Both diagonals by the law of cosines, plus the angle where they cross
  • Parallelogram-law check so you can see the diagonals agree with the sides
  • Detects the rhombus, rectangle and square special cases and says so
  • Scaled drawing with both diagonals shown

How to use the Parallelogram Calculator

  1. 1

    Choose what you know: two sides and the included angle, a base with a height, or two sides and a diagonal.

  2. 2

    Pick your measurement unit and type in side a and side b.

  3. 3

    Enter the third value — the angle in degrees, the perpendicular height, or the known diagonal.

  4. 4

    Read the area, perimeter and both interior angles in the summary tiles.

  5. 5

    Check the diagonals and the parallelogram-law verification in the results panel, then copy the lot.

About the Parallelogram Calculator

The ByteTools Parallelogram Calculator solves the shape from whichever three measurements you happen to have. Give it two sides and the included angle and it returns the area a·b·sin θ; give it a base and a perpendicular height and it recovers the angle from sin θ = h ÷ b; give it two sides and one diagonal and it uses the law of cosines to find the angle first.

Every result comes with the full picture: perimeter 2(a + b), both perpendicular heights, both diagonals from √(a² + b² ∓ 2ab·cos θ), the angle where the diagonals cross, and a check of the parallelogram law d₁² + d₂² = 2(a² + b²) so you can see the numbers are self-consistent. The shape is drawn to scale with both diagonals marked.

All the trigonometry runs in your browser with JavaScript — nothing is uploaded, nothing is stored, and the page works offline once it has loaded. The tool also recognises the special cases, telling you when your figures actually describe a rhombus, a rectangle or a square, and it refuses impossible combinations with a message explaining the valid range rather than printing a meaningless answer.

Frequently asked questions

How do you find the area of a parallelogram without the height?

Use area = a × b × sin θ, where a and b are the two side lengths and θ is the angle between them. With sides of 6 and 4 and an included angle of 60°, the area is 24 × sin 60° = 20.78. This is the same as base × height, because b × sin θ is the height.

How do you calculate the diagonals of a parallelogram?

The law of cosines gives both: the diagonal opposite the angle θ is √(a² + b² − 2ab·cos θ) and the other is √(a² + b² + 2ab·cos θ). For sides 6 and 4 at 60° that is 5.29 and 8.72. The two diagonals are equal only when the parallelogram is a rectangle.

What is the parallelogram law?

It says the sum of the squares of the two diagonals equals twice the sum of the squares of the two sides: d₁² + d₂² = 2(a² + b²). The calculator prints both sides of that equation so you can confirm your figures hang together — for sides 6 and 4 both come to 104.

Does a parallelogram have two different heights?

Yes. There is one perpendicular height measured onto side a and a different one measured onto side b, unless the two sides happen to be equal. Both equal the area divided by the side they stand on, and the calculator reports each of them.

Why can't I enter any diagonal I like?

A diagonal has to close the triangle formed with the two sides, so it must be longer than the difference of the sides and shorter than their sum. If you go outside that range the tool tells you the valid window instead of returning a nonsense angle.

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