Parallelogram Calculator

How it works

A parallelogram is a four-sided figure whose opposite sides are parallel and equal: a rectangle pushed over sideways. Its area is base times height, exactly like a rectangle's, because slicing the overhanging triangle off one end and moving it to the other end turns it into one. The height is the perpendicular distance between the two bases, not the slanted side.

If you know the angle between the base and the side instead of the height, the height is the side times the sine of that angle, so the area is a × b × sin θ. If you know the height, the angle is recovered from asin(h ÷ b). A parallelogram has two different angles that add up to 180°; either one describes the same shape, so the results list the acute and obtuse angles together.

The two diagonals are found with the law of cosines in the triangles they cut off: the diagonal opposite the acute angle, joining the two obtuse corners, is the short one, √(a² + b² − 2ab cos θ), and the one joining the two acute corners is the long one with a plus sign. They always satisfy the parallelogram law, p² + q² = 2(a² + b²).

Formula

area      = a × h = a × b × sin θ
perimeter = 2 (a + b)
h = b × sin θ          θ = asin(h ÷ b)
p = √(a² + b² + 2ab cos θ)     q = √(a² + b² − 2ab cos θ)     (θ the acute angle)

Example

A parallelogram with base 8 ft, side 5 ft and height 4 ft has an area of 8 × 4 = 32 ft² and a perimeter of 2 × (8 + 5) = 26 ft. Its acute angle is asin(4 ÷ 5) ≈ 53.13°, the obtuse one 126.87°, and with cos 53.13° = 0.6 the diagonals are √(64 + 25 + 48) = √137 ≈ 11.70 ft and √(64 + 25 − 48) = √41 ≈ 6.40 ft.

The same 8 ft and 5 ft sides meeting at 60° give a height of 5 × sin 60° ≈ 4.33 ft, an area of 8 × 5 × sin 60° ≈ 34.64 ft², and diagonals √(89 + 40) ≈ 11.36 ft and √(89 − 40) = 7 ft.

Assumptions and limitations

• Opposite sides are parallel and equal. For a general four-sided figure the sides and one angle do not fix the shape.
• The angle is in degrees. Either of the parallelogram's two angles gives the same figure; results list the acute one first.
• All lengths are in the unit you chose. The calculator labels the results with it and does not convert between units.
• Results are rounded for display only.

Frequently asked questions