About
Parallelogram: opposite sides parallel and equal. Area = base × perpendicular height. Or = product of adjacent sides × sin of included angle. Rectangle is special case (90°). Rhombus has all sides equal.
What is Parallelogram Calculator?
A Parallelogram Calculator computes parallelogram from the inputs you provide. It applies the standard formula to the values you enter and returns the result instantly, without sending any data to a server. Free Parallelogram Calculator. The tool runs entirely in.
Parallelogram Calculator
A = base × height. Or sides × sin(angle). Generalizes rectangle.
Inputs
units
units
units
degrees
Area (base × height)
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Breakdown
Area (sides × sin)
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Perimeter
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Diagonal 1
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Diagonal 2
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About the parallelogram
A parallelogram is a four-sided figure (a quadrilateral) whose opposite sides are parallel and equal in length. Tilt a rectangle so its angles are no longer square and you have a parallelogram: the sides keep their lengths and stay parallel, but the corners become a pair of acute and a pair of obtuse angles. Rectangles, rhombuses, and squares are all special cases of the parallelogram, which is why the area formula here covers every one of them.
The single most important fact about a parallelogram's area is that it depends on the perpendicular height, not the length of the slanted side. As you shear a parallelogram (push the top sideways while keeping the base and the vertical height fixed), the side gets longer and the figure looks bigger, but the area does not change at all. This is the same reason a stack of cards has the same volume whether you push it into a slant or leave it square, a principle known as Cavalieri's principle.
This calculator returns the area two ways (base times height, and sides times the sine of the included angle), the perimeter, and the lengths of both diagonals, so you can cross-check whichever inputs you have.
How the formulas work
The area, perimeter, and diagonals all come from standard plane geometry:
Area (base x height) A = b x h
Area (sides x angle) A = a x b x sin(theta)
Perimeter P = 2 x (a + b)
Diagonals (law of cos) d1 = sqrt(a^2 + b^2 - 2ab cos(theta))
d2 = sqrt(a^2 + b^2 + 2ab cos(theta))
- b is the base and h is the perpendicular distance to the opposite side, not the slanted side length.
- The two area formulas agree because the perpendicular height equals the side times the sine of the angle: h = b x sin(theta).
- theta is the included angle between the two adjacent sides. At 90 degrees the figure is a rectangle and sin(theta) = 1.
- The diagonals are unequal in a general parallelogram; they become equal only when theta = 90 degrees (a rectangle).
Worked example
A parallelogram has a base of 10 units, two adjacent sides of 10 and 7 units, and an included angle of 60 degrees. The perpendicular height is 6 units.
- Area from base and height: A = b x h = 10 x 6 = 60 square units.
- Area from sides and angle: A = a x b x sin(60 degrees) = 10 x 7 x 0.866 = 60.6 square units (matches, within rounding, once height equals 7 x sin 60).
- Perimeter: P = 2 x (10 + 7) = 34 units.
- Shorter diagonal: d1 = sqrt(10^2 + 7^2 - 2 x 10 x 7 x cos 60) = sqrt(149 - 70) = sqrt(79) = 8.89 units.
- Longer diagonal: d2 = sqrt(149 + 70) = sqrt(219) = 14.80 units.
Result: The area is about 60 square units, the perimeter is 34 units, and the diagonals measure 8.89 and 14.80 units. The two diagonals differ because the corners are not square.
Reference: special cases of the parallelogram
| Shape | Defining condition | Area formula |
|---|---|---|
| General parallelogram | Opposite sides parallel and equal | b x h, or ab sin(theta) |
| Rectangle | All angles 90 degrees | length x width |
| Rhombus | All four sides equal | (d1 x d2) / 2, or s^2 sin(theta) |
| Square | Equal sides and 90-degree angles | side^2 |
Every row is a parallelogram, so the base-times-height formula applies to all of them; the simpler formulas are shortcuts for the special cases.
Common pitfalls
- Using the slanted side as the height. The most common mistake. Height is the perpendicular distance between the base and the opposite side, always shorter than the slanted side (unless the figure is a rectangle).
- Degrees versus radians. The sine and cosine forms need the angle in the unit your tool expects. This calculator takes degrees; a spreadsheet's SIN function expects radians (multiply degrees by pi/180).
- Assuming the diagonals are equal. Only rectangles have equal diagonals. In a tilted parallelogram one diagonal is noticeably longer than the other.
- Mixing the two side lengths. The trigonometric area uses the two adjacent sides and the angle between them. Plugging in a diagonal or a non-adjacent measurement gives a wrong answer.
- Forgetting the parallelogram law. A quick sanity check: d1^2 + d2^2 should equal 2(a^2 + b^2). If it does not, an input is off.
Related calculators
Frequently asked questions
What is the formula for the area of a parallelogram?
Area equals base times perpendicular height: A = b x h. The height must be the perpendicular distance between the base and the opposite side, not the length of the slanted side. If you only know the two adjacent side lengths and the angle between them, use the trigonometric form A = a x b x sin(theta), which gives the same answer.
Why is the area base times height and not base times side?
Because you can slice a triangle off one end of the parallelogram and slide it to the other end to form a rectangle of the same base and the same perpendicular height. That rectangle has area base times height, and rearranging preserved the area, so the parallelogram has the same area. The slanted side is longer than the height, so using it would overstate the area.
How do I find the area if I only know the sides and the angle?
Use A = a x b x sin(theta), where a and b are the two adjacent side lengths and theta is the included angle between them. The perpendicular height equals b x sin(theta), so this formula is just base times height in disguise. At theta = 90 degrees sin is 1 and the parallelogram is a rectangle.
How are the diagonals of a parallelogram calculated?
The two diagonals follow the law of cosines: d1 = sqrt(a^2 + b^2 - 2ab cos(theta)) and d2 = sqrt(a^2 + b^2 + 2ab cos(theta)). Unlike a rectangle, the diagonals of a general parallelogram are unequal. The sum of their squares equals twice the sum of the squares of the sides, a relationship known as the parallelogram law.
Is a rectangle a parallelogram?
Yes. A rectangle is a parallelogram whose angles are all 90 degrees, so its height equals its side and area = base x height = length x width. A rhombus is a parallelogram with all four sides equal, and a square is both a rectangle and a rhombus. The base-times-height area formula covers every one of these cases.