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How to use
Use the parallelogram solver to calculate area, perimeter, height and both diagonals from two adjacent side lengths and their included angle.
- Enter the two side lengths.
- Enter the included angle in degrees between 0 and 180.
- Choose a length unit.
- Review area, perimeter, height and diagonals.
How it is calculated
Example
For a=8, b=5 and θ=60°, area is about 34.641 and perimeter is 26. For the same a=8, b=5, θ=60° example, height on base a is 5sin60°≈4.330127; diagonals are √(64+25+40)=√129≈11.357817 and √(64+25−40)=7. At θ=120°, area, perimeter and height stay the same while the diagonal values swap. Enter the included interior angle between sides, not a diagonal angle.
Important notes
The reported height is b·sinθ relative to base a. As the angle approaches 0° or 180°, the area approaches zero.
Worked examples and interpreting results
A parallelogram with base 10, height 4 and adjacent side 6 has area 40 and perimeter 32. Height is the perpendicular distance between the bases, not the sloping side length.
| Case | Calculation | Result |
|---|---|---|
| Area from base and height | 10 × 4 | 40 |
| Perimeter from adjacent sides | 2 × (10 + 6) | 32 |
| Longer diagonal for a=8, b=5, θ=60° | √(8² + 5² + 2 × 8 × 5 × cos60°) = √129 | 11.357817 |
| Shorter diagonal for a=8, b=5, θ=60° | √(8² + 5² − 2 × 8 × 5 × cos60°) = √49 | 7 |
How to check the result
Side lengths alone do not determine area without a height or angle. Use the height associated with your chosen base. Switching the base to the adjacent side requires its corresponding height.
Parallelogram area and perimeter
A parallelogram's area is its base multiplied by perpendicular height. Its perimeter equals twice the sum of two adjacent side lengths.
- Base 8 and height 5: 8 × 5 = 40
- Sides 8 and 6: 2 × (8 + 6) = 28
- Area 60 and base 12: 60 / 12 = 5
Important note
Height is perpendicular to the base and is not necessarily the length of a sloping side.
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Frequently asked questions
How do I solve a parallelogram from two sides and an angle?
Use sides a and b with included angle θ. Area is ab·sinθ, perimeter is 2(a+b), and the diagonals follow from the cosine rule.
Is a rectangle a parallelogram?
Yes. It is the 90° special case.
Are the diagonals always equal?
No; they are equal for the rectangle case.
How is area calculated?
A=ab·sinθ.
Is the height equal to the slanted side?
Usually not. Height is the perpendicular distance between parallel sides, while the slanted side can be longer. With a side and included angle, the corresponding height uses the side length times the sine of the angle.
Is the sloping side the height?
Usually no. Height is perpendicular to the base and equals bsinθ; it equals b only at θ=90°. Sides 8 and 5 at 60° give height about 4.330127, not 5.
Can I enter an angle of 0° or 180°?
Not for a nondegenerate shape. The angle must be greater than 0° and less than 180°; at either endpoint the sides become collinear and area is zero.