Parallelogram Area
Calculator
Results
- Area
- 34.641016
- Perimeter
- 26
- Height
- 4.330127
Results
| Area | 34.641016 |
| Perimeter | 26 |
| Height | 4.330127 |
formula-map diagram
- Area
- 34.641016
- Perimeter
- 26
- Height
- 4.330127
Formula map
Formula
A = a × b × sin(θ)= 34.641016151378
Note
Simplified model: these are the exact results of the stated idealised geometric formulas, assuming perfect shapes, exact inputs and consistent units. Real measurements carry error, and results such as the cyclic-quadrilateral area only hold when the figure truly satisfies the stated condition. Verify independently before relying on these figures.
More in Advanced geometry
See all →Frequently asked questions
How is a parallelogram's area calculated from two sides and an angle?+
Area equals the product of two adjacent side lengths times the sine of the included angle between them: A = ab × sin(θ). This works because b × sin(θ) effectively gives the perpendicular height relative to side a.
Why does the formula use sine rather than cosine of the angle?+
Sine is used because it converts the slanted side into the perpendicular height component needed for area (base times height), whereas cosine would give the horizontal projection, which isn't relevant to area. At exactly 90 degrees, sin(90°) = 1, correctly reducing the formula to the familiar length × width for a rectangle.
What happens to the area if the angle between the sides is very small, close to zero?+
As the angle approaches zero, sin(θ) approaches zero too, so the parallelogram's area shrinks toward zero even though the side lengths stay fixed. Geometrically, this makes sense because the shape becomes an increasingly squashed sliver as the angle closes.
Does it matter which angle of the parallelogram I use, since it has two different angle measures?+
No, it doesn't matter — a parallelogram's two distinct angles are always supplementary (they sum to 180°), and since sin(θ) = sin(180° − θ), both angles give exactly the same area result. You can use either angle interchangeably.
How is this formula related to the more familiar base-times-height formula?+
They're the same formula in different forms: b × sin(θ) calculates the true perpendicular height from the slanted side and angle, so ab·sin(θ) is really just base × height in disguise. This side-angle version is more convenient when you know the sides and angle but haven't directly measured the perpendicular height.