Parallelogram Area Calculator
Find the area of any parallelogram using the method that matches what you know: base and perpendicular height, two adjacent sides and the included angle, or both diagonals and the angle between them. Flip between metric and imperial or choose degrees for angles. The calculator also returns the perimeter, and it shows every step of the working.
Formula
Worked example
Mode 1: base = 8 cm, height = 5 cm, so A = 8 x 5 = 40 cm². Mode 2: side a = 8 cm, side b = 6 cm, angle = 60 deg, so A = 8 x 6 x sin(60°) = 48 x 0.866 = 41.57 cm² and perimeter = 2 x (8 + 6) = 28 cm. Mode 3: diagonals 10 cm and 8 cm, angle between them = 60 deg, so A = 0.5 x 10 x 8 x sin(60°) = 40 x 0.866 = 34.64 cm².
Why area is base times perpendicular height
A parallelogram and a rectangle that share the same base and the same perpendicular height enclose exactly the same area. You can see why with a simple cut-and-slide: slice a right triangle off one slanted end of the parallelogram and move it to the other end, and the shape becomes a rectangle of width equal to the base and height equal to the perpendicular gap between the two parallel sides. Because nothing was added or removed during that rearrangement, the area is unchanged, so A = base x height. The crucial point is that height means the straight perpendicular distance between the parallel sides, not the length of the slanted edge that connects them.
Using two sides and an interior angle
When you know the lengths of both pairs of parallel sides (a and b) and the interior angle between them, the perpendicular height equals b x sin(angle). Substituting into the base-height formula gives A = a x b x sin(angle). This formula also lets you recover the perimeter as P = 2(a + b) and the perpendicular height as h = b x sin(angle). If your angle is obtuse (greater than 90 degrees), sin still gives a positive result, so the same formula covers all interior angles from just above 0 to just below 180 degrees. A rectangle is simply the special case where the angle is exactly 90 degrees and sin(90°) = 1.
Using the two diagonals and the angle between them
A third formula applies when you know both diagonal lengths (e and f) and the angle at which they cross. The diagonals of a parallelogram always bisect each other, splitting the shape into four triangles. Each triangle has two sides that are half a diagonal, and the included angle is either the crossing angle or its supplement. Pairing opposite triangles shows that A = (1/2) x e x f x sin(angle). The angle here is one of the two angles formed at the crossing point; because opposite angles at a crossing are equal and supplementary pairs share the same sine, it does not matter which of the two crossing angles you enter.
Perimeter and height as bonus outputs
In the two-sides-and-angle mode the calculator also returns the perimeter and the perpendicular height. Perimeter is straightforward: a parallelogram has two pairs of equal opposite sides, so P = 2a + 2b. The perpendicular height is h = b x sin(angle), since the slanted side b, the height h, and the horizontal offset form a right triangle. These values are useful for practical tasks such as fencing a parallelogram-shaped plot or checking whether a leaning frame meets a minimum clearance requirement.
Parallelogram area examples (base x height)
| Base | Height | Area |
|---|---|---|
| 4 | 3 | 12.0000 |
| 8 | 5 | 40.0000 |
| 10 | 7 | 70.0000 |
| 12.5 | 6 | 75.0000 |
Every area below uses A = base x height. The slant of the sides is irrelevant to the area.
Frequently asked questions
Do I use the slanted side or the perpendicular height to find the area?
Always use the perpendicular height, the straight-line distance measured at a right angle between the two parallel sides. The slanted edge is longer than the true height, so multiplying the base by it would overstate the area. If you only know the slanted side, switch to the "two sides and angle" mode and enter the slanted side as side b along with the interior angle.
What if I only know the side lengths and the interior angle?
Select the "two sides and included angle" mode. Enter side a (one pair of parallel sides), side b (the other pair), and the interior angle between them. The calculator uses A = a x b x sin(angle), also returning the perimeter and perpendicular height.
How does the diagonals formula work?
When you know both diagonal lengths e and f and the angle at which they cross, use A = (1/2) x e x f x sin(angle). The diagonals of a parallelogram bisect each other, creating four triangles whose areas sum to the total area. You can enter either of the two crossing angles because supplementary angles share the same sine value.
Is a rectangle just a special parallelogram?
Yes. A rectangle is a parallelogram whose interior angles are all 90 degrees. Because sin(90°) = 1, the formula A = a x b x sin(angle) reduces to the familiar length x width. A rhombus is the special case where all four sides are equal. A square is both, with all equal sides and all 90-degree angles.
Why does the perimeter not appear in the base-and-height mode?
In that mode you supply only the base and the perpendicular height. Knowing the height and the base tells you the area, but not the length of the slanted side. Without both side lengths the perimeter cannot be computed. Switch to the "two sides and angle" mode to get the perimeter.