A parallelogram is a four-sided shape whose opposite sides are parallel. That single condition makes opposite sides equal, opposite angles equal, and the two diagonals cut each other in half. Rectangles, rhombuses and squares are all special parallelograms. You meet the general, leaning version in angled parking stalls, slanted floor tiles, a bookcase racked out of square, and the force diagrams of physics class. This parallelogram calculator solves the shape from four different starting points and lists its area, heights, angles and diagonals.
How to use the parallelogram calculator
- In I know the, choose Two sides and the angle between them, Two sides and the height, Two sides and one diagonal, or Base and height only (area).
- Enter Base (a) and, in the first three modes, Side (b), the slanted side. Then add the angle θ between them, the height h (the perpendicular distance between the two a sides), or a diagonal d.
- Pick the Units, and choose degrees or radians under Angle unit (it applies to the angle you enter and to the angles in the results).
- Read the area at the top. Below it are the perimeter, the height on base a, the height on side b, the smaller and larger angle, and the long and short diagonals. Base and height only returns just the area, because the slant is unknown.
Parallelogram formulas
| Quantity | Formula |
|---|---|
| Perimeter | P = 2(a + b) |
| Height on base a | h = b sin θ |
| Height on side b | a sin θ |
| Diagonals | √(a² + b² ± 2ab cos θ) |
| Angle from the height | sin θ = h ÷ b |
| Angle from a diagonal | cos θ = (a² + b² − d²) ÷ 2ab |
Why the area is base × height
Drop a perpendicular from one top corner down to the base. It cuts off a right triangle. Slide that triangle to the other end and it fits exactly against the opposite slanted side, turning the parallelogram into a rectangle with the same base a and the same height h. Nothing was added or removed, so the area is a × h.
The slanted side only matters through the height it produces, h = b sin θ. Keep the sides fixed at a = 10 cm and b = 6 cm and let the shape lean:
| Angle θ | Height on base a | Area | Perimeter |
|---|---|---|---|
| 90° | 6 cm | 60 cm² | 32 cm |
| 75° | 5.80 cm | 57.96 cm² | 32 cm |
| 60° | 5.20 cm | 51.96 cm² | 32 cm |
| 45° | 4.24 cm | 42.43 cm² | 32 cm |
| 30° | 3 cm | 30 cm² | 32 cm |
| 15° | 1.55 cm | 15.53 cm² | 32 cm |
The perimeter never changes, but the area drops toward zero. That is what happens to a bookcase with no back panel when it racks sideways: every board keeps its length, yet the cabinet gets lower and holds less.
Worked example: an angled parking stall
A parking lot paints its stalls at 60° to the aisle. Each stripe is 18 ft long, and the stripes are 10.4 ft apart measured along the aisle. How much pavement does one stall use, and how wide is it for a car?
- Choose Two sides and the angle between them: base a = 10.4 ft (along the aisle), side b = 18 ft (the stripe), θ = 60°, units in feet.
- Area: 10.4 × 18 × sin 60° ≈ 162.1 ft².
- Height on side b: 10.4 × sin 60° ≈ 9.01 ft, the true width between stripes measured square to them.
- Height on base a: 18 × sin 60° ≈ 15.59 ft, how far the stall reaches back from the aisle.
- Diagonals: about 24.89 ft and 15.65 ft.
The 10.4 ft spacing along the aisle gives only about 9 ft of usable width.
Diagonals and the parallelogram law
The diagonals of a parallelogram bisect each other but are equal only in a rectangle. Their squares always add up to the squares of all four sides: d1² + d2² = 2(a² + b²). For a = 10, b = 6 and θ = 60°, that is 14² + 8.72² ≈ 196 + 76 = 272 = 2 × (100 + 36), a quick check on measured work. In Two sides and one diagonal mode, the calculator applies the law of cosines to the triangle formed by a, b and that diagonal, so the diagonal must be shorter than a + b and longer than the difference between them.
For four equal sides, use the rhombus calculator. If only one pair of sides is parallel, switch to the trapezoid calculator.
Frequently asked questions
Why isn't the area of a parallelogram side × side?
Multiplying the two sides only works when the corners are 90°. As a parallelogram leans over, its sides keep their length but the distance between the parallel sides shrinks, and the area shrinks with it. The correct formula is base × perpendicular height, which is the same as a × b × sin θ.
How do I find the height of a parallelogram?
Multiply the slanted side by the sine of the angle between the sides: h = b sin θ. With b = 6 cm and θ = 60°, the height on base a is 6 × 0.866 ≈ 5.20 cm. A parallelogram has two heights, one for each pair of parallel sides, and the calculator lists both.
How long are the diagonals of a parallelogram?
Use d = √(a² + b² ± 2ab cos θ): the plus sign gives the long diagonal and the minus sign the short one. For a = 10, b = 6 and θ = 60° they are 14 and about 8.72. The two diagonals are equal only when the angles are 90°.
Can I find the perimeter from base and height only?
No. Base and height fix the area but not the length of the slanted side, so infinitely many parallelograms share them. Choose a mode that includes side b to get the perimeter, angles and diagonals.
Can I enter the angle in radians?
Yes. The Angle unit menu lets you pick degrees or radians for the angle you type and for the angles in the results. Adjacent angles of a parallelogram always add to 180° (π radians), so the results show both the smaller and the larger angle.