Parallelogram Calculator: Area, Height, Diagonals

Two ways to solve a parallelogram

A parallelogram has two pairs of parallel, equal-length sides. Depending on what you already know, this calculator solves it two ways:

1. Base and height — when you only know the base $b$ and the perpendicular height $h$:

$K = bh$

This gives the area directly, but nothing else (perimeter, angles, and diagonals all depend on the side length and angle, which base and height alone don't determine).

2. Two sides and the included angle — when you know both sides $a, b$ and the base angle $A$ (in degrees), every measure follows:

$h = a\sin(A) \qquad K = ab\sin(A) \qquad P = 2a + 2b$

$B = 180° - A \qquad p = \sqrt{a^2+b^2-2ab\cos A} \qquad q = \sqrt{a^2+b^2+2ab\cos A}$

Where the height formula comes from

Drop a perpendicular from a top vertex down to the base line. That perpendicular has length $h$, and together with the slanted side $a$ and the angle $A$ between them, it forms a right triangle: $h = a\sin(A)$. Substituting into $K = bh$ gives the equivalent area formula $K = ab\sin(A)$ directly from the two sides and the angle, without needing to compute the height as a separate step.

Where the diagonal formulas come from

Each diagonal is the third side of a triangle formed by sides $a$ and $b$ meeting at one of the two angles. The law of cosines gives:

• Short diagonal $p$: opposite the acute angle, using $-2ab\cos A$
• Long diagonal $q$: opposite the obtuse angle, using $+2ab\cos A$

Since $A + B = 180°$, $\cos B = -\cos A$ — so both formulas are really the same law of cosines applied to the two different triangles, differing only by that sign.

Worked example: base b = 6, height h = 4

$K = 6 \times 4 = 24$

✅ Area = 24. With just base and height, that's the only measure this mode can produce.

Worked example: sides a = 5, b = 6, angle A ≈ 53.13°

$h = 5 \times \sin(53.13°) \approx 4.0000 \qquad K = 6 \times 4 = 24.0000$

$P = 2(5) + 2(6) = 22 \qquad B = 180° - 53.13° = 126.87°$

$p = \sqrt{25+36-60\cos(53.13°)} \approx 5.0000 \qquad q = \sqrt{25+36+60\cos(53.13°)} \approx 9.8489$

Worked example: the rectangle special case, A = 90°

With the same sides a = 5, b = 6 but A = 90°: sin(90°) = 1, so height = side = 5, and area = 30. Both diagonals collapse to the same value, √(25+36) = √61 ≈ 7.8102, because cos(90°) = 0 removes the ± term entirely — exactly matching the rectangle formula.

Common mistakes

• Using the side instead of the height. K = ab only equals the true area when A = 90° (a rectangle); otherwise it overstates the area, since sin(A) < 1.
• Mixing up the diagonal signs. The short diagonal subtracts the cosine term, the long diagonal adds it — reversing them swaps which diagonal is which.
• Forgetting angles are supplementary, not equal. B = 180° − A, not B = A. Only the OPPOSITE angle (C) equals A.

Where this shows up

• Land surveying: an irregularly leaning plot of land is often modeled as a parallelogram to estimate its area from base and height measurements.
• Physics: the parallelogram of forces uses this exact shape to add two vectors — the resultant is the diagonal of the parallelogram formed by the two force vectors as sides.
• Structural engineering: shear deformation turns rectangular cross-sections into parallelograms, and the diagonal formulas describe how much the diagonal bracing lengthens or shortens.

A parallelogram generalizes the rectangle (a right-angle special case) and is generalized in turn by the rhombus (equal-sides special case) — all three share the same core area and diagonal relationships, just with different constraints on the angle or side lengths.