Parallelogram & Rhombus Calculator

Enter two adjacent sides and the angle between them to get the area, height, perimeter and both diagonals, drawn to scale. Equal sides make a rhombus, a right angle makes a rectangle, and both together make a square — the calculator detects each. You can also click a value on the drawing to change it.

Sides & angle

parallelogram

Enter both sides and the angle between them.

°
Click an amber value to edit it · drawn true to scale

Results

Parallelogram formulas

The area is the base times the perpendicular height — and that height is the slanted side times the sine of the angle. The diagonals come from the law of cosines on the two triangles inside.

Area · A = a b sin θ = base × height


Height · h = b sin θ


Diagonals · √(a² + b² ∓ 2ab cos θ)

Rhombus, rectangle, square

These are all special parallelograms. Equal sides give a rhombus (whose diagonals meet at right angles); a 90° angle gives a rectangle; both at once give a square.

A useful check

The two diagonals always satisfy d₁² + d₂² = 2(a² + b²) — handy for checking a result by hand.

Working with parallelograms

A parallelogram is a four-sided shape whose opposite sides are parallel and equal. Slant a rectangle over and you have one; the rhombus, rectangle and square are all special cases of it. This calculator takes the two side lengths and the angle between them — enough to fix the shape completely — and returns the area, the perpendicular height, the perimeter and both diagonals, drawn to scale.

The parts of a parallelogram

Worked example

Take a parallelogram with sides a = 8 and b = 6 and an angle of θ = 60° between them.

Step 1 — height (perpendicular, not the slant)
h = b sin θ = 6 × sin 60° = 6 × 0.8660 = 5.196
Step 2 — area, base times height
A = a × h = 8 × 5.196 = 41.57 (same as a b sin θ)
Step 3 — perimeter
P = 2(a + b) = 2(8 + 6) = 28
Step 4 — diagonals, by the law of cosines
d₁ = √(a²+b²−2ab cos θ) = √52 = 7.21 d₂ = √(a²+b²+2ab cos θ) = √148 = 12.17
Result

Height 5.196, area 41.57, perimeter 28, diagonals 7.21 and 12.17. As a check, 7.21² + 12.17² = 200, which equals 2(8² + 6²) — the parallelogram law holds.

Formulas this calculator uses

Area · A = a b sin θ

Height · h = b sin θ

Perimeter · P = 2(a + b)

Diagonals · √(a² + b² ∓ 2ab cos θ)

For a rhombus (a = b), the diagonals simplify to 2a sin(θ/2) and 2a cos(θ/2), they cross at right angles, and the area can also be found as half their product: ½ d₁ d₂.

Frequently asked questions

How do you find the area of a parallelogram?

Multiply the base by the perpendicular height: A = base × height. If you know the two sides and the angle between them instead, the area is a b sin θ, since the height equals b sin θ. Sides 8 and 6 at 60° give an area of about 41.57.

What is the height of a parallelogram?

The height is the perpendicular distance between the two parallel sides — not the length of the slanted side. If the slant side is b and the angle is θ, the height is h = b sin θ. It's always less than or equal to b.

How do you find the diagonals of a parallelogram?

Use the law of cosines on the triangles formed by two sides and a diagonal: d = √(a² + b² ∓ 2ab cos θ). The minus sign gives the shorter diagonal, the plus sign the longer. They always satisfy d₁² + d₂² = 2(a² + b²).

What is the difference between a parallelogram and a rhombus?

A rhombus is a parallelogram with all four sides equal. So every rhombus is a parallelogram, but only those with equal sides qualify. A rhombus also has the special feature that its diagonals cross at right angles.

How do you find the area of a rhombus from its diagonals?

For a rhombus, the area is half the product of the two diagonals: A = ½ d₁ d₂. This works because the diagonals are perpendicular and bisect each other, splitting the rhombus into four right triangles.

Is a square a parallelogram?

Yes. A square is a parallelogram with all sides equal and all angles 90° — it's both a rhombus and a rectangle at once. Enter equal sides and a 90° angle and this calculator labels the shape a square.