Starlight Tools

Parallelogram Calculator — Area, Perimeter, Height and Diagonals

Enter two adjacent sides and their included angle—or a base, adjacent side and perpendicular height—to calculate the parallelogram’s complete geometry.

Enter parallelogram measurements

Parallelogram measurement diagram A slanted parallelogram with adjacent sides a and b, included angle theta, perpendicular height h, and two dashed diagonals. base a side b height h θ diagonal 1 diagonal 2

Use the interior angle where side a meets side b. Angles are entered in degrees.

A positive base length
The side next to base a
Greater than 0° and less than 180°
Choose 0 to 10

Use one unit: enter both sides and height, when used, in the selected unit. Changing the selector changes labels only; it does not convert entered values. Area is shown in square units.

Parallelogram results

Enter valid measurements and select Calculate to see area, perimeter, heights, diagonals, angles and calculation steps.

Parallelogram formulas

Let a and b be adjacent sides, θ their included angle, ha the perpendicular height to base a, hb the height to base b, and d₁ and d₂ the diagonals.

Area

A = aha = ab sin θ

A base times its perpendicular height. Square the selected length unit.

Perimeter

P = 2(a + b)

Opposite sides are equal, so each adjacent side occurs twice.

Heights

ha = b sin θ
hb = a sin θ

Each height depends on which side is chosen as the base.

Diagonal 1

d₁ = √(a² + b² + 2ab cos θ)

This is the vector-sum diagonal for the entered angle.

Diagonal 2

d₂ = √(a² + b² − 2ab cos θ)

This is the vector-difference diagonal. The two diagonals bisect each other.

Angle from height

θ = sin⁻¹(ha/b)

The supplementary angle 180° − θ has the same height and area.

Determination rule: base and height alone do not determine a unique parallelogram. The adjacent side or equivalent extra information is needed to find the perimeter and diagonals.

Worked examples

Two sides and a 60° angle

For a = 8 cm, b = 5 cm and θ = 60°, area is 34.64 cm², perimeter is 26 cm, and height to base a is 4.33 cm.

Base, side and height

For a = 10 m, b = 6 m and ha = 3 m, area is 30 m². The principal angle is 30°, with a 150° supplementary solution.

Rectangle special case

For a = 12 in, b = 5 in and θ = 90°, both diagonals are 13 in, area is 60 in², and perimeter is 34 in.

Method, precision and privacy

Method:
Euclidean geometry, trigonometry and the law of cosines

Numerical policy:
Full browser precision is used until display rounding

Input limits:
Positive finite lengths up to 10150; angles strictly between 0° and 180°

Last reviewed:

All values stay in your browser. The calculator does not send, save or include your measurements in analytics. Very thin parallelograms near 0° or 180° are mathematically valid, but displayed values may round to zero at low precision.

Parallelogram calculator FAQs

How do you find the area of a parallelogram?

Multiply a base by its perpendicular height: A = ah. With adjacent sides a and b and their included angle θ, use A = ab sin θ.

How do you find the perimeter?

Add the adjacent sides and double the result: P = 2(a + b). The angle and height do not affect perimeter.

How are the two diagonals calculated?

Use d₁ = √(a² + b² + 2ab cos θ) and d₂ = √(a² + b² − 2ab cos θ). At 90°, the diagonals are equal because the parallelogram is a rectangle.

Can base and height determine all the other values?

No. They determine area, but many parallelograms can share that base and height. An adjacent side or equivalent information is needed for perimeter and diagonals.

Why must height be no greater than side b?

Because ha = b sin θ and sine cannot exceed 1. If height equals side b, the included angle is 90°.

Why does height mode show two possible angles?

An acute angle and its supplementary obtuse angle have the same sine. They therefore produce the same area and heights; the two diagonal lengths remain the same pair but exchange labels.

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