$Parallelogram$

Parallelogram

Solution with formulas: Parallelogram d1 = 12 cm, d2 = 8 cm, φ = 60°: S ≈ 41.5692 cm², P ≈ 28.0186 cm, a ≈ 5.2915 cm, b ≈ 8.7178 cm

What is known:

Side and height

Two sides and the angle between them

Diagonals and the angle between them

Two sides and a diagonal

Diagonal d₁

Diagonal d₂

Angle between the diagonals φ, °

Units:

Lengths are positive numbers (whole or decimal), angles are in degrees.

Answer:

S ≈ 41.5692 cm², P ≈ 28.0186 cm, a ≈ 5.2915 cm, b ≈ 8.7178 cm

Solution:

Drawing to scale:

diagonalsaltitude

Given: d₁ = 12 cm, d₂ = 8 cm, φ = 60°.

The diagonals bisect each other. Find the sides with the law of cosines from the triangles formed by the halves of the diagonals:

1. Side a:

a = √(d₁² + d₂² − 2d₁d₂ cos φ)2 ≈ 5.2915 cm

2. Side b:

b = √(d₁² + d₂² + 2d₁d₂ cos φ)2 ≈ 8.7178 cm

3. Angle β:

β = 180° − α = 180° − 115.6934° ≈ 64.3066°

4. Area:

S = d₁ · d₂ · sin φ2 = 12 · 8 · sin 60°2 ≈ 41.5692 cm²

5. Perimeter:

P = 2(a + b) = 2 · (5.2915 + 8.7178) ≈ 28.0186 cm

6. Height to side a:

h_a = Sa = 41.56925.2915 ≈ 7.8558 cm

7. Height to side b:

h_b = Sb = 41.56928.7178 ≈ 4.7683 cm

All values:

Value	Result
Side a, a	≈ 5.2915 cm
Side b, b	≈ 8.7178 cm
Angle α, α	≈ 115.6934°
Angle β, β	≈ 64.3066°
Area, S	≈ 41.5692 cm²
Perimeter, P	≈ 28.0186 cm
Diagonal d₁, d₁	12 cm
Diagonal d₂, d₂	8 cm
Height to side a, h_a	≈ 7.8558 cm
Height to side b, h_b	≈ 4.7683 cm

Got it? Now solve one yourself.

Test yourself

The parallelogram calculator finds the area, perimeter, diagonals, heights and angles of a parallelogram. It can be given by a side and its height, two sides and the included angle, the diagonals and the angle between them, or two sides and a diagonal.

Area formulas

What is known	Formula
Side and height	S = a · h_a
Two sides and an angle	S = a · b · sin α
Diagonals and the angle between them	S = ½ · d₁ · d₂ · sin φ

Diagonals and sides

The diagonals are found with the law of cosines: d₁² = a² + b² − 2ab cos α, d₂² = a² + b² + 2ab cos α.

The sum of the squares of the diagonals equals the sum of the squares of all sides: d₁² + d₂² = 2(a² + b²).

Example. a = 10 cm, b = 6 cm, α = 30°.
S = 10 · 6 · sin 30° = 30 cm², P = 2 · (10 + 6) = 32 cm
d₁ = √(100 + 36 − 120 · cos 30°) ≈ 5.66 cm, d₂ ≈ 15.49 cm

Properties of a parallelogram

Opposite sides are equal and parallel, opposite angles are equal.
Adjacent angles add up to 180°.
The diagonals bisect each other and divide the parallelogram into four triangles of equal area.