Parallelogram
Solution with formulas: Parallelogram a = 10, b = 6, α = 30°: S = 30, P = 32, d1 ≈ 5.6637, d2 ≈ 15.4894
Drawing to scale:
Given: a = 10, b = 6, α = 30°.
1. Angle β:
β = 180° − α = 180° − 30° = 150°
2. Area:
S = a · b · sin α = 10 · 6 · sin 30° = 30
3. Perimeter:
P = 2(a + b) = 2 · (10 + 6) = 32
4. Diagonal d₁:
d1 = √(a² + b² − 2ab cos α) = √(10² + 6² − 2 · 10 · 6 · cos 30°) ≈ 5.6637
5. Diagonal d₂:
d2 = √(a² + b² + 2ab cos α) = √(10² + 6² + 2 · 10 · 6 · cos 30°) ≈ 15.4894
6. Height to side a:
ha = Sa = 3010 = 3
7. Height to side b:
hb = Sb = 306 = 5
All values:
| Value | Result |
|---|---|
| Side a, a | 10 |
| Side b, b | 6 |
| Angle α, α | 30° |
| Angle β, β | 150° |
| Area, S | 30 |
| Perimeter, P | 32 |
| Diagonal d₁, d1 | ≈ 5.6637 |
| Diagonal d₂, d2 | ≈ 15.4894 |
| Height to side a, ha | 3 |
| Height to side b, hb | 5 |
Got it? Now solve one yourself.
Test yourselfThe 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 · ha |
| 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.
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