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# Parallelogram calculator - diagonals, area, perimeter, sides

Parallelogram calculator will help you calculate the long diagonal of the parallelogram, the short diagonal of the parallelogram, the side length, heights, area of the parallelogram.

## Longer diagonal of the parallelogram calculator

### Longer diagonal of the parallelogram from the sides and the angle α (law of cosines)

$$f= \sqrt{a^2+2ab \cos \alpha\ +b^2}$$

### Longer diagonal of the parallelogram from the sides and the angle β (law of cosines)

$$f= \sqrt{a^2-2ab \cos \beta\ +b^2}$$

### Longer diagonal of the parallelogram from the sides and the shorter diagonal

$$f= \sqrt{2a^2 + 2b^2 - e^2}$$

### Longer diagonal of the parallelogram from the field, the shorter diagonal and the angle between the diagonals

$$f= \frac{2S}{e\cdot \sin \gamma} = \frac{2S}{e\cdot \sin \delta}$$

## Shorter diagonal of the parallelogram calculator

### Shorter diagonal of the parallelogram from the sides and the angle α (law of cosines)

$$e= \sqrt{a^2-2ab \cos \alpha\ +b^2}$$

### Shorter diagonal of the parallelogram from the sides and the angle β (law of cosines)

$$e= \sqrt{a^2+2ab \cos \beta\ +b^2}$$

### Shorter diagonal of the parallelogram from the sides and the longer diagonal

$$e= \sqrt{2a^2 + 2b^2 - f^2}$$

### Shorter diagonal of the parallelogram from the field, the longer diagonal and the angle between the diagonals

$$e= \frac{2S}{f\cdot \sin \gamma} = \frac{2S}{f\cdot \sin \delta}$$

## Height of the parallelogram calculator

### Height of the parallelogram from the side and the area

$$h_1=\frac{S}{a}; h_2=\frac{S}{b}$$

### Height of the parallelogram from the side and angle

$$h_1=b \cdot \sin \alpha; h_2=a \cdot \sin \alpha$$

### Height of a parallelogram from perimeter, side, and angle

$$h_1=\frac{(L-2a) \cdot \sin \alpha}{2}; h_2=\frac{(L-2b) \cdot \sin \alpha}{2}$$

## Area of the parallelogram calculator

### Area of the parallelogram from the side and height

$$S=a\cdot h_1; S=b\cdot h_2$$

### Area of the parallelogram from the sides and angle(α) or (β)

$$S=a\cdot b \cdot \sin \alpha; S=a\cdot b \cdot \sin \beta$$

### Area of a parallelogram from diagonals and angle(γ) or (δ)

$$S=\frac {e \cdot f \cdot \sin \gamma}{2}; S=\frac {e \cdot f \cdot \sin \delta}{2}$$

## Perimeter of the parallelogram calculator

### Perimeter of the parallelogram from the sides

$$L = 2a+2b = 2(a+b)$$

### Perimeter of the parallelogram from the side and diagonals

$$L = 2a+\sqrt{2e^2+2f^2-4a^2};$$ $$L = 2b+\sqrt{2e^2+2f^2-4b^2}$$

### Perimeter of the parallelogram from the side, height and angle α

$$L = 2(a + \frac {h_1}{\sin \alpha});$$ $$L = 2(b + \frac {h_2}{\sin \alpha})$$

## Sides of the parallelogram calculator

### Sides of the parallelogram from diagonals and angle of intersection of diagonals

$$a=\frac {\sqrt{e^2+f^2+2ef\cdot\cos \gamma}}{2}; a=\frac {\sqrt{e^2+f^2-2ef\cdot\cos \delta}}{2}$$ $$b=\frac {\sqrt{e^2+f^2-2ef\cdot\cos \gamma}}{2}; b=\frac {\sqrt{e^2+f^2+2ef\cdot\cos \delta}}{2}$$

### Side of the parallelogram from the other side and diagonals

$$a=\frac {\sqrt {2e^{2}+2f^{2}-4b^2}}{2}$$ $$b=\frac {\sqrt {2e^{2}+2f^{2}-4a^2}}{2}$$

### Sides of the parallelogram from height and angle α

$$a=\frac {h_2}{\sin \alpha}$$ $$b=\frac {h_1}{\sin \alpha}$$

Parallelogram - information

Parallelogram - a quadrilateral whose opposite pairs of sides have the same length and are parallel. The diagonals of the parallelogram intersect exactly at half their length. The opposite angles are equal. The sum of the angles on the same side is 180°.
A special case of a parallelogram is a rhombus (all sides of the same length) and a straight angle (all right angles), and a square (all sides of the same length and right angles).

It has the following properties:
1. The parallelogram is a convex figure.
2. The sum of the measures of all interior angles is 2Π (360°), and the sum of the measures of two adjacent interior angles is Π, $$\alpha + \beta = 180°$$ is: $$\alpha = 180° - \beta$$ $$\beta = 180° - \alpha$$
3. The sum of the measures of two adjacent angles at which the diagonals intersect is Π, $$\gamma + \delta = 180°$$ as a result of: $$\gamma = 180° - \delta$$ $$\delta = 180° - \gamma$$
4. The point of intersection of the diagonals of the parallelogram divides each of them into halves.
5. Formula on the longer diagonal of the parallelogram from the sides and the angle α
6. $$f= \sqrt{a^2+2ab \cos \alpha\ +b^2}$$
7. Formula on the longer diagonal of the parallelogram from the sides and the angle β
8. $$f= \sqrt{a^2-2ab \cos \beta\ +b^2}$$
9. Formula on the longer diagonal of the parallelogram from the sides and the shorter diagonal
10. $$f= \sqrt{2a^2 + 2b^2 - e^2}$$
11. Formula for the longest diagonal of the parallelogram from the field, the shorter diagonal and the angle between the diagonals
12. $$f= \frac{2S}{e\cdot \sin \gamma} = \frac{2S}{e\cdot \sin \delta}$$
13. Formula for the shorter diagonal of the parallelogram from the sides and the angle α
14. $$e= \sqrt{a^2-2ab \cos \alpha\ +b^2}$$
15. Formula for the shorter diagonal of the parallelogram from the sides and the angle β
16. $$e= \sqrt{a^2+2ab \cos \beta\ +b^2}$$
17. Formula for the shorter diagonal of the parallelogram from the sides and the longer diagonal
18. $$e= \sqrt{2a^2 + 2b^2 - f^2}$$
19. Formula for the shorter diagonal of the parallelogram from the field, the longer diagonal and the angle between the diagonals
20. $$e= \frac{2S}{f\cdot \sin \gamma} = \frac{2S}{f\cdot \sin \delta}$$
21. Formula for the height of the parallelogram from the side and the area
22. $$h_1=\frac{S}{a}; h_2=\frac{S}{b}$$
23. Formula for the height of the parallelogram from the side and angle
24. $$h_1=b \cdot \sin \alpha; h_2=a \cdot \sin \alpha$$
25. Formula for the height of a parallelogram from perimeter, side, and angle
26. $$h_1=\frac{(L-2a) \cdot \sin \alpha}{2}; h_2=\frac{(L-2b) \cdot \sin \alpha}{2}$$
27. Formula for the area of the parallelogram from the side and height
28. $$S=a\cdot h_1; S=b\cdot h_2$$
29. Formula for the area of the parallelogram from the sides and angle(α) or (β)
30. $$S=a\cdot b \cdot \sin \alpha; S=a\cdot b \cdot \sin \beta$$
31. Formula for the area of a parallelogram from diagonals and angle(γ) or (δ)
32. $$S=\frac {e \cdot f \cdot \sin \gamma}{2}; S=\frac {e \cdot f \cdot \sin \delta}{2}$$
33. Formula for the perimeter of the parallelogram from the sides
34. $$L = 2a+2b = 2(a+b)$$
35. Formula for the perimeter of the parallelogram from the side and diagonals
36. $$L = 2a+\sqrt{2e^2+2f^2-4a^2};$$ $$L = 2b+\sqrt{2e^2+2f^2-4b^2}$$
37. Formula for the perimeter of the parallelogram from the side, height and angle α
38. $$L = 2(a + \frac {h_1}{\sin \alpha});$$ $$L = 2(b + \frac {h_2}{\sin \alpha})$$
39. Formula for the sides of the parallelogram from diagonals and angle of intersection of diagonals
40. $$a=\frac {\sqrt{e^2+f^2+2ef\cdot\cos \gamma}}{2}; a=\frac {\sqrt{e^2+f^2-2ef\cdot\cos \delta}}{2}$$ $$b=\frac {\sqrt{e^2+f^2-2ef\cdot\cos \gamma}}{2}; b=\frac {\sqrt{e^2+f^2+2ef\cdot\cos \delta}}{2}$$
41. Formula for the side of the parallelogram from the other side and diagonals
42. $$a=\frac {\sqrt {2e^{2}+2f^{2}-4b^2}}{2}$$ $$b=\frac {\sqrt {2e^{2}+2f^{2}-4a^2}}{2}$$
43. Formula for the sides of the parallelogram from height and angle α $$a=\frac {h_2}{\sin \alpha}$$ $$b=\frac {h_1}{\sin \alpha}$$

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