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# Deltoid, kite - diagonals, area, perimeter, sides

The calculator will help you calculate deltoid (kite) diagonals, side lengths, area, perimeter and radius of the inscribed circle. Each size can be calculated using many formulas, just indicate what data we have.

## Shorter diagonal of the deltoid (kite)

### Shorter diagonal of the deltoid on the side (a) and the angle α

$$e=a\cdot 2\sin\left(\frac{\alpha}{2}\right)$$

### Shorter diagonal of the deltoid on the side (b) and the angle β

$$e=b\cdot 2\sin\left(\frac{\beta}{2}\right)$$

### Shorter diagonal of the deltoid on the sides, the longer diagonal and the angle γ

$$e=\frac{2\cdot a\cdot b\cdot \sin\gamma}{f}$$

### Shorter diagonal of the deltoid from the radius of the inscribed circle and the angle α & γ

$$e=\frac{2r\cdot cos\left(\frac{\gamma+\alpha-180^{\circ}}{2}\right)}{sin\left(\frac{\gamma}{2}\right)}$$

## Longer diagonal of the deltoid (kite)

### Longer diagonal of the deltoid on sides (a) (b) and angles α & β

$$f=a\cdot cos\left(\frac{\alpha}{2}\right)+ b\cdot cos\left(\frac{\beta}{2}\right)$$

### Longer diagonal of the deltoid on the sides (a) (b) and the shorter diagonal

$$f=\sqrt{a^2-\left(\frac{e}{2}\right)^2}+\sqrt{b^2-\left(\frac{e}{2}\right)^2}$$

### Longer diagonal of the deltoid on the sides, the shorter diagonal and the angle γ

$$f=\frac{2\cdot a\cdot b\cdot \sin\gamma}{e}$$

### Longer diagonal of the deltoid on the sides and angle γ

$$f=\sqrt{a^2+b^2-2\cdot a \cdot b \cdot \cos\gamma}$$

### Longer diagonal of the deltoid on the side (a) and angle β & γ

$$f=\frac{a\cdot \sin\gamma}{\sin\left(\frac{\beta}{2}\right)}$$

## The length of the radius of the circle inscribed in the deltoid (kite)

### The radius of the circle inscribed in the deltoid from the diagonal and the angle α & γ

$$r=\frac{e\cdot \sin\left(\cfrac{\gamma}{2}\right)}{2\cdot\cos\left(\cfrac{\gamma+\alpha-180^\circ}{2}\right)}$$

### Length of the radius of the circle inscribed in the deltoid on the sides and diagonals

$$r= \frac{e\cdot f}{2a+2b}$$

## Surface area of the deltoid (kite)

### Area of the deltoid from sides (a)(b) and angles α & β

$$S=\frac{a^2\cdot \sin\alpha}{2}+\frac{b^2\cdot\sin\beta}{2}$$

### Area of the deltoid on the sides (a)(b) and the angle γ

$$S=a\cdot b\cdot \sin\gamma$$

### Area of the deltoid from the diagonals

$$S=\frac{e\cdot f}{2}$$

## Circumference of a deltoid (kite)

### Circumference of the deltoid on the sides

$$L = 2a + 2b$$

### Circumference of the deltoid from the shorter diagonal and the angle(α) & (β)

$$L = \frac{e}{\sin\left(\frac{\beta}{2}\right)}+\frac{e}{\sin\left(\frac{\alpha}{2}\right)}$$

## Sides of the deltoid (kite)

### Side (a) of the deltoid from the shorter diagonal and angle α

$$a=\frac{e}{2\cdot\sin\left(\cfrac{\alpha}{2}\right)}$$

### Side (b) of the deltoid from the shorter diagonal and angle β

$$b=\frac{e}{2\cdot\sin\left(\cfrac{\beta}{2}\right)}$$

Deltoid, kite - information

Deltoid - is a quadrilateral whose four sides can be grouped into two pairs of equal length adjacent sides. The sides of the same length have a common vertex. Deltoid can be convex or concave. When the internal angle between the shorter sides of the deltoid is greater than 180 °, the deltoid is concave, otherwise the deltoid is convex. A concave deltoid is sometimes called an "dart" or "arrowhead" and is a kind of pseudo-triangle.

A convex deltoid has the following properties:
1. The sum of the measures of all interior angles is 2Π $$\alpha+\beta+2\cdot\gamma=360^\circ$$
2. Formula for the shorter diagonal of the side deltoid (a) and the angle α
3. $$e=a\cdot 2\sin\left(\frac{\alpha}{2}\right)$$
4. Formula for the shorter diagonal of the deltoid on the side (b) and the angle β
5. $$e=b\cdot 2\sin\left(\frac{\beta}{2}\right)$$
6. Formula for the shorter diagonal of the deltoid on the sides, the longer diagonal and the angle γ
7. $$e=\frac{2\cdot a\cdot b\cdot \sin\gamma}{f}$$
8. Formula for the shorter diagonal of the deltoid from the radius of the inscribed circle and the angle α i γ
9. $$e=\frac{2r\cdot cos\left(\frac{\gamma+\alpha-180^{\circ}}{2}\right)}{sin\left(\frac{\gamma}{2}\right)}$$
10. Formula for the longer diagonal of the deltoid on the sides (a) (b) and angles α & β
11. $$f=a\cdot cos\left(\frac{\alpha}{2}\right)+ b\cdot cos\left(\frac{\beta}{2}\right)$$
12. Formula for the longer diagonal of the deltoid on the sides (a) (b) and the shorter diagonal
13. $$f=\sqrt{a^2-\left(\frac{e}{2}\right)^2}+\sqrt{b^2-\left(\frac{e}{2}\right)^2}$$
14. Formula for the longer diagonal of the deltoid on the sides, the shorter diagonal and the angle γ
15. $$f=\frac{2\cdot a\cdot b\cdot \sin\gamma}{e}$$
16. Formula for the longer diagonal of the deltoid from the sides and the angle γ
17. $$f=\sqrt{a^2+b^2-2\cdot a \cdot b \cdot \cos\gamma}$$
18. Formula for the longer diagonal of the deltoid on the side (a) and the angle β & γ
19. $$f=\frac{a\cdot \sin\gamma}{\sin\left(\frac{\beta}{2}\right)}$$
20. Formula for the radius of the circle inscribed in the deltoid from the diagonal and the angle α & γ
21. $$r=\frac{e\cdot \sin\left(\cfrac{\gamma}{2}\right)}{2\cdot\cos\left(\cfrac{\gamma+\alpha-180^\circ}{2}\right)}$$
22. Formula for the radius of the circle inscribed in the deltoid of the sides and diagonals
23. $$r= \frac{e\cdot f}{2a+2b}$$
24. Formula for the area of the deltoid on the sides (a)(b) and angles α & β
25. $$S=\frac{a^2\cdot \sin\alpha}{2}+\frac{b^2\cdot\sin\beta}{2}$$
26. Formula for the area of the deltoid from the sides (a)(b) and the angle γ
27. $$S=a\cdot b\cdot \sin\gamma$$
28. Formula for the area of the deltoid from the diagonals
29. $$S=\frac{e\cdot f}{2}$$
30. Formula for the perimeter of the deltoid on the sides
31. $$L = 2a + 2b$$
32. Formula for the perimeter of a deltoid with a shorter diagonal and angle(α) & (β)
33. $$L = \frac{e}{\sin\left(\frac{\beta}{2}\right)}+\frac{e}{\sin\left(\frac{\alpha}{2}\right)}$$
34. Pattern on the side (a) of the deltoid with the shorter diagonal and angle α
35. $$a=\frac{e}{2\cdot\sin\left(\cfrac{\alpha}{2}\right)}$$
36. Pattern to the side of the (b) deltoid with the shorter diagonal and angle β
37. $$b=\frac{e}{2\cdot\sin\left(\cfrac{\beta}{2}\right)}$$