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# Deltoid, kite concave, dart, arrowhead - diagonals, area, perimeter, sides

Calculate the diagonals of a concave deltoid, kite, side lengths, surface area, perimeter and radius of the inscribed circle. Each size can be calculated using many formulas, just indicate what data we have.

## First diagonal of concave deltoid (concave kite, dart, arrowhead)

### First diagonal of concave deltoid on side (a) and angle α

$$e=a\cdot 2\sin\left(\frac{\alpha}{2}\right);$$ $$e=2\cdot\sqrt{a^2-g^2 }$$ where $$g=a\cdot \sin \left(\frac{\alpha}{2}-90^\circ\right)$$

### First diagonal of concave deltoid side (b) and angle β

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

### First diagonal of the concave deltoid on the sides, the second diagonal and the angle γ

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

### First diagonal of the concave deltoid on the sides, the second diagonal and the angle α

$$e=2\cdot\sqrt{b^2-(f+g)^2}$$ gdzie $$g=a\cdot \sin \left(\frac{\alpha}{2}-90^\circ\right)$$

## Second diagonal of the concave deltoid (concave kite, dart, arrowhead)

### Second diagonal of concave deltoid on sides (a) (b) and angles α & β

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

### Second diagonal of the concave deltoid on the sides (a) (b) and the first diagonal

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

### Second diagonal of concave deltoid with sides, first diagonal and angle γ

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

### Second diagonal of the concave deltoid on the sides and angle γ

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

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

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

### Second diagonal of concave deltoid with sides, first diagonal and angle α

$$f=\sqrt{b^2-\left(\frac{e}{2}\right)^2}-g$$ where $$g=a\cdot \sin \left(\frac{\alpha}{2}-90^\circ\right)$$

## Surface area of the concave deltoid (concave kite, dart, arrowhead)

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

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

### Area of the concave deltoid from the diagonals

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

## Concave deltoid circumference (concave kite, dart, arrowhead)

### Concave deltoid circumference from the sides

$$L = 2a + 2b$$

### Concave deltoid circumference 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 concave deltoid (concave kite, dart, arrowhead)

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

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

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

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

### Side (a) of the concave deltoid on the side (b) and angles α & β

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

### Side (b) of the concave deltoid on the side (a) and angles α & β

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

# Concave deltoid, concave kite, dart, arrowhead

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.

Concave deltoid, also known as an concave kite, dart, arrowhead - is a deltoid in which the internal angle between the shorter sides is greater than 180 °.

Concave deltoid has the following properties:
1. The sum of the measures of all the internal angles of the deltoid is 2Π $$\alpha+\beta+2\cdot\gamma=360^\circ$$
2. Formula for the first diagonal of the concave deltoid on the side (a) and the angle α
3. $$e=a\cdot 2\sin\left(\frac{\alpha}{2}\right);$$ $$e=2\cdot\sqrt{a^2-g^2 }$$ gdzie $$g=a\cdot \sin \left(\frac{\alpha}{2}-90^\circ\right)$$
4. Formula for the first diagonal of the concave deltoid on the side (b) and the angle β
5. $$e=b\cdot 2\sin\left(\frac{\beta}{2}\right)$$
6. Formula for the first diagonal of the concave deltoid on the sides, the second diagonal and the angle γ
7. $$e=\frac{2\cdot a\cdot b\cdot \sin\gamma}{f}$$
8. Formula for the first diagonal of the concave deltoid on the sides, the second diagonal and the angle α
9. $$e=2\cdot\sqrt{b^2-(f+g)^2}$$ gdzie $$g=a\cdot \sin \left(\frac{\alpha}{2}-90^\circ\right)$$
10. Second diagonal of concave deltoid on 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 second diagonal of the concave deltoid from the sides (a) (b) and the first diagonal
13. $$f=\sqrt{b^2-\left(\frac{e}{2}\right)^2}-\sqrt{a^2-\left(\frac{e}{2}\right)^2}$$
14. Formula for the second diagonal of the concave deltoid from the sides, the first diagonal and the angle γ
15. $$f=\frac{2\cdot a\cdot b\cdot \sin\gamma}{e}$$
16. Formula for the second diagonal of the concave 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 second diagonal of the concave 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 second diagonal of the concave deltoid from the sides, the first diagonal and the angle α
21. $$f=\sqrt{b^2-\left(\frac{e}{2}\right)^2}-g$$ gdzie $$g=a\cdot \sin \left(\frac{\alpha}{2}-90^\circ\right)$$
22. Formula for Area of the concave deltoid on the sides (a) (b) and angles α & β
23. $$S=\frac{a^2\cdot \sin\alpha}{2}+\frac{b^2\cdot\sin\beta}{2}$$
24. Formula for the Area of the concave deltoid on the sides (a)(b) and the angle γ
25. $$S=a\cdot b\cdot \sin\gamma$$
26. Formula for the Area of the concave deltoid from the diagonals
27. $$S=\frac{e\cdot f}{2}$$
28. Formula for the perimeter of the concave deltoid on the sides
29. $$L = 2a + 2b$$
30. Formula for the circumference of the concave deltoid from the shorter diagonal and the angle(α) & (β)
31. $$L = \frac{e}{\sin\left(\frac{\beta}{2}\right)}+\frac{e}{\sin\left(\frac{\alpha}{2}\right)}$$
32. Formula on the side (a) of the concave deltoid with the shorter diagonal and angle α
33. $$a=\frac{e}{2\cdot\sin\left(\cfrac{\alpha}{2}\right)}$$
34. Formula on the side (b) of the concave deltoid with the shorter diagonal and angle β
35. $$b=\frac{e}{2\cdot\sin\left(\cfrac{\beta}{2}\right)}$$
36. Formula for the Side (a) of the deltoid on the concave side (b) and angles α & β
37. $$a=\frac{b\cdot \sin\left(\cfrac{\beta}{2}\right)}{\sin\left(\cfrac{\alpha}{2}\right)}$$
38. Formula on the side (b) of the concave deltoid on the side (a) and angles α & β
39. $$b=\frac{a\cdot \sin\left(\cfrac{\alpha}{2}\right)}{\sin\left(\cfrac{\beta}{2}\right)}$$