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

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