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# Isosceles trapezoid calculator - diagonals, area, perimeter, sides lenght

Isosceles trapezoid calculator will help you calculate diagonal of the isosceles, the side length, height, area, perimeter and the radius of the circumscribed circle of the isosceles.

## Diagonal of isosceles trapezoid

### Diagonal of isosceles trapezoid from sides (a) i (c) and angle α

$$d=\sqrt{a^2+c^2-2ac\cdot \cos(\beta)}$$

### Diagonal of isosceles trapezoid from sides (b) i (c) and angle β

$$d=\sqrt{b^2+c^2-2bc\cdot \cos(\beta)}$$

### Diagonal of isosceles trapezoid from sides (a)(b)(c)

$$d= \sqrt{a\cdot b+c^2}$$

### Diagonal of isosceles trapezoid from area and angle γ

$$d= \frac{2S}{\sin \gamma}$$

## Radius of the circle circumscribed by an isosceles trapezoid

### Radius of the circle circumscribed from the diagonal (d) and angle α or β

$$R=\frac{d}{2\sin \alpha}$$ $$R=\frac{d}{2\sin \beta}$$

### Radius of the circle circumscribed from the leg (c), diagonal (d) and height (h)

$$R=\frac{c\cdot d}{2h}$$

### Radius of the circle circumscribed from sides (a)(b)(c)

$$R= c\cdot\sqrt{\frac{a\cdot b + c^2}{4c^2-(a-b)^2}}$$

## Height of the isosceles trapezoid

### Height of the trapezoid from sides (a)(b) and area

$$h=\frac{2S}{a+b}$$

### Height of the trapezoid from leg (c) and angle

$$h=c \cdot \sin \alpha = c \cdot \sin \beta$$

### Height of the trapezoid from sides

$$h= \frac{\sqrt{4c^2 - (a-b)^2 }}{2}$$

## Area of the isosceles trapezoid

### Area of the trapezoid from bases (a)(b) and height (h)

$$S={\frac {a+b}{2}}\cdot h$$

### Area of the trapezoid from sides and angle(α)

$$S=\frac{(a+b)\cdot c\cdot \sin\alpha}{2}$$

### Area of the trapezoid from sides (a)(b)(c)

$$S=\frac{\sqrt{(a+b)^2\cdot(a-b+2c)\cdot(b-a+2c)}}{4}$$

### Area of the trapezoid z diagonal (d) and angle(γ)

$$S=\frac{d^2}{2}\cdot\sin\gamma$$

## Perimeter of the isosceles trapezoid

### Perimeter of the trapezoid from sides

$$L = a + b + 2c$$

### Perimeter of the trapezoid from bases, height and angle(α)

$$L = a+b+\frac{2h}{\sin(\alpha)}$$

## Legs and bases isosceles trapezoid

### Base (a) trapezoid from sides (b)(c) and height (h)

$$a=b+2\sqrt{c^2-h^2}$$

### Base (b) trapezoid from sides (a)(c) and height (h)

$$b=a-2\sqrt{c^2-h^2}$$

### Leg (c) trapezoid from sides (a)(b) and height (h)

$$c=\sqrt{(\frac{a-b}{2})^2 +h^2}$$

## Midline (median , midsegment) of the isosceles trapezoid

### Midline (median , midsegment) of the isosceles trapezoid

$$m=\frac {a+b}{2}$$

Isosceles trapezoid - Information

Trapezoid – a quadrilateral having at least one pair of parallel sides; The (selected) pair of parallel sides are called the bases, the remaining sides are called the legs, the distance between the bases is called the height of the trapezoid. Some colloquial definitions define a trapezoid as a quadrilateral with only one pair of parallel sides and, accordingly, the parallelogram is not a trapezoid. The sum of the angles on the same legs is 180°.

Isosceles trapezoid is a trapezoid with sides of equal length. If such a trapezoid is not a non-rectangular parallelogram, then it has an axis of symmetry: their common symmetry passing through the centers of the bases. In this case, the angles between the legs and a given base are equal, and the opposite angles add up to 180 °; hence it can then be inscribed in a circle.

Isosceles trapezoid has the following properties:
1. Trapez jest figurą wypukłą.
2. Sum of the measures of all interior angles is 2Π (360 °), and the sum of the measures of two adjacent internal angles lying on the same leg is Π, $$\alpha + \beta = 180°$$
3. Formula for the angle α $$\alpha = \arccos( \frac{(\frac{a-b}{2})^2 + c^2-h^2 }{2c\cdot\frac{a-b}{2}} )$$
4. Formula for the diagonal of a isosceles trapezoid from sides (a) and (c) and angle α
5. $$d=\sqrt{a^2+c^2-2ac\cdot \cos(\beta)}$$
6. Formula for the diagonal of a isosceles trapezoid from sides (b) i (c) and angle β
7. $$d=\sqrt{b^2+c^2-2bc\cdot \cos(\beta)}$$
8. Formula for the diagonal of a isosceles trapezoid from sides (a)(b)(c)
9. $$d= \sqrt{a\cdot b+c^2}$$
10. Formula for the diagonal of a isosceles trapezoid from area and angle γ
11. $$d= \frac{2S}{\sin \gamma}$$
12. Formula for the radius of the circle described on the isosceles trapezoid from diagonal (d) and angle α or β
13. $$R=\frac{d}{2\sin \alpha}$$ $$R=\frac{d}{2\sin \beta}$$
14. Formula for the radius of a circle described on an isosceles trapezoid from leg (c), diagonal (d) and height (h)
15. $$R=\frac{c\cdot d}{2h}$$
16. Formula for the radius of a circle described on an isosceles trapezoid from sides (a)(b)(c)
17. $$R= c\cdot\sqrt{\frac{a\cdot b + c^2}{4c^2-(a-b)^2}}$$
18. Formula for the height of the isosceles trapezoid from sides (a)(b) and area
19. $$h=\frac{2S}{a+b}$$
20. Formula for the height of the isosceles trapezoid from leg (c) and angle
21. $$h=c \cdot \sin \alpha = c \cdot \sin \beta$$
22. Formula for the height of the isosceles trapezoid from sides
23. $$h= \frac{\sqrt{4c^2 - (a-b)^2 }}{2}$$
24. Formula for the area of the isosceles trapezoid from bases (a)(b) and height (h)
25. $$S={\frac {a+b}{2}}\cdot h$$
26. Formula for the area of the isosceles trapezoid from sides and angle(α)
27. $$S=\frac{(a+b)\cdot c\cdot \sin\alpha}{2}$$
28. Formula for the area of the isosceles trapezoid from sides (a)(b)(c)
29. $$S=\frac{\sqrt{(a+b)^2\cdot(a-b+2c)\cdot(b-a+2c)}}{4}$$
30. Formula for the area of the isosceles trapezoid z diagonal (d) and angle(γ)
31. $$S=\frac{d^2}{2}\cdot\sin\gamma$$
32. Formula for the perimeter of the isosceles trapezoid from sides
33. $$L = a + b + 2c$$
34. Formula for the perimeter of the isosceles trapezoid from bases, height and angle(α)
35. $$L = a+b+\frac{2h}{\sin(\alpha)}$$
36. Formula for the side (a) isosceles trapezoid from sides (b)(c) and height (h)
37. $$a=b+2\sqrt{c^2-h^2}$$
38. Formula for the side (b) isosceles trapezoid from sides (a)(c) and height (h)
39. $$b=a-2\sqrt{c^2-h^2}$$
40. Formula for the leg (c) isosceles trapezoid from sides (a)(b) and height (h)
41. $$c=\sqrt{(\frac{a-b}{2})^2 +h^2}$$
42. Formula for the midline (median , midsegment) of the trapezoid $$m=\frac {a+c}{2}$$