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
$$ d=\sqrt{a^2+c^2-2ac\cdot \cos(\beta)} $$
5. Formula for the diagonal of a isosceles trapezoid from sides (b) i (c) and angle β
$$ d=\sqrt{b^2+c^2-2bc\cdot \cos(\beta)} $$
6. Formula for the diagonal of a isosceles trapezoid from sides (a)(b)(c)
$$ d= \sqrt{a\cdot b+c^2} $$
7. Formula for the diagonal of a isosceles trapezoid from area and angle γ
$$ d= \frac{2S}{\sin \gamma} $$
8. Formula for the radius of the circle described on the isosceles trapezoid from diagonal (d) and angle α or β
$$ R=\frac{d}{2\sin \alpha} $$ $$ R=\frac{d}{2\sin \beta} $$
9. Formula for the radius of a circle described on an isosceles trapezoid from leg (c), diagonal (d) and height (h)
$$ R=\frac{c\cdot d}{2h} $$
10. Formula for the radius of a circle described on an isosceles trapezoid from sides (a)(b)(c)
$$ R= c\cdot\sqrt{\frac{a\cdot b + c^2}{4c^2-(a-b)^2}} $$
11. Formula for the height of the isosceles trapezoid from sides (a)(b) and area
$$ h=\frac{2S}{a+b} $$
12. Formula for the height of the isosceles trapezoid from leg (c) and angle
$$ h=c \cdot \sin \alpha = c \cdot \sin \beta $$
13. Formula for the height of the isosceles trapezoid from sides
$$ h= \frac{\sqrt{4c^2 - (a-b)^2 }}{2} $$
14. Formula for the area of the isosceles trapezoid from bases (a)(b) and height (h)
$$ S={\frac {a+b}{2}}\cdot h $$
15. Formula for the area of the isosceles trapezoid from sides and angle(α)
$$ S=\frac{(a+b)\cdot c\cdot \sin\alpha}{2} $$
16. Formula for the area of the isosceles trapezoid from sides (a)(b)(c)
$$ S=\frac{\sqrt{(a+b)^2\cdot(a-b+2c)\cdot(b-a+2c)}}{4} $$
17. Formula for the area of the isosceles trapezoid z diagonal (d) and angle(γ)
$$ S=\frac{d^2}{2}\cdot\sin\gamma $$
18. Formula for the perimeter of the isosceles trapezoid from sides
$$ L = a + b + 2c $$
19. Formula for the perimeter of the isosceles trapezoid from bases, height and angle(α)
$$ L = a+b+\frac{2h}{\sin(\alpha)} $$
20. Formula for the side (a) isosceles trapezoid from sides (b)(c) and height (h)
$$ a=b+2\sqrt{c^2-h^2} $$
21. Formula for the side (b) isosceles trapezoid from sides (a)(c) and height (h)
$$ b=a-2\sqrt{c^2-h^2} $$
22. Formula for the leg (c) isosceles trapezoid from sides (a)(b) and height (h)
$$ c=\sqrt{(\frac{a-b}{2})^2 +h^2} $$
23. Formula for the midline (median , midsegment) of the trapezoid $$ m=\frac {a+c}{2} $$