Trapezoid

A trapezoid is a Quadrilatère, Polygone at four sides, having at least two parallel opposite sides. These two parallel sides are called small base and great base .

Example of trapezoid

With this definition, the quadrilaterals ABCD and ABDC of the figure are both of the trapezoids (of which the sides AB and CD are parallel).

Certain authors impose like additional condition the convexity of the quadrilateral, which results in excluding the “trapezoids crossed” such as ABDC .

Properties

A convex quadrilateral is a trapezoid if and only if it has an even consecutive angles of sum equal to 180 degrees or π radians. The sum of the two other angles is then the same one. For example: Here the two pairs of angles have as a top (has, D) and (B, C).

Caution: In a trapezoid the sum of two consecutive angles is not always equal to 180 degrees. (example: the angles of tops has and B)

Particular cases

• a trapezoid is described as right-angled as soon as it has two sides of the same longor and a right angle.

A trapezoid is described as isosceles when it checks one of the following equivalent properties:

• the two adjacent angles at the same base are equal.
• Two opposite sides are of the same length.
• the two bases of the trapezoid have the same mediator, and this one is an axis of symmetry of the trapezoid.

A convex trapezoid whose bases have even length is a parallelogram

Surface of the trapezoid

The surface of the convex trapezoid is worth the product its height by the half the sum of its bases.

I.e., H the height is , has the first base, and B the second.

$\backslash \; frac\; \{H\; (B\; +\; a)\}\; \{2\}$

This can be shown easily by noticing that the trapezoid is a rectangle to which one joins two triangles.

Another formula gives the surface of the trapezoid when only the four lengths are known has , B , C , D on the four sides:

$\backslash \; frac\; \{a+c\}\; \{4\; (ac)\}\backslash \; sqrt\; \{(a+b-c+d)\; (a-b-c+d)\; (a+b-c-d)\; (-\; a+b+c+d)\}$

Attention, for this formula has and C represent the two with dimensions parallels of the trapezoid and has longest of both.

Method of the trapezoids

principal Article: Method of the trapezoids

The method of integration approximate, known as of the trapezoids , described by Isaac Newton and his pupil Roger Dimensions, consists in replacing the successive arcs of curve M _I M _{I +1} by the segments: it is a linear Interpolation.

The method of the trapezoids is more precise than the method elementary, known as of the Rectangle S, corresponding to the sums of Riemann, consisting in replacing the function given by a function in staircase.

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