word trapezoid geometry used to refer to the quadrangle, which is characterized by certain properties.Furthermore, it has several meanings.The architecture used to refer to symmetrical doors, windows and buildings built wide at the base and tapering to the top (in the Egyptian style).In sports - is exercise equipment, in fashion - the dress, coat or other specific type of clothing cut and style.
word "trapezoid" comes from the Greek, translated into Russian means "table" or "table food."In Euclidean geometry, the so-called convex quadrilateral having one pair of opposite sides which are necessarily parallel to each other.It should be recalled several definitions in order to find the area of a trapezoid.The parallel sides of the polygon are called bases, and the other two - side.The height of the trapezoid is the distance between the bases.Central line is considered to be a line connecting the midpoints of side.All these concepts (the base, the height, the middle line and the sides) are the elements of a polygon, which is a special case of the quadrangle.
therefore entitled to claim that the area of a trapezoid can be found on a formula intended for a quadrilateral: S = ½ • (a + ƀ) • ħ.Where S - is the area, a, and ƀ - it's lower and upper warping, ħ - the height, dropped out of the corner adjacent to the upper base, perpendicular to the lower base.That is S is equal to half the product of the amount of base and the height.For example, if the base trapezium - 6 and 2 mm, and its height - 15 mm, its area will be equal to: S = ½ • (2 + 6) = 60 • 15 mm².
Using the known properties of the quadrilateral, you can calculate the area of a trapezoid.In one of the most important statements said that the middle line (denoted by the letter μ, and the base of the letters a and ƀ) equal to half the sum of the bases, which she always parallel.That is, μ = ½ (a + ƀ).Thus, substituting the known calculation formula S quadrangle, the middle line, we can write the formula for the calculation in a different form: S = μ • ħ.In the case where the middle line - 25 cm, height - 15 cm, the area of a trapezoid is equal to: S = 25 • 15 = 375 cm².
According to the well-known property of the polygon with two parallel sides, is the basis, to inscribe a circle of radius r it can be provided that the sum of the bases will necessarily equal the sum of its sides.If, moreover, the trapezoid is an isosceles (i.e., equal to each other side thereof: c = d), and the known angle at the base α, it is possible to find what is the area of the trapezoid using the formula: S = 4r² / sinα, and forspecial case when α = 30 °, S = 8r².For example, if the angle at one of the bases is 30 °, and the inscribed circle with a radius of 5 dm, then the area of the polygon will be equal to: S = 8 • 5² = 200 dm².
You can also find the area of a trapezoid, breaking it into pieces, calculate the area of each and adding these values.It is best to consider three options:
- sides and angles at the base are equal.In this case, an isosceles trapezoid are called.
- If one side forms right angles with the base, ie, perpendicular to it, then this will be called a rectangular trapezoid.
- Quadrilateral, which are parallel to the two sides.In this case, the parallelogram can be considered as a special case.
For an isosceles trapezoid area is the sum of two equal areas of right-angled triangles S1 = S2 (their height equal to the height of the trapezoid ħ, and the base of the triangle half the difference between the base of the trapezoid ½ [a - ƀ]) and rectangle area S3 (one side of it is topbase ƀ, and the other - the height of ħ).From which it follows that the area of a trapezoid S = S1 + S2 + S3 = ¼ (a - ƀ) • ħ + ¼ (a - ƀ) • ħ + (ƀ • ħ) = ½ (a - ƀ) • ħ + (ƀ• ħ).For a rectangular area of a trapezoid is the sum of the areas of the triangle and the quadrangle: S = S1 + S3 = ½ (a - ƀ) • ħ + (ƀ • ħ).
curvilinear trapezoid in the scope of this article, the area of a trapezoid, in this case is calculated using integrals.
