The word trapezoid is used in geometry to mean a quadrangle characterized by certain properties. In addition, it has a few more meanings. In architecture, it is 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, this is a gymnastic apparatus; in fashion, a dress, coat, or other type of clothing of a certain cut and style.
The word "trapezoid" itself came from Greek, translated into Russian meaning "table" or "table, food." In Euclidean geometry, a convex quadrangle is called so, having one pair of opposite sides that are necessarily parallel to each other. Several definitions should be recalled in order to find the area of the trapezoid. The parallel sides of this polygon are called bases, and the other two are called sides. The height of the trapezoid is the distance between the bases. The middle line is considered to be the line connecting the midpoints of the sides of the side. All these concepts (bases, height, middle line and sides) are elements of a polygon, which is a special case of a quadrangle.
Therefore, the statement is valid that the area of the trapezoid can be found by the formula intended for the quadrangle: S = ½ • (a + ƀ) • ħ. Here S is the area, a and ƀ are the lower and upper warps, ħ is the height omitted from the angle adjacent to the upper base, perpendicular to the lower base. That is, S is equal to half the product of the sum of the bases by the height. For example, if the base of the trapezoid is 6 and 2 mm and its height is 15 mm, then its area will be: S = ½ • (6 + 2) • 15 = 60 mm².
Using the known properties of this quadrangle, the area of the trapezoid can be calculated. One important statement says that the middle line (denoted by the letter µ, and the bases by the letters a and ƀ) is equal to half the sum of the bases to which it is always parallel. That is, µ = ½ (a + ƀ). Thus, substituting the middle line in the well-known calculation formula S of the quadrangle, we can write the formula for the calculation in another form: S = µ • ħ. For the case when the middle line is 25 cm and the height is 15 cm, the area of the trapezoid is: S = 25 • 15 = 375 cm².
According to the well-known property of a polygon with two parallel sides that are the base, you can enter a circle with a radius r into it, provided that the sum of the bases will necessarily equal the sum of its lateral sides. If, in addition, the trapezoid is isosceles (that is, its sides are equal to each other: c = d), and the angle at the base α is known, then you can find what the area of the trapezoid is by the formula: S = 4r² / sinα, and for special case when α = 30 °, S = 8r². For example, if the angle at one of the bases is 30 °, and a circle with a radius of 5 dm is inscribed, then the area of such a polygon will be: S = 8 • 5² = 200 dm².
You can also find the area of the trapezoid by breaking it into figures, calculating the area of each and adding these values. This is best considered for three possible options:
- The sides and angles at the base are equal. In this case, the trapezoid is usually called isosceles.
- If one side forms right angles with the bases, that is, is perpendicular to them, then such a trapezoid will be called rectangular.
- A quadrangle in which two sides are parallel. In this case, a parallelogram can be considered as a special case.
For an isosceles trapezoid, the area is the sum of two identical areas of right-angled triangles S1 = S2 (their height is equal to the height of the trapezoid ħ, and the base of the triangles is half the difference between the bases of the trapezoid ½ [a - ƀ]) and the area of the rectangle S3 (one side of it is equal to the upper base ƀ, and the other - height ħ). It follows that the area of the trapezoid S = S1 + S2 + S3 = ¼ (a - ƀ) • ħ + ¼ (a - ƀ) • ħ + (ƀ • ħ) = ½ (a - ƀ) • ħ + (ƀ • ħ). For a rectangular trapezoid, the area is the sum of the areas of the triangle and the quadrangle: S = S1 + S3 = ½ (a - ƀ) • ħ + (ƀ • ħ).
The curvilinear trapezoid was not considered in this article; the trapezoid area in this case is calculated using the integrals.