Trapezium area

The word trapezium is used in geometry to denote a quadrangle characterized by certain properties. In addition, it has several more meanings. In architecture it is used to designate symmetrical doors, windows and buildings, built broad at the base and tapering to the top (in the Egyptian style). In sport - a gymnastic shell, in fashion - a dress, coat or other kind of clothing of a certain cut and style.

The very word "trapezium" came from the Greek, translated into Russian meaning "table" or "table, food." In Euclidean geometry, this is the so-called convex quadrilateral having one pair of opposite sides, which are necessarily parallel to each other. It should be remembered several definitions in order to find the area of the trapezium. The parallel sides of this polygon are called the bases, and the other two are called the lateral sides. The height of the trapezoid is the distance between the bases. The middle line is considered to be a line connecting the middle sides of the side. All these concepts (bases, height, middle line and sides) are elements of a polygon, which is a particular case of a quadrangle.

Therefore, it is legitimate to assert that the area of the trapezoid can be found from the formula for the quadrilateral: S = ½ • (a + ƀ) • . Here S is the area, a and ƀ are the lower and upper legs, ħ is the height dropped from the angle adjacent to the upper base, perpendicular to the lower base. That is, S equals half the product of the sum of the bases by the height. For example, if the trapezium's bases are 6 and 2 mm high and its height is 15 mm, then its area will be: S = ½ • (6 + 2) • 15 = 60 mm².

Using the known properties of this quadrilateral, one can calculate the area of the trapezoid. In one of the important statements it is said that the middle line (we denote it by the letter μ, and the bases by the letters a and ) equals half the sum of the bases, to which it is always parallel. That is, μ = ½ (a + ). Thus, substituting in the known formula for calculating S of a quadrilateral, the middle line, we can write the formula for calculation in another form: S = μ • . 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 known property of a polygon with two parallel sides that are a base, you can enter a circle with radius r into it, provided that the sum of the bases is necessarily equal to the sum of its lateral sides. If, in addition, the trapezium is isosceles (that is, its lateral sides are equal c = d), and also the angle at the base α is known, then it is possible to find what the trapezium area is equal to by the formula: S = 4r² / sinα, and for Particular 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 polygon will be equal to: S = 8 • 5² = 200 dm².

You can also find the area of the trapezoid by dividing it into shapes, calculating the area of each and adding these values. This is better to consider for three possible options:

1. The sides and angles at the base are equal. In this case, the trapezium is called isosceles.
2. If one side forms straight angles with bases, that is, perpendicular to them, then such a trapezoid will be called rectangular.
3. Quadrilateral, which has two sides parallel. In this case, the parallelogram can be considered as a special case.

For an isosceles trapezium, the area consists of the sum of two identical areas of rectangular triangles S1 = S2 (their height is equal to the height of the trapezium ħ, and the bases of the triangles are half the difference in the bases of the trapezium ½ [a - ƀ]) and the area of the rectangle S3 (one side of it is equal to the upper base ƀ, And the other to the height ħ). It follows from this that the area of the trapezium is S = S1 + S2 + S3 = 0 (a-ƀ) • ħ + 0 (a-ƀ) • ħ + (ƀ · ħ) = ½ (a-ƀ) • ħ + (ƀ • ħ). For a rectangular trapezoid, the area consists of the sum of the areas of the triangle and the quadrilateral: S = S1 + S3 = ½ (a-ƀ) • ħ + (ƀ • ħ).

The curvilinear trapezoid in this paper was not considered, the area of the trapezium in this case is calculated with the help of integrals.