How to find the area of ​​a quadrangle?

If on a plane several lines are drawn sequentially so that each following begins at the place where the previous one ended, then we get a broken line. These segments are called links, and their intersection points are called vertices. When the end of the last segment intersects with the starting point of the first, you get a closed broken line dividing the plane into two parts. One of them is finite, and the second is infinite.

A simple closed line together with the part of the plane (the one that is finite) enclosed in it is called a polygon. The line segments are sides, and the angles formed by them are vertices. The number of sides of any polygon is equal to the number of its vertices. A figure that has three sides is called a triangle, and four are called a quadrangle. The polygon is numerically characterized by such a value as the area, which shows the size of the figure. How to find the area of ​​a quadrangle? This is taught by the branch of mathematics - geometry.

To find the area of ​​a quadrangle, you need to know what type it belongs to - convex or non-convex? The convex polygon all lies relatively straight (and it necessarily contains any of its sides) on one side. In addition, there are such types of quadrangles as a parallelogram with pairwise equal and parallel opposite sides (varieties of it: a rectangle with right angles, a rhombus with equal sides, a square with all right angles and four equal sides), a trapezoid with two parallel opposite sides and deltoid with two pairs of adjacent sides that are equal.

The areas of any polygon are found using the general method, which is to divide it into triangles, for each calculate the area of ​​an arbitrary triangle and add the results. Any convex quadrangle is divided into two triangles, non-convex - into two or three triangles, its area in this case can be made up of the sum and difference of the results. The area of ​​any triangle is calculated as half the product of the base (a) by the height (ħ) drawn to the base. The formula that is used in this case for the calculation is written as: S = ½ • a • ħ.

How to find the area of ​​a quadrangle, for example, a parallelogram? You need to know the length of the base (a), the length of the side (ƀ) and find the sine of the angle α formed by the base and side (sinα), the formula for the calculation will look: S = a • ƀ • sinα. Since the sine of the angle α is the product of the base of the parallelogram by its height (ħ = ƀ) - a line perpendicular to the base, then its area is calculated by multiplying its height by the height: S = a • ħ. This formula is also suitable for calculating the area of ​​a rhombus and a rectangle. Since the rectangle ƀ of the rectangle coincides with the height of ħ, its area is calculated by the formula S = a • ƀ. The square area, because a = ƀ, will be equal to the square of its side: S = a • a = a². The area of ​​the trapezoid is calculated as half the sum of its sides, multiplied by the height (it is drawn to the base of the trapezoid perpendicularly): S = ½ • (a + ƀ) • ħ.

How to find the area of ​​a quadrangle if the lengths of its sides are unknown, but its diagonals (e) and (f) are known, as well as the sine of the angle α? In this case, the area is calculated as half the product of its diagonals (lines that connect the vertices of the polygon), multiplied by the sine of the angle α. The formula can be written in the following form: S = ½ • (e • f) • sinα. In particular, the area of ​​the rhombus in this case will be equal to half the product of the diagonals (lines connecting the opposite corners of the rhombus): S = ½ • (e • f).

How to find the area of ​​a quadrangle that is not a parallelogram or a trapezoid, it is usually called an arbitrary quadrangle. The area of ​​such a figure is expressed through its half-perimeter (Ρ is the sum of two sides with a common vertex), the sides a, ƀ, c, d and the sum of two opposite angles (α + β): S = √ [(Ρ - a) • (Ρ - ƀ) • (Ρ - c) • (Ρ - d) - a • ƀ • c • d • cos² ½ (α + β)].

If the quadrangle is inscribed in a circle, and φ = 180 °, then the Brahmagupta formula (an Indian astronomer and mathematician who lived in 6-7 centuries AD) is used to calculate its area: S = √ [(Ρ - a) • (Ρ - ƀ) • (Ρ - c) • (Ρ - d)]. If the quadrangle is described by a circle, then (a + c = ƀ + d), and its area is calculated: S = √ [a • ƀ • c • d] • sin1 (α + β). If the quadrangle is simultaneously described by one circle and inscribed in another circle, then the following formula is used to calculate the area: S = √ [a • ƀ • c • d].


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