How to find the area of a quadrilateral?

If you draw a series of segments in a plane in such a way that each subsequent one starts at the point where the previous one ends, you get a broken line. These segments are called links, and the places of their intersection are tops. When the end of the last segment intersects with the initial point of the first, we 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 enclosed in it (the one that is finite) is called a polygon. The segments are sides, and the angles formed by them are the 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 is a quadrangle. The polygon is numerically characterized by a size such as the area that shows the size of the figure. How to find the area of a quadrilateral? This is taught by the section of mathematics - geometry.

To find the area of a quadrilateral, you need to know to what type it relates - convex or non-convex? A convex polygon lies entirely with respect to a straight line (and it necessarily contains one of its sides) along one side. In addition, there are also such types of quadrilaterals as a parallelogram with pairwise equal and parallel opposite sides (its forms: 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 The deltaoid with two pairs of adjacent sides, which are equal.

The areas of any polygon are found using the general method, which is to break it into triangles, calculate the area of an arbitrary triangle for each and add the results. Any convex quadrangle is divided into two triangles, nonconvex - by two or three triangles, its area in this case can be composed 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 bottom. The formula, which is used in this case for calculation, is written as: S = ½ • a • .

How to find the area of a quadrilateral, 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 the 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), the line is perpendicular to the base, then its area is calculated by multiplying its base by the height: S = a • . To calculate the area of a diamond and a rectangle, this formula also fits. Since at the rectangle the side ƀ coincides with the height ħ, its area is calculated by the formula S = a • . The square of the square, 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 trapezium perpendicularly): S = ½ • (a + ƀ) • ħ.

How to find the area of a quadrilateral 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 (the 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 = 1 • (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 is usually called an arbitrary quadrilateral. The area of such a figure is expressed through its half -perimeter (P is the sum of two sides with a common vertex), sides a, ƀ, c, d and the sum of two opposite angles (α + β): S = √ [(P - a) • (P - Ƀ) • (P - c) • (P - d) - a • ƀ • c • d • cos ½ (α + β)].

If the quadrangle is inscribed in a circle, and φ = 180 °, then the Brahmagupta formula is used to calculate its area (Indian astronomer and mathematician who lived in 6-7 centuries AD): S = √ [(P - a) • (P - ƀ) • (P - c) • (P - d)]. If the quadrilateral is circumscribed, then (a + c = ƀ + d), and its area is calculated: S = √ [a · ƀ · c · d] · sin ½ (α + β). If the quadrilateral 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].