If the plane has consistently draw some segments so that one should start at the point where the previous one ended, we get a broken line.These segments are called links, and places of their intersection - tops.When the end of the last segment intersects 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 infinite.
simple closed curve with the enclosed part of the plane (that which is finite) is called a polygon.The segments are parties, and the angles formed by them - tops.The number of sides of any polygon is the number of vertices.A figure which has three sides, called triangle, and four - quadrangle.Polygon is characterized by a numerical value, as the area that shows the size of the figure.How to find the area of the quadrilateral?This section teaches mathematics - geometry.
To find the area of the quadrilateral, you need to know what type it is - convex or nonconvex?A convex polygon is all relative to the line (and it must contain any of the parties) on the same side.In addition, there are some kinds of quadrangles as a parallelogram with mutually equal and parallel to the opposite side (the variety of its: a rectangle with right angles, lozenge with equal sides, the square with all the right angles and four equal sides), a trapezoid with two parallel opposite sides anddeltoid with two pairs of adjacent sides that are equal.
area of any polygon are using a common method, which is to divide it into triangles, each to calculate the area of a triangle and fold arbitrary results.Any convex quadrilateral is divided into two triangles, nonconvex - two or three of the triangle area, in this case it may be composed of the sum and difference results.The area of any triangle is calculated as half of the base product of (a) to the height (ħ), carried out by the base.The formula which 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?It is necessary to know the length of the base (a), a side length (ƀ) and find the sine of the angle α, formed by the base and the side (sinα), the formula for the calculation will appear: S = a • ƀ • sinα.Since the sine of the angle α is the product of the base of the parallelogram on the height (ħ = ƀ) - a line perpendicular to the base, its area is calculated by multiplying the height of its base: S = a • ħ.To calculate the area of a rhombus and a rectangle also fits this formula.Since the rectangle side ƀ coincides with the height of ħ, its area is calculated according to the formula S = a • ƀ.The area of the square, because a = ƀ, will be equal to the square of its side: S = a • a = a².The area of a trapezoid is calculated as half the sum of its sides times the height (it is held perpendicular to the base of the trapezoid): S = ½ • (a + ƀ) • ħ.
How to find the area of the quadrangle, if the length of its sides is unknown, but known for its diagonal (e) and (f), and 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 this form: S = ½ • (e • f) • sinα.In particular rhombus area in this case will be equal to half the product of the diagonals (the lines connecting opposite corners of a rhombus): S = ½ • (e • f).
How to find the area of the quadrangle, which is not a parallelogram or trapezoid, it is commonly referred to as an arbitrary rectangle.The area of the figure is expressed through his semiperimeter (Ρ - the sum of the two sides with a common vertex), the part of 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 a quadrilateral inscribed in a circle, and φ = 180 °, in order to calculate its area used formula Brahmagupta (Indian astronomer and mathematician who lived in 6-7 centuries AD): S = √ [(Ρ - a) • (Ρ -ƀ) • (Ρ - c) • (Ρ - d)].If a quadrilateral circumscribed circle, then (a + c = ƀ + d), and its area is calculated: S = √ [a • ƀ • c • d] • sin ½ (α + β).If the quadrilateral is both described a circle and an inscribed circle to another, then calculate the area using the following formula: S = √ [a • ƀ • c • d].
