Finding the area of the trapezoid is one of the main actions that allows you to solve many problems of geometry. Also, in KIM in mathematics of the Unified State Examination and the Unified State Examination there are many problems for the solution of which it is necessary to know how to look for the area of this geometric figure. This article will consider all the trapezoid area formulas.
What is this figure?
Before considering all the trapezoid area formulas, you need to know what it is, because without a clear definition it is impossible to correctly use the formulas and properties of this figure. A trapezoid is a quadrangle, the two sides of which are opposite each other, and if you continue them to infinite lines, they will never intersect (these sides are the bases of the figure). The other two sides can have blunt and sharp corners and are called lateral (in this case, if its lateral sides are the same, and the corners at the base are pairwise equal to each other, then such a trapezoid is called isosceles). All area formulas of this quadrilateral are considered below.
All trapezoid area formulas
In geometry, there are many formulas for finding the area of figures, which is both a plus and a minus. How to find the area of the trapezoid?
- Through the diagonals and the vertical angle. To do this, multiply half the product of the diagonals by the angle between them.
- Trapezoid area through the base and height. Half the sum of the bases, multiply by the height of the trapezoid drawn to one of the bases.
- With the help of all parties. Divide the sum of the bases in half and multiply by the root. Under the root: the side in the square minus the fraction, in the numerator of which is the difference in the bases in the square plus the difference in the sides, each of which is squared and in the denominator is the difference in the bases, multiplied by two.
- Through height and median. Divide the sum of the bases of the trapezoid in half and multiply by the height drawn to the base of the figure.
- For an isosceles trapezoid also has its own formula for finding the area. To find the area of this figure, multiply the square of the radius by four and divide by the sine of the angle alpha.
Trapezium bisector properties
Like the bisector of an isosceles triangle, drawn to the base, a straight line, dividing the angle in half, this figure has its own properties that are useful in solving problems in geometry.
- Bisectors with sides that are not parallel to each other are perpendiculars (from this property it follows that they form a right triangle, the hypotenuse of which is the side of this figure).
- The point of their intersection on the side that is the base of this figure belongs to another base (it follows from this property that such straight lines of obtuse angles form an isosceles triangle at the base).
- The bisector cuts off a segment of the same length from the base as the side (from this property it follows that it forms an isosceles triangle with the base, the side and the trapezoid base will be the sides, and the bisector - the base of the isosceles triangle).
Conclusion
In this article, all the trapezoid area formulas were proposed. Most of them are not considered in the textbooks of geometry, but at the same time they are all necessary for the successful solution of problems.