Before you find the area of the trapezoid, you must give its definition.
A trapezoid is a geometric figure with four corners, in which two sides are parallel to each other, and the other two are not. Two sides that are parallel to each other are called bases, and non-parallel sides. If the sides that are lateral are equal, the trapezoid will be called isosceles. If at the intersection they form a right angle, it is rectangular.
In algebra there is also the concept of a curvilinear trapezoid - by it we mean a figure bounded on one side by the x axis, and on the other, by a graph of the function y = f (x) b and defined on the interval [a; b]
How to find the area of the trapezoid
Such a geometric figure is calculated by the formula S = 0.5 * (a + b) * h, where a and b are the lengths of the trapezoid bases, and h is its height.
Example. A trapezoid is given, one base of which is 2 cm, the second is 3 cm, and the height is 4 cm. We calculate the area by the formula, we get the result: S = 0, 5 * (2 + 3) * 4 = 12 cm2.
From the same formula it follows that, knowing the area of this figure, its height, the length of one of the sides, you can find the length of the other. The second option - knowing the lengths of the sides and the area of the trapezoid, you can find its height.
Example. A trapezium is given, in which one base is 3 times longer than the other. The height of the figure is 3 cm, the area is 24 cm2. It is required to find the length of both bases.
Decision. The area is calculated by the following formula S = 0.5 * (a + c) * h. From the conditions of the problem it is clear that one side is 3 times larger than the other, therefore, a = 3c. Replace a in the formula and get S = 0.5 * (3c + c) * h = 0.5 * 4c * h. As a result, we get S = 2v * h, that is, = S / 2h. We substitute the digital values and get in = 6 cm, and = 18 cm.
However, this is not the only way to determine the area of this figure. According to the second method, before you find the area of the trapezoid, you can divide it into simple geometric shapes: a rectangle and two triangles (or one triangle, if we are talking about a rectangular trapezoid). In this case, the total area will be calculated as the sum of the areas of these figures. Alternatively, you can enter it in a rectangle, the side of which will be equal to the length of the larger of the bases. In this case, the area of the trapezoid is defined as the difference between the areas of the rectangle and the triangles.
How to find the area of a rectangular trapezoid? Earlier it was said that a rectangular trapezoid can be called a trapezoid, in which the base (let's call it a) and the side with intersect, forming a prima angle. Accordingly, in the indicated figure, the abso side c will be the height. Then, knowing the length of all 3 sides, you can find the area of the figure S = 0.5 * (a + c) * s.
The simplest formula is as follows: S = k * h, where k is the length of the midline of the trapezoid, h is its height. The problem is that in practice, measuring the length of the bases is easier than finding the middle line. And it is as follows:
Given: non-equilateral, non-rectangular trapezoid of the ATSD, in which the sides of the AB and SD are the bases. Before finding the area of the trapezoid, it is necessary to divide the segments AC and VD into 2 equal parts, designating the points of intersection with the letters G and K. Then the straight line drawn parallel to the bases will be the middle line of the trapezoid m.
Another special case is when the trapezoid is equilateral. For it, all the indicated formulas are suitable (of course, except for formulas for a rectangular one). Its area can be determined by knowing the angle between the bases. The formula is as follows: S = (a + b) * c * sin (x) * 0.5, where a and b are the length of the bases, c is the length of the side, and x is the angle between them.
Sometimes it becomes necessary to determine the area of a given figure not only in geometry, but also in algebra according to the coordinate system. In this regard, students have a question how to find the area of a trapezoid by coordinates. The principle of calculation is the same - the lengths of the sides are determined as the difference in the coordinates of the points of the bases, the height is calculated, and the area is calculated using the first formula. The height will be considered a straight line drawn from the corner of one of the bases to another base.
To determine the area of a curved trapezoid, use the integral.