How to find the height of a trapezoid?

In our life very often we have to deal with the application of geometry in practice, for example, in construction. Among the most common geometric figures there is a trapeze. And in order for the project to be successful and beautiful, you need a correct and accurate calculation of the elements for such a figure.

What is a trapezoid? It is a convex quadrilateral that has a pair of parallel sides, called the bases of the trapezium. But there are two other sides connecting these bases. They are called lateral. One of the questions concerning this figure is: "How to find the height of the trapezoid?" It is immediately necessary to pay attention that the height is a segment that determines the distance from one base to the other. There are several ways to determine this distance, depending on known quantities.

1. The values of both bases are known, we denote them by b and k, as well as the area of this trapezium. Using known quantities, finding the height of the trapezium in this case is very easy. As is known from geometry, the area of the trapezoid is calculated as the product of half the sum of the bases and the height. From this formula one can easily derive the desired quantity. To do this, you need to divide the area by half of the sum of the bases. In the form of formulas this will look like this:

S = ((b + k) / 2) * h, hence h = S / ((b + k) / 2) = 2 * S / (b + k)

2. The length of the middle line is known, denoted by d, and the area. For those who do not know, the middle line is the distance between the middle of the sides. How to find the height of the trapezium in this case? According to the trapezoidal property, the middle line corresponds to half the sum of the bases, that is, d = (b + k) / 2. Again, we resort to the area formula. Replacing half the sum of the bases by the value of the midline, we get the following:

S = d * h

As you can see from the obtained formula it is very easy to deduce the height. Dividing the area by the value of the midline, we find the desired value. We write this by the formula:

H = S / d

3. The length of one side (b) and the angle formed between this side and the largest base are known. The answer to the question of how to find the height of the trapeze is also in this case. Consider the trapezoid ABCD, where AB and CD are sides, with AB = b. The greatest reason is AD. The angle formed by AB and AD is denoted by α. From point B we lower the height h to the base AD. Now consider the resulting triangle ABF, which is rectangular. The side AB is the hypotenuse, and the BF-leg. From the property of a right triangle, the ratio of the value of the leg and the value of the hypotenuse corresponds to the sine of the angle opposite the leg (BF). Therefore, proceeding from the foregoing, to calculate the height of the trapezium, we multiply the value of the known side and the sine of the angle α. In the form of a formula, it looks like this:

H = b * sin (α)

4. Similarly, the case is considered if the side-side size and the angle are known, denote it by β, formed between this side and the smaller base. When solving such a problem, the angle between the known side and the height is 90 ° - β. From the property of triangles - the ratio of the length of the leg and the hypotenuse corresponds to the cosine of the angle located between them. From this formula it is easy to deduce the height:

H = b * cos (β-90 °)

5. How to find the height of a trapezoid if only the radius of the inscribed circle is known? From the definition of a circle, it touches one point of each base. In addition, these points are in line with the center of the circle. From this it follows that the distance between them is the diameter and, at the same time, the height of the trapezium. Looks like that:

H = 2 * r

6. There are often problems in which it is necessary to find the height of an isosceles trapezium. Recall that the trapezoid, which has equal sides, is called isosceles. How to find the height of an isosceles trapezoid? At perpendicular diagonals, the height is equal to half the sum of the bases.

But, what if the diagonals are not perpendicular? Consider the isosceles trapezoid ABCD. According to its properties, the bases are parallel. From this it follows that the angles at the bases will also be equal. We draw two heights BF and CM. Proceeding from the above, we can say that the triangles ABF and DCM are equal, that is, AF = DM = (AD-BC) / 2 = (bk) / 2. Now, starting from the condition of the problem, we determine the known quantities, and only then find Height, taking into account all the properties of an isosceles trapezoid.