Trapezium is a quadrilateral with two parallel and two non-parallel sides. To calculate its perimeter, you need to know the dimensions of all sides of the trapezoid.
The data in the tasks can be different.
You will need
- - calculator-
- - tables of sines, cosines and tangents-
- - paper-
- - drawing accessories.
The simplest version of the problem - when given allThe sides of the trapezium. In this case, they just need to be folded. The following formula can be used: p = a + b + c + d, where p is the perimeter, and the letters a, b, c and d denote the sides opposite the corners, denoted by corresponding uppercase letters.
There is given an isosceles trapezoid, enoughFold two of its bases and add to them the double size of the side. That is, the perimeter in this case is calculated by the formula: p = a + c + 2b, where b is the side of the trapezium, and a is the base.
Calculations will be somewhat longer ifOne of the sides must be calculated. For example, we know a long base, adjacent corners and height. You need to calculate the short base and side. To do this, draw the trapezoid ABCD, from the upper corner B draw the height BE. You will get the triangle ABE. You know the angle A, respectively, you know its sine. In the task data, the height BE is also indicated, which is simultaneously the leg of the right triangle opposite the corner known to you. To find the hypotenuse AB which is simultaneously the side of the trapezoid, it is enough to divide BE into sinA. Similarly, find the length of the second side. For this, it is necessary to draw a height from the other upper corner, that is, CF.
Now you know the greater ground and the sides. To calculate the perimeter of this small, we need another size of a smaller base. Accordingly, in the two triangles formed inside the trapezium, it is necessary to find the dimensions of the segments AE and DF. This can be done, for example, through the cosines of the angles A and D known to you. Cosine is the ratio of the adjacent leg to the hypotenuse. To find the cathet, you need to multiply the hypotenuse by the cosine. Next, calculate the perimeter by the same formula as in the first step, that is, having folded all sides.
Another option: Given two grounds, the height and one side, you need to find the second side. This is also better done using trigonometric functions. To do this, draw a trapezoid. Let's say you know the bases of AD and BC, as well as the side of AB and the height of BF. From these data, you can find the angle A (through the sine, that is, the ratio of the height to the known side), the segment AF (through the cosine or tangent, since the angle is already known to you.) Recall also the properties of the trapezium angles - the sum of the corners adjacent to one side is 180 °.
Draw the height of CF. You've got another rectangular triangle in which you need to find the hypotenuse CD and the DF cathet. Start with a leg. Subtract the length of the upper base from the length of the bottom base, and from the result obtained, the length of the already known segment AF. Now in the right triangle CFD you know the two legs, that is, you can find the tangent of the angle D, and on it - and the angle itself. After this, it will remain through the sine of the same angle to calculate the side of the CD, as already described above.