Sometimes the question of how to find the area of an isosceles triangle arises not only for schoolchildren or students, but also in real, practical life. For example, during construction, it becomes necessary to finish the façade under the roof. How to calculate the amount of material you need?
Often these tasks are faced by masters who work with fabric or leather. Indeed, many of the details that will have to be cut to the master have just the shape of an isosceles triangle.
So, there are several ways to help find the area of an isosceles triangle. The first is calculating it by its base and height.
For the solution, we need to construct for clarity the triangle MNP with the base MN and height PO. Now we’ll finish something in the drawing: from point P draw a line parallel to the base, and from point M draw a line parallel to height. We call the intersection point Q. To find out how to find the area of an isosceles triangle, we need to consider the resulting quadrilateral MOPQ, in which the side of the triangle MP given to us is already its diagonal.
We prove first that it is a rectangle. Since we built it ourselves, we know that the sides of MO and OQ are parallel. And the sides of QM and OP are also parallel. The angle POM is straight, so the angle OPQ is also straight. Therefore, the resulting quadrilateral is a rectangle. Find its area is not difficult, it is equal to the product of PO by OM. OM is the half base of this MPN triangle. It follows that the area of the rectangle we constructed is equal to the half-product of the height of the rectangular triangle at its base.
The second stage of the task set before us, how to determine the area of a triangle, is the proof of the fact that the rectangle we obtained in terms of area corresponds to this isosceles triangle, that is, the area of the triangle is also equal to the semi-product of the base and height.
To begin with, let's compare the triangle PON and PMQ. They are both rectangular, since the right angle in one of them is formed by the height, and the right angle in the other is the angle of the rectangle. The hypotenuses in them are sides of an isosceles triangle, therefore, are also equal. The legs of PO and QM are also equal as the parallel sides of the rectangle. This means that the area of the triangle PON and the triangle PMQ are equal.
The area of the QPOM rectangle is equal to the areas of the PQM and MOP triangles in total. Replacing the superstructure triangle QPM with the triangle PON, we obtain in total the triangle given to us to derive the theorem. Now we know how to find the area of an isosceles triangle by the base and height - calculate their semi-product.
But you can learn how to find the area of an isosceles triangle on the base and side. There are also two options: the theorem of Heron and Pythagoras. Consider a solution using the Pythagorean theorem. For example, take the same isosceles triangle PMN with height PO.
In a right triangle, POM MP is the hypotenuse. Its square is the sum of the squares of PO and OM. And since OM is half the base that we know, we can easily find OM and square the number. Having subtracted the resulting number from the hypotenuse square, we find out what is the square of the other leg, which in the isosceles triangle is the height. Having found the square root of the difference and knowing the height of the right triangle, we can give an answer to the task before us.
You just need to multiply the height by the base and divide the result in half. Why this should be done, we explained in the first version of the proof.
It happens that you need to make calculations on the side and corner. Then we find the height and the base, using the formula with sines and cosines, and, again, multiply them and divide the result in half.