How to find the area of a triangle

If you need to find the area of a triangle, do not worry that you have long forgotten all that teachers put in your head at school. Our article will tell you how to solve this issue, and in many ways.

To begin with, remember that the triangle is a figure that is formed by crossing three straight lines. The three points where the lines intersect are the vertices of the figure, and the segments opposite to them are the edges of the triangle. There are several special types of triangles (isosceles, rectangular, equilateral), the areas of which we will also look for.

How to calculate the area of a triangle according to the general formula

For the most general case, the area of a given geometric figure is calculated by the following formula: Area = ½ of the length of one side of the figure multiplied by the length of the height dropped to this side.

Find the area of a triangle if we know all three of its sides

In the event that you know all three sides of the triangle, then its area you can find using the formula of Geron. To begin with, we find a semi-perimeter of a triangle, by adding the lengths of all its three sides and dividing by two. Then we find the square of the area, according to the following formula: SS = p (p-a) (p-b) (p-c), where a, b, c are the lengths of the sides of the figure, and p is the half-perimeter. To find the area, simply extract the square root of the resulting value.

Find the area of the triangle, if we know its hypotenuse, the catheter and the angle formed by them

To do this, we use a trigonometric plate and the following formula:

S = 1/2 * a * b * sinB, where a and b are a cat with a hypotenuse, and B is the angle formed at their intersection.

According to this formula, we can find the area of an ordinary triangle, and equilateral, and isosceles, and rectangular.

Find the area of the triangle if we know the cathetus and the angle opposite to it

We apply the formula: S = 1/2 (a * a) / (2tgB), where a is a known cathetus, and B is the opposite angle.

We find the area of the triangle, if we know only the hypotenuse and the cathet

First we find the value FF = 1/2 (a * a - a * a). Then we extract the root (F) from this number and substitute it into the formula to find the area of the triangular figure: S = a * F. Here, a is the cathetus, and c is the hypotenuse.

Find the area of the triangle, if we know one of the acute angles and the hypotenuse

Known values of the problem value we substitute in the formula: S = 1/2 (в * в) * cosA * sinA *. Here the acute angle is A, and in the hypotenuse.

Find the area of a triangle with respect to the coordinates of the vertices

If you are given the coordinates of three points, which are the vertices of a triangular figure, by the condition of the problem , then you can also calculate the area.

So, you are given the vertices A (x1, y1), D (x2, y2), B (x3, y3). To find the area, we use the following formula: S = 1/2 ((x1-x3) (y2-y3) - (x2-x3) (y1-y3)). Remember that the module is taken from the value that you calculate in brackets, because some points can have coordinates with the minus sign.

Also you can act in a different way.

Method 1. We first find the lengths of all sides of the triangular figure, and then use the Heron formula, which was described above. First we find the squares of the sides according to the following formulas:

AB * AB = (x1-x2) (x1-x2) + (y1-y2) (y1-y2);

BV * BV = (x2-x3) (x2-x3) + (y2-y3) (y2-y3);

BA * BA = (x3-x1) (x3-x1) + (y3-y1) (y3-y1).

We find the half perimeter of a triangular figure:

P = 1 \ 2 (AB + BB + BA)

Now substitute the values in the formula:

SS = p (p-AB) (p-BB) (p-BA). This turned out square in the square. We extract the root from the value and find, finally, what we were looking for.

By the way, for the sake of curiosity, you can calculate the area by coordinates in the above two ways. Then you will know that the final values will differ slightly. This is because the result obtained during the first calculation will have a rounded value, rather than the result obtained with the help of the Heron formula. Thus, to obtain more accurate data, it is recommended to use the second method.