Triangle is one of the most commongeometric figures, with which we are already acquainted in primary school. With the question of how to find the area of a triangle, every schoolboy encounters in geometry lessons. So, what are the features of finding the area of this figure can be distinguished? In this article, we will review the basic formulas needed to perform this task, and also we will look at the types of triangles.
You can find the area of a triangle in completely different ways, because in geometry there is not one kind of figure that contains three angles. These types include:
Let us consider in more detail each of the existing types of triangles.
Such a geometric figure is considered the mostcommon for solving geometric problems. When there is a need to draw an arbitrary triangle, this option comes to the rescue.
In the acute-angled triangle, as the name implies, all angles are sharp and in total are 180 °.
Such a triangle is also very common,However, it occurs somewhat less often than acute. For example, when solving triangles (i.e., several of its sides and angles are known and it is necessary to find the remaining elements), sometimes it is required to determine whether the angle is blunt or not. The cosine of the obtuse angle is a negative number.
In an obtuse triangle, the magnitude of one of the angles exceeds 90 °, so the remaining two angles can take small values (for example, 15 ° or altogether 3 °).
To find the area of a triangle of this type, you need to know some nuances, which we'll talk about further.
A regular polygon is a figure,Including n angles, in which all sides and angles are equal. So is the right triangle. Since the sum of all the angles of the triangle is 180 °, each of the three angles is 60 °.
The right triangle, due to its property, is also called an equilateral figure.
It is also worth noting that only one circle can be inscribed in a regular triangle, and only one circle can be described near it, and their centers are located at the same point.
In addition to the equilateral type, one can also single outan isosceles triangle, slightly different from it. In such a triangle, the two sides and the two angles are equal to each other, and the third side (to which the equal angles adjoin) is the base.
The figure shows an isosceles triangle DEF, whose angles D and F are equal, and DF is the base.
A rectangular triangle is named so because one of its corners is a straight line, that is 90 °. The other two angles in total are 90 °.
The largest side of such a triangle, which lies opposite the angle of 90 °, is the hypotenuse, while the other two sides are the legs. For a given type of triangle, the Pythagorean theorem is applicable:
The sum of the squares of the lengths of the legs is equal to the square of the length of the hypotenuse.
The figure shows the rectangular triangle BAC with the hypotenuse AC and the legs AB and BC.
To find the area of a triangle with a right angle, you need to know the numerical values of its legs.
Let us pass to the formulas for finding the area of a given figure.
In geometry, two formulas can be distinguished whichare suitable for finding the area of most types of triangles, namely for an acute-angled, obtuse, regular and isosceles triangles. We will analyze each of them.
This formula is universal forfinding the area, the figure we are considering. To do this, it is sufficient to know the length of the side and the length of the height to it. The formula itself (half the product of the base to the height) looks like this:
S = ½ * A * H,
where A is the side of this triangle, and H is the height of the triangle.
For example, to find the area of the acute-angled triangle ACB, you must multiply its side AB by the height of the CD and divide the resulting value by two.
However, it is not always easy to find an areatriangle in this way. For example, in order to use this formula for an obtuse triangle, it is necessary to continue one of its sides and only after that to hold the height to it.
In practice, this formula is used more often than others.
This formula, like the previous one, is suitable formost triangles and in its meaning is a consequence of the formula for finding the area along the side and height of the triangle. That is, the formula under consideration can be easily derived from the previous one. Its formulation looks like this:
S = ½ * sinO * A * B,
where A and B are the sides of the triangle, and O is the angle between the sides A and B.
Recall that the sine of the angle can be seen in a special table named after the outstanding Soviet mathematician V. M. Bradis.
And now we turn to other formulas, suitable only for exceptional types of triangles.
In addition to the universal formula including the need to draw a height in a triangle, the area of a triangle containing a right angle can be found from its legs.
Thus, the area of a triangle containing a right angle is half the product of its legs, or:
S = ½ * a * b,
where a and b are the legs of a right triangle.
This kind of geometric figures differs in that,that its area can be found at the indicated value of only one side of it (since all sides of the regular triangle are equal). So, having met the task of "finding the area of a triangle, when the sides are equal", we need to use the following formula:
S = A2* √3 / 4,
where A is the side of an equilateral triangle.
The last option for finding the area of the triangle is Heron's formula. In order to use it, you need to know the lengths of the three sides of the figure. Heron's formula looks like this:
S = √p · (p - a) · (p - b) · (p - c),
where a, b and c are the sides of this triangle.
Sometimes the problem is given:"The area of the right triangle is to find the length of its side". In this case, we need to use the formula for finding the area of a regular triangle already known to us and derive from it the value of the side (or its square):
A2 = 4S / √3.
In the problems of GIA in mathematics there are many formulas. In addition, it is often necessary to find the area of a triangle on checkered paper.
In this case, it is most convenient to draw a height to one side of the figure, determine the length of the figure by the cells and use the universal formula for finding the area:
S = ½ * A * H.
So, after studying the formulas presented in the article, you will not have any problems with finding the area of a triangle of any kind.
