A triangle is one of the most common geometric shapes that we get to know in elementary school. The question of how to find the area of a triangle is encountered by every student in geometry classes. So, what features of finding the area of a given figure can be distinguished? In this article, we will consider the basic formulas necessary to complete this task, and also analyze the types of triangles.
Types of Triangles
You can find the area of a triangle in completely different ways, because more than one type of figure with three angles is distinguished in geometry. These types include:
- An acute-angled triangle.
- Obtuse.
- Equilateral (correct).
- Right triangle.
- Isosceles.
Let us consider in more detail each of the existing types of triangles.
Acute triangle
Such a geometric figure is considered the most common when solving geometric problems. When it becomes necessary to draw an arbitrary triangle, this option comes to the rescue.
In an acute-angled triangle, as the name implies, all angles are sharp and add up to 180 °.
Obtuse triangle
Such a triangle is also very common, but it is somewhat less common than acute-angled. For example, when solving triangles (that is, several of its sides and angles are known and the remaining elements need to be found), sometimes it is necessary to determine whether the angle is obtuse or not. The oblique angle cosine is a negative number.
In an obtuse triangle, the value of one of the angles exceeds 90 °, so the remaining two angles can take small values (for example, 15 ° or even 3 °).
To find the area of a triangle of this type, you need to know some of the nuances, which we will talk about later.
Regular and isosceles triangles
A regular polygon is a figure that includes n angles, in which all sides and angles are equal. That is the regular triangle. Since the sum of all the angles of the triangle is 180 °, then each of the three angles is 60 °.
A regular 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 around it, and their centers are located at one point.
In addition to the equilateral type, you can also select an isosceles triangle, slightly different from it. In such a triangle, the two sides and two angles are equal to each other, and the third side (to which equal angles are adjacent) is the base.
The figure shows an isosceles triangle DEF, whose angles D and F are equal, and DF is the base.
Right triangle
A right-angled triangle is so named because one of its angles is straight, that is, equal to 90 °. The other two angles add up to 90 °.
The largest side of such a triangle, lying against an angle of 90 °, is the hypotenuse, while the other two sides are legs. For this type of triangles, the Pythagorean theorem is applicable:
The sum of the squared lengths of the legs is equal to the squared length of the hypotenuse.
The figure shows a right triangle BAC with hypotenuse AC and 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.
We turn to the formulas for finding the area of this figure.
Basic formulas for finding the area
In geometry, two formulas can be distinguished that are suitable for finding the area of most types of triangles, namely for acute-angled, obtuse, regular and isosceles triangles. We will analyze each of them.
Side and height
This formula is universal for finding the area considered by us figures. To do this, it is enough to know the length of the side and the length of the height drawn to it. The formula itself (half the height of the base) is as follows:
S = ½ * A * H,
where A is the side of the given triangle, and H is the height of the triangle.
For example, to find the area of an acute-angled triangle ACB, you need to multiply its side AB by the height CD and divide the resulting value by two.
However, it is not always easy to find the area of a triangle in this way. For example, to use this formula for an obtuse triangle, you need to continue one of its sides and only after that draw a height to it.
In practice, this formula is used more often than others.
On two sides and a corner
This formula, like the previous one, is suitable for most 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 considered formula can be easily derived from the previous one. Its wording 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 of A and B.
Recall that the sine of the angle can be found in a special table named after the outstanding Soviet mathematician V. M. Bradis.
Now let's move on to other formulas suitable only for exceptional types of triangles.
Area of a right triangle
In addition to the universal formula, which includes the need to draw height in a triangle, the area of a triangle containing a right angle can be found by its legs.
So, 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.
Right triangle
This type of geometric shapes differs in that its area can be found with the indicated value of only one of its sides (since all sides of a regular triangle are equal). So, having met the task of “finding the area of a triangle when the sides are equal”, you need to use the following formula:
S = A 2 * √3 / 4,
where A is the side of an equilateral triangle.
Heron's formula
The last option for finding the area of a triangle is the Heron 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 a given triangle.
Sometimes in a task it is given: "the area of a regular triangle is to find the length of its side." In this case, we need to use the formula we already know for finding the area of a regular triangle and derive from it the value of the side (or its square):
A 2 = 4S / √3.
Exam Tasks
In the problems of the GIA in mathematics, there are many formulas. In addition, often enough it is necessary to find the area of the triangle on checkered paper.
In this case, it is most convenient to draw a height to one of the sides of the figure, determine its length from the cells, and use the universal formula to find the area:
S = ½ * A * H.
So, after studying the formulas presented in the article, you will not have problems finding the area of a triangle of any kind.