Triangle is one of the most common geometric shapes, which we are already familiar with in elementary school. The question of how to find the area of a triangle is faced by every student in geometry lessons. So, what are the features of finding the area of \u200b\u200ba given figure can be distinguished? In this article, we will consider the basic formulas necessary to complete such a task, and also analyze the types of triangles.
Types of triangles
You can find the area of a triangle in completely different ways, because in geometry there is more than one type of figure containing three angles. These species include:
- Acute triangle.
- Obt-angled.
- Equilateral (correct).
- Right triangle.
- Isosceles.
Let's take a closer look at each of the existing types of triangles.
Acutetriangle
Such a geometric figure is considered the most common in solving geometric problems. When it becomes necessary to draw an arbitrary triangle, this option comes to the rescue.
In an acute triangle, as the name implies, all angles are acute and add up to 180°.
Obt-angled triangle
This triangle is also very common, but is somewhat less common than the acute-angled one. For example, when solving triangles (that is, you know several of its sides and angles and you need to find the remaining elements), sometimes you need to determine whether the angle is obtuse or not. The cosine of an obtuse angle 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 nuances, which we will talk about later.
Regular and isosceles triangles
A regular polygon is a figure that includes n angles and all sides and angles are equal. This is the right triangle. Since the sum of all the angles of a triangle is 180°, 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 ina regular triangle can only be inscribed with one circle and only one circle can be circumscribed around it, and their centers are located at one point.
Besides the equilateral type, one can also distinguish an isosceles triangle, which differs slightly from it. In such a triangle, two sides and two angles are equal to each other, and the third side (to which equal angles adjoin) is the base.
The figure shows an isosceles triangle DEF, the angles D and F of which are equal, and DF is the base.
Right triangle
A right-angled triangle is named so because one of its angles is a right angle, that is, equal to 90°. The other two angles add up to 90°.
The largest side of such a triangle, lying opposite the angle of 90°, is the hypotenuse, while the other two of its sides are the legs. For this type of triangles, 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 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.
Let's move on to the formulas for finding the area of this figure.
Basic area formulas
In geometry, there are two formulas that are suitable for finding the area of most types of triangles, namely for acute-angled, obtuse-angled, regular andisosceles triangles. Let's analyze each of them.
By side and height
This formula is universal for finding the area of the figure we are considering. 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 product of the base and the height) looks like this:
S=½AH, 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 this way. For example, to use this formula for an obtuse-angled 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 by the side and height of a triangle. That is, the formula under consideration can be easily derived from the previous one. Her wording looks like this:
S=½sinOAB, where A and B are sides of a triangle and O is the angle between sides A and B.
Recall that the sine of an angle can be viewed in a special table named after the outstanding Soviet mathematician V. M. Bradis.
And 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 a height in a triangle, the area of a triangle containing a right angle can be found by its legs.
Thus, the area of a triangle containing a right angle is half the product of its legs, or:
S=½ab, where a and b are the legs of a right triangle.
Regular Triangle
This type of geometric figures differs in that its area can be found with the specified value of only one of its sides (since all sides of a regular triangle are equal). So, having met with the task of "find the area of a triangle when the sides are equal", you need to use the following formula:
S=A2√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 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 task given: "the area of a regular triangle - find the length of its side." In this case, you need to use the already known formula for finding the area of a regular triangle and derive the value of the side (or its square) from it:
A2=4S / √3.
Exam Problems
In GIA tasksThere are many formulas in mathematics. 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 the height to one of the sides of the figure, determine its length by cells and use the universal formula for finding the area:
S=½AH.
So, after studying the formulas presented in the article, you will not have problems finding the area of a triangle of any kind.
