How can you find the area of a triangle. Calculating the area of a polygon from the coordinates of its vertices Determining the area of a triangle from the coordinates of its vertices
The coordinate method, proposed in the 17th century by the French mathematicians R. Descartes (1596-1650) and P. Fermat (1601-1665), is a powerful apparatus that allows one to translate geometric concepts into algebraic language. This method is based on the concept of a coordinate system. We will consider calculating the area of a polygon from the coordinates of its vertices in a rectangular coordinate system.
Area of a triangle
Theorem 1. If is the area of the triangle
then the equality is true
we will call it the determinant of the area of a triangle.
Proof. Let the vertices of the triangle be located in the first coordinate quadrant. There are two possible cases.
Case 1. The direction (or, or) of the location of the vertices of the triangle coincides with the direction of movement of the end of the clock hand (Fig. 1.30).
Since the figure is a trapezoid.
Similarly we find that
By performing algebraic transformations
we get that:
In equality (1.9) the determinant of area is, therefore, there is a minus sign in front of the expression, since.
Let's show that. Indeed, here
(the area of a rectangle with a base and height is greater than the sum of the areas of rectangles with bases and heights; (Fig. 1.30), whence
Case 2. The indicated directions in case 1 are opposite to the direction of movement of the end of the clock hand (Fig. 1.31)
since the figure is a trapezoid, and
Where. Indeed, here
The theorem is proven when the vertices of the triangle are located in the first coordinate quadrant.
Using the concept of modulus, equalities (1.9) and (1.10) can be written as follows:
Note 1. We derived formula (1.8) by considering the simplest arrangement of vertices, shown in Figures 1.30 and 1.31; however, formula (1.8) is true for any arrangement of vertices.
Consider the case depicted in Figure 1.32.
Therefore, by performing simple geometric transformations:
we get again what, where
Area of n-gon
A polygon can be convex or non-convex; the vertex numbering order is considered negative if the vertices are numbered in a clockwise direction. A polygon that does not have self-intersection of sides will be called simple. For simple it is n-gon the following is true
Theorem 2. If is the area of a prime n-gon, where, then the equality is true
we will call the determinant of the area of a prime n-gon.
Proof. There are two possible cases.
Case 1. n-gon - convex. Let us prove formula (1.11) using the method of mathematical induction.
For it has already been proven (Theorem 1). Let us assume that it is true for n-gon; let us prove that it remains valid for convex ( n+1)-gon.
Let's add one more vertex to the polygon (Fig. 1.33).
Thus, the formula is valid for ( n+1)-gon, and, therefore, the conditions of mathematical induction are satisfied, i.e. formula (1.11) for the case of a convex n-gon has been proven.
Case 2. n-gon - non-convex.
In any non-convex n-gon one can draw a diagonal lying inside it, and therefore the proof of case 2 for a non-convex n-gon is similar to the proof for a convex n-gon.
Note 2. Expressions for are not easy to remember. Therefore, to calculate its values, it is convenient to write down the coordinates of the first, second, third, ..., in a column. n-th and again the first vertices n-gon and multiply according to the scheme:
The signs in column (1.12) must be arranged as indicated in diagram (1.13).
Note 3. When composing column (1.12) for a triangle, you can start from any vertex.
Note 4. When compiling column (1.12) for n-gon () it is necessary to follow the sequence of writing out the coordinates of the vertices n-gon (it doesn’t matter which vertex to start the traversal from). Therefore, calculating the area n-gon should begin with the construction of a “rough” drawing.
A triangle is one of the most common geometric shapes, which we become familiar with in elementary school. Every student faces the question of how to find the area of a triangle in geometry lessons. So, what features of finding the area of a given figure can be identified? In this article we will look at 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 types include:
- Obtuse.
- Equilateral (correct).
- Right triangle.
- Isosceles.
Let's take a closer look at each of the existing types of triangles.
This geometric figure is considered the most common when solving geometric problems. When the need arises to draw an arbitrary triangle, this option comes to the rescue.
In an acute triangle, as the name suggests, all the angles are acute and add up to 180°.
This type of triangle is also very common, but is somewhat less common than an acute triangle. For example, when solving triangles (that is, several of its sides and angles are known and you need to find the remaining elements), sometimes you need to determine whether the angle is obtuse or not. Cosine is a negative number.
B, 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 whose sides and angles are all equal. This is what a regular triangle is. Since the sum of all the angles of a 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 the same point.
In addition to the equilateral type, one can also distinguish an isosceles triangle, which is slightly different from it. In such a triangle, 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 triangle is so named because one of its angles is right, that is, equal to 90°. The other two angles add up to 90°.
The largest side of such a triangle, lying opposite the 90° angle, is the hypotenuse, while the remaining two sides are the legs. For this type of triangle, the Pythagorean theorem applies:
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 a given figure.
Basic formulas for finding area
In geometry, there are two formulas that are suitable for finding the area of most types of triangles, namely for acute, obtuse, regular and isosceles triangles. Let's look at 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) is as follows:
where A is the side of a given triangle, and H is the height of the triangle.
For example, to find the area of an acute 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 triangle, you need to extend one of its sides and only then draw an altitude to it.
In practice, this formula is used more often than others.
On both sides and 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 side and height of a triangle. That is, the formula in question 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 sides A and B.
Let us recall that the sine of an angle can be viewed in a special table named after the outstanding Soviet mathematician V. M. Bradis.
Now let's move on to other formulas that are suitable only for exceptional types of triangles.
Area of a right triangle
In addition to the universal formula, which includes the need to find the altitude 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:
where a and b are the legs of a right triangle.
Regular triangle
This type of geometric figure is different 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, when faced with 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 the 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 a given triangle.
Sometimes the problem 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.
Examination tasks
There are many formulas in GIA problems in mathematics. In addition, quite often it is 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 from the cells and use the universal formula for finding the area:
So, after studying the formulas presented in the article, you will not have any problems finding the area of a triangle of any kind.
