Find the area of tr. How to find the area of a triangle. Triangle formulas. Special case: right triangle
How can you remember from school curriculum in geometry, a triangle is a figure formed from three line segments connected by three points that do not lie on one straight line. The triangle forms three corners, hence the name of the figure. The definition may be different. A triangle can also be called a polygon with three corners, the answer is also correct. Triangles are divided by the number of equal sides and by the angles in the figures. So such triangles are distinguished as isosceles, equilateral and versatile, as well as rectangular, acute-angled and obtuse-angled, respectively.
There are a lot of formulas for calculating the area of a triangle. Choose how to find the area of a triangle, i.e. which formula to use, only you. But it is worth noting only some of the notation that is used in many formulas for calculating the area of a triangle. So remember:
S is the area of the triangle,
a, b, c are the sides of the triangle,
h is the height of the triangle,
R is the radius of the circumscribed circle,
p is a semi-perimeter.
Here are some basic notation that may come in handy if you completely forgot your geometry course. Below will be given the most understandable and not complicated options for calculating the unknown and mysterious area of a triangle. It is not difficult and will be useful both for you at home and for helping your children. Let's remember how to calculate the area of a triangle as easy as shelling pears:
In our case, the area of the triangle is: S = ½ * 2.2 cm * 2.5 cm = 2.75 sq. Cm. Remember that area is measured in square centimeters (cm2).
Rectangular triangle and its area.
A right-angled triangle is a triangle with one angle equal to 90 degrees (therefore it is called a right angle). A right angle is formed by two perpendicular lines (in the case of a triangle, two perpendicular segments). In a right-angled triangle, there can be only one right angle, because the sum of all the angles of any one triangle is 180 degrees. It turns out that the other 2 angles must divide the remaining 90 degrees, for example 70 and 20, 45 and 45, etc. So, you remembered the main thing, it remains to find out how to find the area of a right-angled triangle. Let's imagine that we have this right triangle, and we need to find its area S.
1. The easiest way to determine the area of a right-angled triangle is calculated using the following formula:
In our case, the area of a right-angled triangle is: S = 2.5 cm * 3 cm / 2 = 3.75 sq. Cm.
In principle, it is no longer necessary to reconcile the area of the triangle in other ways, since in everyday life, only this one will come in handy and help. But there are also options for measuring the area of a triangle through acute angles.
2. For other methods of calculation, you must have a table of cosines, sines and tangents. Judge for yourself, here are some options for calculating the areas of a right-angled triangle you can still use:
We decided to use the first formula and with small blots (we drew in a notebook and used the old ruler and protractor), but we got the right calculation:
S = (2.5 * 2.5) / (2 * 0.9) = (3 * 3) / (2 * 1.2). We got the following results 3.6 = 3.7, but taking into account the shift of the cells, we can forgive this nuance.
Isosceles triangle and its area.
If you are faced with the task of calculating the formula isosceles triangle, then the easiest way is to use the main one, and as it is considered the classic formula for the area of a triangle.
But first, before finding the area of an isosceles triangle, we will find out what kind of figure it is. An isosceles triangle is a triangle with two sides of the same length. These two sides are called lateral sides, the third side is called the base. Do not confuse an isosceles triangle with an equilateral one, i.e. a regular triangle with all three sides equal. In such a triangle, there are no special tendencies for angles, more precisely, for their size. However, the angles at the base in an isosceles triangle are equal, but different from the angle between equal sides... So, you already know the first and main formula, it remains to find out what other formulas for determining the area of an isosceles triangle are known.
Area formula is necessary to determine the area of a figure, which is a real-valued function defined on a certain class of figures in the Euclidean plane and satisfying 4 conditions:
- Positiveness - The area cannot be less than zero;
- Normalization - a square with a side of one has an area of 1;
- Congruence - congruent shapes have equal area;
- Additivity - the area of the union of 2 figures without common interior points is equal to the sum of the areas of these figures.
| Geometric figure | Formula | Drawing |
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The result of adding the distances between the midpoints of opposite sides of a convex quadrilateral will be equal to its semiperimeter. |
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Circle sector. The area of a sector of a circle is equal to the product of its arc and half the radius. |
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Segment of a circle. To obtain the area of the ASB segment, it is sufficient to subtract the area of the triangle AOB from the area of the AOB sector. |
S = 1/2 R (s - AC) |
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The area of an ellipse is equal to the product of the lengths of the major and minor semiaxes of the ellipse by the number pi. |
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Ellipse. Another option for calculating the area of an ellipse is through two of its radii. |
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Triangle. Through the base and height. Formula for the area of a circle in terms of its radius and diameter. |
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Square. Through his side. The area of a square is equal to the square of the length of its side. |
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Square. Through its diagonals. The area of a square is half the square of the length of its diagonal. |
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Regular polygon. To determine the area of a regular polygon, it is necessary to split it into equal triangles, which would have a common vertex in the center of the inscribed circle. |
S = r p = 1/2 r n a |
Area of a triangle. Many geometry problems associated with the calculation of areas use formulas for the area of a triangle. There are several of them, here we will look at the main ones.It would be too easy to list these formulas and it would be of no use whatsoever. We will analyze the origin of the main formulas, those that are used most often.
Before you get acquainted with the derivation of formulas, be sure to look at the article about.After studying the material, you can easily restore the formulas in memory (if suddenly they "fly out" at the right moment).
The first formula
The diagonal of the parallelogram splits it into two triangles of equal area:
Therefore, the area of the triangle will be equal to half the area of the parallelogram:
Area of a triangle formula
* That is, if we know any side of the triangle and the height lowered to this side, then we can always calculate the area of this triangle.
Formula two
As already stated in the article on the area of a parallelogram, the formula has the form:
The area of a triangle is equal to half of its area, which means:
* That is, if we know any two sides in a triangle and the angle between them, we can always calculate the area of such a triangle.
Heron's formula (third)
This formula is difficult to deduce and you don't need it. Look how beautiful she is, we can say that she herself is remembered.
* If three sides of a triangle are given, then using this formula we can always calculate its area.
Formula four
Where r- radius of the inscribed circle
* If the three sides of the triangle and the radius of the inscribed circle are known, then we can always find the area of this triangle.
Formula five
Where RIs the radius of the circumscribed circle.
* If the three sides of a triangle and the radius of the circumscribed circle around it are known, then we can always find the area of such a triangle.
The question arises: if the three sides of a triangle are known, then isn't it easier to find its area using Heron's formula!
Yes, it is easier, but not always, sometimes it’s difficult. This is due to the extraction of the root. In addition, these formulas are very convenient to apply in problems where the area of a triangle is given, its sides and it is required to find the radius of the inscribed or circumscribed circle. Such tasks are included in the Unified State Exam.
Let's take a look at the formula separately:
It is a special case of the formula for the area of a polygon in which a circle is inscribed:
Let's consider it using the example of a pentagon:
Connect the center of the circle with the vertices of this pentagon and drop the perpendiculars from the center to its sides. We get five triangles, whereby the omitted perpendiculars are the radii of the inscribed circle:
The area of the pentagon is:
Now it is clear that if we are talking about a triangle, then this formula takes the form:
Formula six
Various formulas can be used to determine the area of a triangle. Of all the methods, the easiest and most often used is to multiply the height by the length of the base and then divide the result by two. However, this method is far from the only one. Below you can read how to find the area of a triangle using different formulas.
Separately, we will consider methods for calculating the area of specific types of a triangle - rectangular, isosceles and equilateral. We accompany each formula with a short explanation that will help you understand its essence.
Universal ways to find the area of a triangle
The following formulas use special conventions. We will decipher each of them:
- a, b, c - the lengths of the three sides of the figure we are considering;
- r is the radius of a circle that can be inscribed in our triangle;
- R is the radius of the circle that can be described around it;
- α - the value of the angle formed by the sides b and c;
- β is the angle between a and c;
- γ - the value of the angle formed by sides a and b;
- h - the height of our triangle, lowered from the angle α to the side a;
- p - half the sum of sides a, b and c.
It is logical why it is possible to find the area of a triangle in this way. The triangle can be easily completed to a parallelogram, in which one side of the triangle will act as a diagonal. The area of a parallelogram is found by multiplying the length of one of its sides by the value of the height drawn to it. The diagonal divides this conventional parallelogram into 2 identical triangles. Therefore, it is quite obvious that the area of our original triangle should be equal to half the area of this auxiliary parallelogram.
S = ½ a b sin γ
According to this formula, the area of a triangle is found by multiplying the lengths of its two sides, that is, a and b, by the sine of the angle formed by them. This formula is logically derived from the previous one. If we drop the height from angle β to side b, then, according to the properties of a right-angled triangle, when multiplying the length of side a by the sine of angle γ, we get the height of the triangle, that is, h.
The area of the figure in question is found by multiplying half the radius of the circle, which can be inscribed into it, by its perimeter. In other words, we find the product of the semiperimeter and the radius of the mentioned circle.
S = a b s / 4R
According to this formula, the value we need can be found by dividing the product of the sides of the figure by 4 radii of the circle described around it.
These formulas are universal, since they make it possible to determine the area of any triangle (versatile, isosceles, equilateral, rectangular). This can be done with the help of more complex calculations, on which we will not dwell in detail.
Areas of triangles with specific properties
How do I find the area of a right triangle? The peculiarity of this figure is that its two sides are simultaneously its heights. If a and b are legs, and c becomes a hypotenuse, then the area is found as follows:
How to find the area of an isosceles triangle? It has two sides with length a and one side with length b. Therefore, its area can be determined by dividing by 2 the product of the square of the side a by the sine of the angle γ.
How do you find the area of an equilateral triangle? In it, the length of all sides is equal to a, and the magnitude of all angles is α. Its height is equal to half the product of the length of side a by the square root of 3. To find the area of a regular triangle, you need to multiply the square of side a by the square root of 3 and divide by 4.
Can be found by knowing the base and height. The whole simplicity of the scheme lies in the fact that the height divides the base a into two parts a 1 and a 2, and the triangle itself into two right-angled triangles, the area of which is obtained and. Then the area of the entire triangle will be the sum of the two indicated areas, and if we take out one second of the height outside the bracket, then in total we get the base back:
A more complicated method for calculations is Heron's formula, for which you need to know all three sides. For this formula, you must first calculate the semiperimeter of the triangle: 
Heron's formula itself implies the square root of the half-perimeter, multiplied alternately by its difference on each side.
The next method, also relevant for any triangle, allows you to find the area of a triangle through two sides and the angle between them. The proof of this follows from the formula with height - we draw the height to any of the known sides and through the sine of the angle α we obtain that h = a⋅sinα. To calculate the area, multiply half the height by the other side.
Another way is to find the area of a triangle by knowing the 2 angles and the side between them. The proof of this formula is quite simple, and can be clearly seen from the diagram.
We lower the height from the vertex of the third corner to the known side and call the resulting segments x, respectively. From the right-angled triangles it can be seen that the first segment x is equal to the product
