As you may remember from school curriculum According to geometry, a triangle is a figure formed from three segments connected by three points that do not lie on the same straight line. A triangle forms three angles, hence the name of the figure. The definition may be different. A triangle can also be called a polygon with three angles, the answer will also be correct. Triangles are divided according to the number of equal sides and the size of the angles in the figures. Thus, triangles are distinguished as isosceles, equilateral and scalene, as well as rectangular, acute and obtuse, 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 is up to you. But it is worth noting only some of the notations that are 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 the semi-perimeter.

Here are the basic notations that may be useful to you if you completely forgot your geometry course. Below are the most understandable and not complex options calculating the unknown and mysterious area of ​​a triangle. It is not difficult and will be useful both for your household needs and for helping your children. Let's remember how to calculate the area of ​​a triangle as easily as possible:

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 (sqcm).

Right triangle and its area.

A right triangle is a triangle in which one angle is equal to 90 degrees (hence called right). A right angle is formed by two perpendicular lines (in the case of a triangle, two perpendicular segments). In a right triangle there can only be one right angle, because... the sum of all angles of any one triangle is equal to 180 degrees. It turns out that 2 other angles should divide the remaining 90 degrees, for example 70 and 20, 45 and 45, etc. So, you remember the main thing, all that remains is to find out how to find the area right triangle. Let's imagine that we have such a right triangle in front of us, and we need to find its area S.

1. The simplest way to determine the area of ​​a right triangle is calculated using the following formula:

In our case, the area of ​​the right triangle is: S = 2.5 cm * 3 cm / 2 = 3.75 sq. cm.

In principle, there is no longer any need to verify the area of ​​the triangle in other ways, because Only this one will be useful and will help in everyday life. But there are also options for measuring the area of ​​a triangle through acute angles.

2. For other calculation methods, you must have a table of cosines, sines and tangents. Judge for yourself, here are some options for calculating the area of ​​a right triangle that can still be used:

We decided to use the first formula and with some minor blots (we drew it in a notebook and used an old ruler and protractor), but we got the correct 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 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 and what is considered the classical formula for the area of ​​a triangle.

But first, before finding the area of ​​an isosceles triangle, let’s find out what kind of figure it is. An isosceles triangle is a triangle in which two sides have the same length. These two sides are called lateral, the third side is called the base. Do not confuse an isosceles triangle with an equilateral triangle, i.e. a regular triangle with all three sides equal. In such a triangle there are no special tendencies to the angles, or rather to 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:

A triangle is like this geometric figure, which consists of three lines connecting at points that do not lie on the same line. The connection points of the lines are the vertices of the triangle, which are designated by Latin letters (for example, A, B, C). The connecting straight lines of a triangle are called segments, which are also usually denoted by Latin letters. The following types of triangles are distinguished:

• Rectangular.
• Obtuse.
• Acute angular.
• Versatile.
• Equilateral.
• Isosceles.

General formulas for calculating the area of ​​a triangle

Formula for the area of ​​a triangle based on length and height

S= a*h/2,
where a is the length of the side of the triangle whose area needs to be found, h is the length of the height drawn to the base.

Heron's formula

S=√р*(р-а)*(р-b)*(p-c),
where √ is Square root, p is the semi-perimeter of the triangle, a,b,c is the length of each side of the triangle. The semi-perimeter of a triangle can be calculated using the formula p=(a+b+c)/2.

Formula for the area of ​​a triangle based on the angle and the length of the segment

S = (a*b*sin(α))/2,
Where b,c is the length of the sides of the triangle, sin(α) is the sine of the angle between the two sides.

Formula for the area of ​​a triangle given the radius of the inscribed circle and three sides

S=p*r,
where p is the semi-perimeter of the triangle whose area needs to be found, r is the radius of the circle inscribed in this triangle.

Formula for the area of ​​a triangle based on three sides and the radius of the circle circumscribed around it

S= (a*b*c)/4*R,
where a,b,c is the length of each side of the triangle, R is the radius of the circle circumscribed around the triangle.

Formula for the area of ​​a triangle using the Cartesian coordinates of points

Cartesian coordinates of points are coordinates in the xOy system, where x is the abscissa, y is the ordinate. The Cartesian coordinate system xOy on a plane is the mutually perpendicular numerical axes Ox and Oy with a common origin at point O. If the coordinates of points on this plane are given in the form A(x1, y1), B(x2, y2) and C(x3, y3 ), then you can calculate the area of ​​the triangle using the following formula, which is obtained from the vector product of two vectors.
S = |(x1 – x3) (y2 – y3) – (x2 – x3) (y1 – y3)|/2,
where || stands for module.

How to find the area of ​​a right triangle

A right triangle is a triangle with one angle measuring 90 degrees. A triangle can have only one such angle.

Formula for the area of ​​a right triangle on two sides

S= a*b/2,
where a,b is the length of the legs. Legs are the sides adjacent to a right angle.

Formula for the area of ​​a right triangle based on the hypotenuse and acute angle

S = a*b*sin(α)/ 2,
where a, b are the legs of the triangle, and sin(α) is the sine of the angle at which the lines a, b intersect.

Formula for the area of ​​a right triangle based on the side and the opposite angle

S = a*b/2*tg(β),
where a, b are the legs of the triangle, tan(β) is the tangent of the angle at which the legs a, b are connected.

How to calculate the area of ​​an isosceles triangle

An isosceles triangle is one that has two equal sides. These sides are called the sides, and the other side is the base. To calculate the area of ​​an isosceles triangle, you can use one of the following formulas.

Basic formula for calculating the area of ​​an isosceles triangle

S=h*c/2,
where c is the base of the triangle, h is the height of the triangle lowered to the base.

Formula of an isosceles triangle based on side and base

S=(c/2)* √(a*a – c*c/4),
where c is the base of the triangle, a is the size of one of the sides of the isosceles triangle.

How to find the area of ​​an equilateral triangle

An equilateral triangle is a triangle in which all sides are equal. To calculate the area of ​​an equilateral triangle, you can use the following formula:
S = (√3*a*a)/4,
where a is the length of the side of the equilateral triangle.

The above formulas will allow you to calculate the required area of ​​the triangle. It is important to remember that to calculate the area of ​​triangles, you need to consider the type of triangle and the available data that can be used for the calculation.

Area of ​​a triangle - formulas and examples of problem solving

Below are formulas for finding the area of ​​an arbitrary triangle which are suitable for finding the area of ​​any triangle, regardless of its properties, angles or sizes. The formulas are presented in the form of a picture, with explanations for their application or justification for their correctness. Also, a separate figure shows the correspondence between the letter symbols in the formulas and the graphic symbols in the drawing.

Note . If the triangle has special properties(isosceles, rectangular, equilateral), you can use the formulas given below, as well as additional special formulas that are valid only for triangles with these properties:

• "Formula for the area of ​​an equilateral triangle"

Triangle area formulas

Explanations for formulas:
a, b, c- the lengths of the sides of the triangle whose area we want to find
r- radius of the circle inscribed in the triangle
R- radius of the circle circumscribed around the triangle
h- height of the triangle lowered to the side
p- semi-perimeter of a triangle, 1/2 the sum of its sides (perimeter)
α - angle opposite to side a of the triangle
β - angle opposite to side b of the triangle
γ - angle opposite to side c of the triangle
h a, h b , h c- height of the triangle lowered to sides a, b, c

Please note that the given notations correspond to the figure above, so that when solving a real geometry problem, it will be visually easier for you to substitute the correct values ​​in the right places in the formula.

• The area of ​​the triangle is half the product of the height of the triangle and the length of the side by which this height is lowered(Formula 1). The correctness of this formula can be understood logically. The height lowered to the base will split an arbitrary triangle into two rectangular ones. If you build each of them into a rectangle with dimensions b and h, then obviously the area of ​​these triangles will be equal to exactly half the area of ​​the rectangle (Spr = bh)
• The area of ​​the triangle is half the product of its two sides and the sine of the angle between them(Formula 2) (see an example of solving a problem using this formula below). Even though it seems different from the previous one, it can easily be transformed into it. If we lower the height from angle B to side b, it turns out that the product of side a and the sine of angle γ, according to the properties of the sine in a right triangle, is equal to the height of the triangle we drew, which gives us the previous formula
• The area of ​​an arbitrary triangle can be found through work half the radius of the circle inscribed in it by the sum of the lengths of all its sides(Formula 3), simply put, you need to multiply the semi-perimeter of the triangle by the radius of the inscribed circle (this is easier to remember)
• The area of ​​an arbitrary triangle can be found by dividing the product of all its sides by 4 radii of the circle circumscribed around it (Formula 4)
• Formula 5 is finding the area of ​​a triangle through the lengths of its sides and its semi-perimeter (half the sum of all its sides)
• Heron's formula(6) is a representation of the same formula without using the concept of semi-perimeter, only through the lengths of the sides
• The area of ​​an arbitrary triangle is equal to the product of the square of the side of the triangle and the sines of the angles adjacent to this side divided by the double sine of the angle opposite to this side (Formula 7)
• The area of ​​an arbitrary triangle can be found as the product of two squares of the circle circumscribed around it by the sines of each of its angles. (Formula 8)
• If the length of one side and the values ​​of two adjacent angles are known, then the area of ​​the triangle can be found as the square of this side divided by the double sum of the cotangents of these angles (Formula 9)
• If only the length of each of the heights of the triangle is known (Formula 10), then the area of ​​such a triangle is inversely proportional to the lengths of these heights, as according to Heron’s Formula
• Formula 11 allows you to calculate area of ​​a triangle based on the coordinates of its vertices, which are specified as (x;y) values ​​for each of the vertices. Please note that the resulting value must be taken modulo, since the coordinates of individual (or even all) vertices may be in the region of negative values

Note. The following are examples of solving geometry problems to find the area of ​​a triangle. If you need to solve a geometry problem that is not similar here, write about it in the forum. In solutions, instead of the "square root" symbol, the sqrt() function can be used, in which sqrt is the square root symbol, and the radical expression is indicated in parentheses.Sometimes for simple radical expressions the symbol can be used √

Task. Find the area given two sides and the angle between them

The sides of the triangle are 5 and 6 cm. The angle between them is 60 degrees. Find the area of ​​the triangle.

Solution.

To solve this problem, we use formula number two from the theoretical part of the lesson.
The area of ​​a triangle can be found through the lengths of two sides and the sine of the angle between them and will be equal to
S=1/2 ab sin γ

Since we have all the necessary data for the solution (according to the formula), we can only substitute the values ​​​​from the problem conditions into the formula:
S = 1/2 * 5 * 6 * sin 60

In the table of values trigonometric functions Let's find and substitute the value of sine 60 degrees into the expression. It will be equal to the root of three times two.
S = 15 √3 / 2

Answer: 7.5 √3 (depending on the teacher’s requirements, you can probably leave 15 √3/2)

Task. Find the area of ​​an equilateral triangle

Find the area of ​​an equilateral triangle with side 3 cm.

Solution .

The area of ​​a triangle can be found using Heron's formula:

S = 1/4 sqrt((a + b + c)(b + c - a)(a + c - b)(a + b -c))

Since a = b = c, the formula for the area of ​​an equilateral triangle takes the form:

S = √3 / 4 * a 2

S = √3 / 4 * 3 2

Answer: 9 √3 / 4.

Task. Change in area when changing the length of the sides

How many times will the area of ​​the triangle increase if the sides are increased by 4 times?

Solution.

Since the dimensions of the sides of the triangle are unknown to us, to solve the problem we will assume that the lengths of the sides are respectively equal to arbitrary numbers a, b, c. Then, in order to answer the question of the problem, we will find the area of ​​the given triangle, and then we will find the area of ​​the triangle whose sides are four times larger. The ratio of the areas of these triangles will give us the answer to the problem.

Below we provide a textual explanation of the solution to the problem step by step. However, at the very end, this same solution is presented in a more convenient graphical form. Those interested can immediately go down the solutions.

To solve, we use Heron’s formula (see above in the theoretical part of the lesson). It looks like this:

S = 1/4 sqrt((a + b + c)(b + c - a)(a + c - b)(a + b -c))
(see first line of picture below)

The lengths of the sides of an arbitrary triangle are specified by the variables a, b, c.
If the sides are increased by 4 times, then the area of ​​the new triangle c will be:

S 2 = 1/4 sqrt((4a + 4b + 4c)(4b + 4c - 4a)(4a + 4c - 4b)(4a + 4b -4c))
(see second line in the picture below)

As you can see, 4 is a common factor that can be taken out of brackets from all four expressions according to general rules mathematics.
Then

S 2 = 1/4 sqrt(4 * 4 * 4 * 4 (a + b + c)(b + c - a)(a + c - b)(a + b -c)) - on the third line of the picture
S 2 = 1/4 sqrt(256 (a + b + c)(b + c - a)(a + c - b)(a + b -c)) - fourth line

The square root of the number 256 is perfectly extracted, so let’s take it out from under the root
S 2 = 16 * 1/4 sqrt((a + b + c)(b + c - a)(a + c - b)(a + b -c))
S 2 = 4 sqrt((a + b + c)(b + c - a)(a + c - b)(a + b -c))
(see fifth line of the picture below)

To answer the question asked in the problem, we just need to divide the area of ​​the resulting triangle by the area of ​​the original one.
Let us determine the area ratios by dividing the expressions by each other and reducing the resulting fraction.

Sometimes in life there are situations when you have to delve into your memory in search of long-forgotten school knowledge. For example, you need to determine the area of ​​a triangular-shaped plot of land, or the time has come for another renovation in an apartment or private house, and you need to calculate how much material will be needed for a surface with a triangular shape. There was a time when you could solve such a problem in a couple of minutes, but now you are desperately trying to remember how to determine the area of ​​a triangle?

Don't worry about it! After all, it is quite normal when a person’s brain decides to transfer long-unused knowledge somewhere to a remote corner, from which sometimes it is not so easy to extract it. So that you don’t have to struggle with searching for forgotten school knowledge to solve such a problem, this article contains various methods that make it easy to find the required area of ​​a triangle.

It is well known that a triangle is a type of polygon that is limited to the minimum possible number of sides. In principle, any polygon can be divided into several triangles by connecting its vertices with segments that do not intersect its sides. Therefore, knowing the triangle, you can calculate the area of ​​almost any figure.

Among all the possible triangles that occur in life, the following particular types can be distinguished: and rectangular.

The easiest way to calculate the area of ​​a triangle is when one of its angles is right, that is, in the case of a right triangle. It is easy to see that it is half a rectangle. Therefore, its area is equal to half the product of the sides that form a right angle with each other.

If we know the height of a triangle, lowered from one of its vertices to the opposite side, and the length of this side, which is called the base, then the area is calculated as half the product of the height and the base. This is written using the following formula:

S = 1/2*b*h, in which

S is the required area of ​​the triangle;

b, h - respectively, the height and base of the triangle.

It is so easy to calculate the area of ​​an isosceles triangle because the height will bisect the opposite side and can be easily measured. If the area is determined, then it is convenient to take the length of one of the sides forming a right angle as the height.

All this is of course good, but how to determine whether one of the angles of a triangle is right or not? If the size of our figure is small, then we can use a construction angle, a drawing triangle, a postcard or another object with a rectangular shape.

But what if we have a triangular plot of land? In this case, proceed as follows: count from the top of the expected right angle on one side the distance is a multiple of 3 (30 cm, 90 cm, 3 m), and on the other side a distance is measured in the same proportion that is a multiple of 4 (40 cm, 160 cm, 4 m). Now you need to measure the distance between the end points of these two segments. If the result is a multiple of 5 (50 cm, 250 cm, 5 m), then we can say that the angle is right.

If the length of each of the three sides of our figure is known, then the area of ​​the triangle can be determined using Heron's formula. In order for it to have a simpler form, a new value is used, which is called semi-perimeter. This is the sum of all the sides of our triangle, divided in half. After the semi-perimeter has been calculated, you can begin to determine the area using the formula:

S = sqrt(p(p-a)(p-b)(p-c)), where

sqrt - square root;

p - semi-perimeter value (p = (a+b+c)/2);

a, b, c - edges (sides) of the triangle.

But what if the triangle has irregular shape? There are two possible ways here. The first of them is to try to divide such a figure into two right triangles, the sum of the areas of which is calculated separately, and then added. Or, if the angle between two sides and the size of these sides are known, then apply the formula:

S = 0.5 * ab * sinC, where

a,b - sides of the triangle;

c is the size of the angle between these sides.

The latter case is rare in practice, but nevertheless, everything is possible in life, so the above formula will not be superfluous. Good luck with your calculations!

You can find over 10 formulas for calculating the area of ​​a triangle on the Internet. Many of them are used in problems with known sides and angles of a triangle. However, there are a number of complex examples where, according to the conditions of the assignment, only one side and angles of a triangle are known, or the radius of a circumscribed or inscribed circle and one more characteristic. In such cases, a simple formula cannot be applied.

The formulas given below will allow you to solve 95 percent of problems in which you need to find the area of ​​a triangle.
Let's move on to consider common area formulas.
Consider the triangle shown in the figure below

In the figure and below in the formulas, the classical designations of all its characteristics are introduced.
a,b,c – sides of the triangle,
R – radius of the circumscribed circle,
r – radius of the inscribed circle,
h[b],h[a],h[c] – heights drawn in accordance with sides a,b,c.
alpha, beta, hamma – angles near the vertices.

Basic formulas for the area of ​​a triangle

1. The area is equal to half the product of the side of the triangle and the height lowered to this side. In the language of formulas, this definition can be written as follows

Thus, if the side and height are known, then every student will find the area.
By the way, from this formula one can derive one useful relationship between heights

2. If we take into account that the height of a triangle through the adjacent side is expressed by the dependence

Then the first area formula is followed by the second ones of the same type

Look carefully at the formulas - they are easy to remember, since the work involves two sides and the angle between them. If we correctly designate the sides and angles of the triangle (as in the figure above), we will get two sides a,b and the angle is connected to the third With (hamma).

3. For the angles of a triangle, the relation is true

The dependence allows you to use the following formulas for the area of ​​a triangle in calculations:

Examples of this dependence are extremely rare, but you must remember that there is such a formula.

4. If the side and two adjacent angles are known, then the area is found by the formula

5. The formula for area in terms of side and cotangent of adjacent angles is as follows

By rearranging the indexes you can get dependencies for other parties.

6. The area formula below is used in problems when the vertices of a triangle are specified on the plane by coordinates. In this case, the area is equal to half the determinant taken modulo.

7. Heron's formula used in examples with known sides of a triangle.
First find the semi-perimeter of the triangle

And then determine the area using the formula

or

It is quite often used in the code of calculator programs.

8. If all the heights of the triangle are known, then the area is determined by the formula

It is difficult to calculate on a calculator, but in the MathCad, Mathematica, Maple packages the area is “time two”.

9. The following formulas use the known radii of inscribed and circumscribed circles.

In particular, if the radius and sides of the triangle, or its perimeter, are known, then the area is calculated according to the formula

10. In examples where the sides and the radius or diameter of the circumscribed circle are given, the area is found using the formula

11. The following formula determines the area of ​​a triangle in terms of the side and angles of the triangle.

And finally - special cases:
Area of ​​a right triangle with legs a and b equal to half their product

Formula for the area of ​​an equilateral (regular) triangle=

= one-fourth of the product of the square of the side and the root of three.