To determine the area of ​​a triangle, you can use different formulas. Of all the methods, the easiest and most frequently used is to multiply the height by the length of the base and then divide the result by two. However this method far from the only one. Below you can read how to find the area of ​​a triangle using different formulas.

Separately, we will look at ways to calculate the area of ​​specific types of triangles - rectangular, isosceles and equilateral. We accompany each formula with a short explanation that will help you understand its essence.

Universal methods for finding the area of ​​a triangle

The formulas below use special notation. 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 the circle that can be inscribed in our triangle;
  • R is the radius of the circle that can be described around it;
  • α is the magnitude of the angle formed by sides b and c;
  • β is the magnitude of the angle between a and c;
  • γ is the magnitude of the angle formed by sides a and b;
  • h is the height of our triangle, lowered from angle α to side a;
  • p – half the sum of sides a, b and c.

It is logically clear why you can find the area of ​​a triangle in this way. The triangle can easily be completed into 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 conditional parallelogram into 2 identical triangles. Therefore, it is quite obvious that the area of ​​our original triangle must 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 lower the height from angle β to side b, then, according to the properties right triangle, when multiplying the length of side a by the sine of the angle γ, we obtain 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 that can be inscribed in it by its perimeter. In other words, we find the product of the semi-perimeter and the radius of the mentioned circle.

S= a b c/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, as they make it possible to determine the area of ​​any triangle (scalene, isosceles, equilateral, rectangular). This can be done using more complex calculations, which we will not dwell on in detail.

Areas of triangles with specific properties

How to 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 the hypotenuse, then we find the area like this:

How to find area isosceles triangle? It has two sides with length a and one side with length b. Consequently, its area can be determined by dividing by 2 the product of the square of side a by the sine of angle γ.

How to 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 and 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.

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 of ​​a 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 for an isosceles triangle, then the easiest way is to use the main and what is considered to be 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:

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!

Square geometric figure - a numerical characteristic of a geometric figure showing the size of this figure (part of the surface limited by the closed contour of this figure). The size of the area is expressed by the number of square units contained in it.

Triangle area formulas

  1. Formula for the area of ​​a triangle by side and height
    Area of ​​a triangle equal to half the product of the length of a side of a triangle and the length of the altitude drawn to this side
  2. Formula for the area of ​​a triangle based on three sides and the radius of the circumcircle
  3. Formula for the area of ​​a triangle based on three sides and the radius of the inscribed circle
    Area of ​​a triangle is equal to the product of the semi-perimeter of the triangle and the radius of the inscribed circle.
    4. where S is the area of ​​the triangle,
    - lengths of the sides of the triangle,
    - height of the triangle,
    - the angle between the sides and,
    - radius of the inscribed circle,
    R - radius of the circumscribed circle,

Square area formulas

  1. Formula for the area of ​​a square by side length
    Square area equal to the square of the length of its side.
  2. Formula for the area of ​​a square along the diagonal length
    Square area equal to half the square of the length of its diagonal.
    S=1 2
    2
    3. where S is the area of ​​the square,
    - length of the side of the square,
    - length of the diagonal of the square.

Rectangle area formula

    Area of ​​a rectangle equal to the product of the lengths of its two adjacent sides

    where S is the area of ​​the rectangle,
    - lengths of the sides of the rectangle.

Parallelogram area formulas

  1. Formula for the area of ​​a parallelogram based on side length and height
    Area of ​​a parallelogram
  2. Formula for the area of ​​a parallelogram based on two sides and the angle between them
    Area of ​​a parallelogram is equal to the product of the lengths of its sides multiplied by the sine of the angle between them.

    a b sin α

    3. where S is the area of ​​the parallelogram,
    - lengths of the sides of the parallelogram,
    - length of parallelogram height,
    - the angle between the sides of the parallelogram.

Formulas for the area of ​​a rhombus

  1. Formula for the area of ​​a rhombus based on side length and height
    Area of ​​a rhombus equal to the product of the length of its side and the length of the height lowered to this side.
  2. Formula for the area of ​​a rhombus based on side length and angle
    Area of ​​a rhombus is equal to the product of the square of the length of its side and the sine of the angle between the sides of the rhombus.
  3. Formula for the area of ​​a rhombus based on the lengths of its diagonals
    Area of ​​a rhombus equal to half the product of the lengths of its diagonals.
    4. where S is the area of ​​the rhombus,
    - length of the side of the rhombus,
    - length of the height of the rhombus,
    - the angle between the sides of the rhombus,
    1, 2 - lengths of diagonals.

Trapezoid area formulas

  1. Heron's formula for trapezoid

    Where S is the area of ​​the trapezoid,
    - lengths of the bases of the trapezoid,
    - lengths of the sides of the trapezoid,

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.