From the opposite vertex) and divide the resulting product by two. This looks like this:

S = ½ * a * h,

Where:
S – area of ​​the triangle,
a is the length of its side,
h is the height lowered to this side.

Side length and height must be presented in the same units of measurement. In this case, the area of ​​the triangle will be obtained in the corresponding “ ” units.

Example.
On one side of a scalene triangle 20 cm long, a perpendicular from the opposite vertex 10 cm long is lowered.
The area of ​​the triangle is required.
Solution.
S = ½ * 20 * 10 = 100 (cm²).

If the lengths of any two sides of a scalene triangle and the angle between them are known, then use the formula:

S = ½ * a * b * sinγ,

where: a, b are the lengths of two arbitrary sides, and γ is the angle between them.

In practice, for example, when measuring land plots, the use of the above formulas is sometimes difficult, since it requires additional construction and measurement of angles.

If you know the lengths of all three sides of a scalene triangle, then use Heron’s formula:

S = √(p(p-a)(p-b)(p-c)),

a, b, c – lengths of the sides of the triangle,
p – semi-perimeter: p = (a+b+c)/2.

If, in addition to the lengths of all sides, the radius of the circle inscribed in the triangle is known, then use the following compact formula:

where: r – radius of the inscribed circle (р – semi-perimeter).

To calculate the area of ​​a scalene triangle and the length of its sides, use the formula:

where: R – radius of the circumscribed circle.

If you know the length of one of the sides of the triangle and three angles (in principle, two are enough - the value of the third is calculated from the equality of the sum of the three angles of the triangle - 180º), then use the formula:

S = (a² * sinβ * sinγ)/2sinα,

where α is the value of the angle opposite to side a;
β, γ – values ​​of the remaining two angles of the triangle.

The need to find various elements, including area triangle, appeared many centuries BC among learned astronomers Ancient Greece. Square triangle can be calculated in various ways using different formulas. The calculation method depends on which elements triangle known.

Instructions

If from the condition we know the values ​​of two sides b, c and the angle formed by them?, then the area triangle ABC is found by the formula:
S = (bcsin?)/2.

If from the condition we know the values ​​of two sides a, b and the angle not formed by them?, then the area triangle ABC is found as follows:
Finding the angle?, sin? = bsin?/a, then use the table to determine the angle itself.
Finding the angle?, ? = 180°-?-?.
We find the area itself S = (absin?)/2.

If from the condition we know the values ​​of only three sides triangle a, b and c, then the area triangle ABC is found by the formula:
S = v(p(p-a)(p-b)(p-c)), where p is the semi-perimeter p = (a+b+c)/2

If from the problem conditions we know the height triangle h and the side to which this height is lowered, then the area triangle ABC according to the formula:
S = ah(a)/2 = bh(b)/2 = ch(c)/2.

If we know the meanings of the sides triangle a, b, c and the radius described about this triangle R, then the area of ​​this triangle ABC is determined by the formula:
S = abc/4R.
If three sides a, b, c and the radius of the inscribed in are known, then the area triangle ABC is found by the formula:
S = pr, where p is the semi-perimeter, p = (a+b+c)/2.

If ABC is equilateral, then the area is found by the formula:
S = (a^2v3)/4.
If triangle ABC– isosceles, then the area is determined by the formula:
S = (cv(4a^2-c^2))/4, where c – triangle.
If triangle ABC is right-angled, then the area is determined by the formula:
S = ab/2, where a and b are legs triangle.
If triangle ABC is a right isosceles triangle, then the area is determined by the formula:
S = c^2/4 = a^2/2, where c is the hypotenuse triangle, a=b – leg.

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• how to measure the area of ​​a triangle

Tip 3: How to find the area of ​​a triangle if the angle is known

Knowing just one parameter (the angle) is not enough to find the area tre square . If there are any additional dimensions, then to determine the area you can choose one of the formulas in which the angle value is also used as one of the known variables. Several of the most frequently used formulas are given below.

Instructions

If, in addition to the size of the angle (γ) formed by the two sides tre square , the lengths of these sides (A and B) are also known, then square(S) of a figure can be defined as half the product of the lengths of the sides and the sine of this known angle: S=½×A×B×sin(γ).

Concept of area

The concept of the area of ​​any geometric figure, in particular a triangle, will be associated with a figure such as a square. For the unit area of ​​any geometric figure we will take the area of ​​a square whose side is equal to one. For completeness, let us recall two basic properties for the concept of areas geometric shapes.

Property 1: If geometric figures are equal, then their areas are also equal.

Property 2: Any figure can be divided into several figures. Moreover, the area of ​​the original figure is equal to the sum of the areas of all its constituent figures.

Let's look at an example.

Example 1

Obviously, one of the sides of the triangle is a diagonal of a rectangle, one side of which has a length of $5$ (since there are $5$ cells), and the other is $6$ (since there are $6$ cells). Therefore, the area of ​​this triangle will be equal to half of such a rectangle. The area of ​​the rectangle is

Then the area of ​​the triangle is equal to

Answer: $15$.

Next, we will consider several methods for finding the areas of triangles, namely using the height and base, using Heron’s formula and area equilateral triangle.

How to find the area of ​​a triangle using its height and base

Theorem 1

The area of ​​a triangle can be found as half the product of the length of a side and the height to that side.

Mathematically it looks like this

$S=\frac(1)(2)αh$

where $a$ is the length of the side, $h$ is the height drawn to it.

Proof.

Consider a triangle $ABC$ in which $AC=α$. The height $BH$ is drawn to this side, which is equal to $h$. Let's build it up to the square $AXYC$ as in Figure 2.

The area of ​​rectangle $AXBH$ is $h\cdot AH$, and the area of ​​rectangle $HBYC$ is $h\cdot HC$. Then

$S_ABH=\frac(1)(2)h\cdot AH$, $S_CBH=\frac(1)(2)h\cdot HC$

Therefore, the required area of ​​the triangle, by property 2, is equal to

$S=S_ABH+S_CBH=\frac(1)(2)h\cdot AH+\frac(1)(2)h\cdot HC=\frac(1)(2)h\cdot (AH+HC)=\ frac(1)(2)αh$

The theorem has been proven.

Example 2

Find the area of ​​the triangle in the figure below if the cell has an area equal to one

The base of this triangle is equal to $9$ (since $9$ is $9$ squares). The height is also $9$. Then, by Theorem 1, we get

$S=\frac(1)(2)\cdot 9\cdot 9=40.5$

Answer: $40.5$.

Heron's formula

Theorem 2

If we are given three sides of a triangle $α$, $β$ and $γ$, then its area can be found as follows

$S=\sqrt(ρ(ρ-α)(ρ-β)(ρ-γ))$

here $ρ$ means the semi-perimeter of this triangle.

Proof.

Consider the following figure:

By the Pythagorean theorem, from the triangle $ABH$ we obtain

From the triangle $CBH$, according to the Pythagorean theorem, we have

$h^2=α^2-(β-x)^2$

$h^2=α^2-β^2+2βx-x^2$

From these two relations we obtain the equality

$γ^2-x^2=α^2-β^2+2βx-x^2$

$x=\frac(γ^2-α^2+β^2)(2β)$

$h^2=γ^2-(\frac(γ^2-α^2+β^2)(2β))^2$

$h^2=\frac((α^2-(γ-β)^2)((γ+β)^2-α^2))(4β^2)$

$h^2=\frac((α-γ+β)(α+γ-β)(γ+β-α)(γ+β+α))(4β^2)$

Since $ρ=\frac(α+β+γ)(2)$, then $α+β+γ=2ρ$, which means

$h^2=\frac(2ρ(2ρ-2γ)(2ρ-2β)(2ρ-2α))(4β^2)$

$h^2=\frac(4ρ(ρ-α)(ρ-β)(ρ-γ))(β^2 )$

$h=\sqrt(\frac(4ρ(ρ-α)(ρ-β)(ρ-γ))(β^2))$

$h=\frac(2)(β)\sqrt(ρ(ρ-α)(ρ-β)(ρ-γ))$

By Theorem 1, we get

$S=\frac(1)(2) βh=\frac(β)(2)\cdot \frac(2)(β) \sqrt(ρ(ρ-α)(ρ-β)(ρ-γ) )=\sqrt(ρ(ρ-α)(ρ-β)(ρ-γ))$

Triangle is one of the most common geometric shapes, which we get acquainted with already 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 absolutely in different ways, because in geometry there is more than one type of figures 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 all sides and angles are 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 geometric figures differs 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.

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.

So easy to calculate area isosceles triangle, since 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 an 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!