How to find the area of ​​a triangle knowing 2 legs. How to calculate the area of ​​a triangle? Formula of an isosceles triangle based on side and base

As you may remember from your school geometry curriculum, 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 uncomplicated options for 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 this 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.

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 of geometric figures.

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 the area of ​​an 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

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

A triangle is three points that do not lie on the same line and three segments that connect them. Otherwise, a triangle is a polygon that has exactly three angles.

These three points are called the vertices of the triangle, and the segments are called the sides of the triangle. The sides of the triangle form three angles at the vertices of the triangle.

An isosceles triangle is one in which two sides are equal. These sides are called lateral, the third side is called the base. In an isosceles triangle, the base angles are equal.

An equilateral or regular triangle is one in which all three sides are equal. All angles of an equilateral triangle are also equal and equal 60°.

The area of ​​an arbitrary triangle is calculated using the formulas: or

The area of ​​a right triangle is calculated by the formula:

The area of ​​a regular or equilateral triangle is calculated using the formulas: or or

Where a,b,c- sides of the triangle, h- height of the triangle, y- angle between sides, R- radius of the circumscribed circle, r- radius of the inscribed circle.

A triangle is the simplest geometric figure, which consists of three sides and three vertices. Due to its simplicity, the triangle has been used since ancient times to take various measurements, and today the figure can be useful for solving practical and everyday problems.

Features of a triangle

The figure has been used for calculations since ancient times, for example, land surveyors and astronomers operate with the properties of triangles to calculate areas and distances. It is easy to express the area of ​​any n-gon through the area of ​​this figure, and this property was used by ancient scientists to derive formulas for the areas of polygons. Constant work with triangles, especially the right triangle, became the basis for an entire branch of mathematics - trigonometry.

Triangle geometry

The properties of the geometric figure have been studied since ancient times: the earliest information about the triangle was found in Egyptian papyri from 4,000 years ago. Then the figure was studied in Ancient Greece and the greatest contributions to the geometry of the triangle were made by Euclid, Pythagoras and Heron. The study of the triangle never ceased, and in the 18th century, Leonhard Euler introduced the concept of the orthocenter of a figure and the Euler circle. At the turn of the 19th and 20th centuries, when it seemed that absolutely everything was known about the triangle, Frank Morley formulated the theorem on angle trisectors, and Waclaw Sierpinski proposed the fractal triangle.

There are several types of flat triangles that are familiar to us from school geometry courses:

• acute - all the corners of the figure are acute;
• obtuse - the figure has one obtuse angle (more than 90 degrees);
• rectangular - the figure contains one right angle equal to 90 degrees;
• isosceles - a triangle with two equal sides;
• equilateral - a triangle with all equal sides.
• There are all kinds of triangles in real life, and in some cases we may need to calculate the area of ​​a geometric figure.

Area of ​​a triangle

Area is an estimate of how much of the plane a figure encloses. The area of ​​a triangle can be found in six ways, using the sides, height, angles, radius of the inscribed or circumscribed circle, as well as using Heron's formula or calculating the double integral along the lines bounding the plane. The simplest formula for calculating the area of ​​a triangle is:

where a is the side of the triangle, h is its height.

However, in practice it is not always convenient for us to find the height of a geometric figure. The algorithm of our calculator allows you to calculate the area knowing:

• three sides;
• two sides and the angle between them;
• one side and two corners.

To determine the area through three sides, we use Heron's formula:

S = sqrt (p × (p-a) × (p-b) × (p-c)),

where p is the semi-perimeter of the triangle.

The area on two sides and an angle is calculated using the classic formula:

S = a × b × sin(alfa),

where alfa is the angle between sides a and b.

To determine the area in terms of one side and two angles, we use the relation that:

a / sin(alfa) = b / sin(beta) = c / sin(gamma)

Using a simple proportion, we determine the length of the second side, after which we calculate the area using the formula S = a × b × sin(alfa). This algorithm is fully automated and you only need to enter the specified variables and get the result. Let's look at a couple of examples.

Examples from life

Paving slabs

Let's say you want to pave the floor with triangular tiles, and to determine the amount of material needed, you need to know the area of ​​\u200b\u200bone tile and the area of ​​​​the floor. Suppose you need to process 6 square meters of surface using a tile whose dimensions are a = 20 cm, b = 21 cm, c = 29 cm. Obviously, to calculate the area of ​​a triangle, the calculator uses Heron’s formula and gives the result:

Thus, the area of ​​one tile element will be 0.021 square meters, and you will need 6/0.021 = 285 triangles for the floor improvement. The numbers 20, 21 and 29 form a Pythagorean triple - numbers that satisfy . And that's right, our calculator also calculated all the angles of the triangle, and the gamma angle is exactly 90 degrees.

School task

In a school problem, you need to find the area of ​​a triangle, knowing that side a = 5 cm, and angles alpha and beta are 30 and 50 degrees, respectively. To solve this problem manually, we would first find the value of side b using the proportion of the aspect ratio and the sines of the opposite angles, and then determine the area using the simple formula S = a × b × sin(alfa). Let's save time, enter the data into the calculator form and get an instant answer

When using the calculator, it is important to indicate the angles and sides correctly, otherwise the result will be incorrect.

Conclusion

The triangle is a unique figure that is found both in real life and in abstract calculations. Use our online calculator to determine the area of ​​triangles of any kind.

Area of ​​a triangle. In many geometry problems involving the calculation of areas, formulas for the area of ​​a triangle are used. There are several of them, here we will look at the main ones.Listing these formulas would be too simple and of no use. We will analyze the origin of the basic formulas, those that are used most often.

Before you read the derivation of the formulas, be sure to look at the article about.After studying the material, you can easily restore the formulas in your memory (if they suddenly “fly out” at the moment you need).

First formula

The diagonal of a parallelogram divides 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 ​​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 looks like:

The area of ​​a triangle is equal to half its area, which means:

*That is, if any two sides in a triangle and the angle between them are known, we can always calculate the area of ​​such a triangle.

Heron's formula (third)

This formula is difficult to derive and it is of no use to you. Look how beautiful she is, you can say that she herself is memorable.

*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 a triangle and the radius of the circle inscribed in it are known, then we can always find the area of ​​this triangle.

Formula five

Where R– radius of the circumscribed circle.

*If the three sides of a triangle and the radius of the circle circumscribed around it are known, then we can always find the area of ​​such a triangle.

The question arises: if three sides of a triangle are known, then isn’t it easier to find its area using Heron’s formula!

Yes, it can be easier, but not always, sometimes complexity arises. This involves extracting the root. In addition, these formulas are very convenient to use in problems where the area of ​​a triangle and its sides are given and you need to find the radius of the inscribed or circumscribed circle. Such tasks are available as part of the Unified State Examination.

Let's look at the formula separately:

It is a special case of the formula for the area of ​​a polygon into which a circle is inscribed:

Let's consider it using the example of a pentagon:

Let us connect the center of the circle with the vertices of this pentagon and lower perpendiculars from the center to its sides. We get five triangles, with the dropped perpendiculars being 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