Instructions

Parties and angles are considered basic elements A. A triangle is completely defined by any of its following basic elements: either three sides, or one side and two angles, or two sides and an angle between them. For existence triangle given by three sides a, b, c, it is necessary and sufficient to satisfy the inequalities called inequalities triangle:
a+b > c,
a+c > b,
b+c > a.

To build triangle on three sides a, b, c, it is necessary from point C of the segment CB = a to draw a circle of radius b using a compass. Then, in the same way, draw a circle from point B with a radius equal to side c. Their intersection point A is the third vertex of the desired triangle ABC, where AB=c, CB=a, CA=b - sides triangle. The problem has , if the sides a, b, c, satisfy the inequalities triangle specified in step 1.

Area S constructed in this way triangle ABC with known sides a, b, c, is calculated using Heron's formula:
S=v(p(p-a)(p-b)(p-c)),
where a, b, c are sides triangle, p – semi-perimeter.
p = (a+b+c)/2

If a triangle is equilateral, that is, all its sides are equal (a=b=c).Area triangle calculated by the formula:
S=(a^2 v3)/4

If the triangle is right-angled, that is, one of its angles is equal to 90°, and the sides forming it are legs, the third side is the hypotenuse. In this case square equals the product of the legs divided by two.
S=ab/2

To find square triangle, you can use one of the many formulas. Choose a formula depending on what data is already known.

You will need

• knowledge of formulas for finding the area of ​​a triangle

Instructions

If you know the size of one of the sides and the value of the height lowered to this side from the angle opposite to it, then you can find the area using the following: S = a*h/2, where S is the area of ​​the triangle, a is one of the sides of the triangle, and h - height, to side a.

There is a known method for determining the area of ​​a triangle if its three sides are known. It is Heron's formula. To simplify its recording, an intermediate value is introduced - semi-perimeter: p = (a+b+c)/2, where a, b, c - . Then Heron's formula is as follows: S = (p(p-a)(p-b)(p-c))^½, ^ exponentiation.

Let's assume that you know one of the sides of a triangle and three angles. Then it is easy to find the area of ​​the triangle: S = a²sinα sinγ / (2sinβ), where β is the angle opposite to side a, and α and γ are angles adjacent to the side.

Video on the topic

Please note

The most general formula that is suitable for all cases is Heron's formula.

Sources:

Tip 3: How to find the area of ​​a triangle based on three sides

Finding the area of ​​a triangle is one of the most common problems in school planimetry. Knowing the three sides of a triangle is enough to determine the area of ​​any triangle. In special cases of equilateral triangles, it is enough to know the lengths of two and one side, respectively.

You will need

• lengths of sides of triangles, Heron's formula, cosine theorem

Instructions

Heron's formula for the area of ​​a triangle is as follows: S = sqrt(p(p-a)(p-b)(p-c)). If we write the semi-perimeter p, we get: S = sqrt(((a+b+c)/2)((b+c-a)/2)((a+c-b)/2)((a+b-c)/2) ) = (sqrt((a+b+c)(a+b-c)(a+c-b)(b+c-a)))/4.

You can derive a formula for the area of ​​a triangle from considerations, for example, by applying the cosine theorem.

By the cosine theorem, AC^2 = (AB^2)+(BC^2)-2*AB*BC*cos(ABC). Using the introduced notations, these can also be written in the form: b^2 = (a^2)+(c^2)-2a*c*cos(ABC). Hence, cos(ABC) = ((a^2)+(c^2)-(b^2))/(2*a*c)

The area of ​​a triangle is also found by the formula S = a*c*sin(ABC)/2 using two sides and the angle between them. The sine of angle ABC can be expressed through it using the basic trigonometric identity: sin(ABC) = sqrt(1-((cos(ABC))^2). By substituting the sine into the formula for the area and writing it out, you can arrive at the formula for the area of ​​the triangle ABC.

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To carry out repair work, it may be necessary to measure square walls This makes it easier to calculate the required amount of paint or wallpaper. For measurements, it is best to use a tape measure or measuring tape. Measurements should be taken after walls were leveled.

You will need

• -roulette;
• -ladder.

Instructions

To count square walls, you need to know the exact height of the ceilings, and also measure the length along the floor. This is done as follows: take a centimeter and lay it over the baseboard. Usually a centimeter is not enough for the entire length, so secure it in the corner, then unwind it to the maximum length. At this point, put a mark with a pencil, write down the result obtained and carry out further measurements in the same way, starting from the last measurement point.

Standard ceilings are 2 meters 80 centimeters, 3 meters and 3 meters 20 centimeters, depending on the house. If the house was built before the 50s, then most likely the actual height is slightly lower than indicated. If you are calculating square for repair work, then a small supply will not hurt - consider based on the standard. If you still need to know the real height, take measurements. The principle is similar to measuring length, but you will need a stepladder.

Multiply the resulting indicators - this is square yours walls. True, when painting or for painting it is necessary to subtract square door and window openings. To do this, lay a centimeter along the opening. If we are talking about a door that you are subsequently going to change, then proceed with the door frame removed, taking into account only square directly to the opening itself. The area of ​​the window is calculated along the perimeter of its frame. After square window and doorway calculated, subtract the result from the total resulting area of ​​the room.

Please note that measuring the length and width of the room is carried out by two people, this makes it easier to fix a centimeter or tape measure and, accordingly, get a more accurate result. Take the same measurement several times to make sure the numbers you get are accurate.

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Finding the volume of a triangle is truly a non-trivial task. The fact is that a triangle is a two-dimensional figure, i.e. it lies entirely in one plane, which means that it simply has no volume. Of course, you can't find something that doesn't exist. But let's not give up! We can accept the following assumption: the volume of a two-dimensional figure is its area. We will look for the area of ​​the triangle.

You will need

• sheet of paper, pencil, ruler, calculator

Instructions

Draw on a piece of paper using a ruler and pencil. By carefully examining the triangle, you can make sure that it really does not have a triangle, since it is drawn on a plane. Label the sides of the triangle: let one side be side "a", the other side "b", and the third side "c". Label the vertices of the triangle with the letters "A", "B" and "C".

Measure any side of the triangle with a ruler and write down the result. After this, restore a perpendicular to the measured side from the vertex opposite to it, such a perpendicular will be the height of the triangle. In the case shown in the figure, the perpendicular "h" is restored to side "c" from vertex "A". Measure the resulting height with a ruler and write down the measurement result.

It may be difficult for you to restore the exact perpendicular. In this case, you should use a different formula. Measure all sides of the triangle with a ruler. After this, calculate the semi-perimeter of the triangle "p" by adding the resulting lengths of the sides and dividing their sum in half. Having the value of the semi-perimeter at your disposal, you can use Heron's formula. To do this, you need to take the square root of the following: p(p-a)(p-b)(p-c).

You have obtained the required area of ​​the triangle. The problem of finding the volume of a triangle has not been solved, but as mentioned above, the volume is not. You can find a volume that is essentially a triangle in the three-dimensional world. If we imagine that our original triangle has become a three-dimensional pyramid, then the volume of such a pyramid will be the product of the length of its base by the area of ​​the triangle we have obtained.

Please note

The more carefully you measure, the more accurate your calculations will be.

Sources:

• Calculator “Everything to everything” - a portal for reference values
• triangle volume in 2019

The three points that uniquely define a triangle in the Cartesian coordinate system are its vertices. Knowing their position relative to each of the coordinate axes, you can calculate any parameters of this flat figure, including those limited by its perimeter square. This can be done in several ways.

Instructions

Use Heron's formula to calculate area triangle. It involves the dimensions of the three sides of the figure, so start your calculations with . The length of each side must be equal to the root of the sum of the squares of the lengths of its projections onto the coordinate axes. If we denote the coordinates A(X₁,Y₁,Z₁), B(X₂,Y₂,Z₂) and C(X₃,Y₃,Z₃), the lengths of their sides can be expressed as follows: AB = √((X₁-X₂)² + (Y₁ -Y₂)² + (Z₁-Z₂)²), BC = √((X₂-X₃)² + (Y₂-Y₃)² + (Z₂-Z₃)²), AC = √((X₁-X₃)² + (Y₁-Y₃)² + (Z₁-Z₃)²).

To simplify calculations, introduce an auxiliary variable - semi-perimeter (P). From the fact that this is half the sum of the lengths of all sides: P = ½*(AB+BC+AC) = ½*(√((X₁-X₂)² + (Y₁-Y₂)² + (Z₁-Z₂)²) + √ ((X₂-X₃)² + (Y₂-Y₃)² + (Z₂-Z₃)²) + √((X₁-X₃)² + (Y₁-Y₃)² + (Z₁-Z₃)²).

A triangle is a geometric figure that consists of three straight lines connecting at points that do not lie on the same straight line. The connecting 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 the 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.

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 be only 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:

Some of the problems in geometry, or more precisely, in planimetry, require finding the area of ​​​​a given figure. The area of ​​any figure can be both the final goal of the problem and an intermediate calculation necessary for substitution into a more complex formula. Often in such problems you are asked to find the area of ​​a triangle. Initial data may vary. In some cases, some side of the triangle and the value of the height drawn to it are known, in others - the perimeter of the triangle, and so on.

Suppose you are asked to find the area of ​​a triangle if three sides are known. To find the area of ​​such a triangle, Heron's formula is used. To determine the area using this formula, you must first calculate the semi-perimeter of the triangle (n). Knowing the meanings of all three sides, this is easy to do. You need to sum up all the sides of the triangle - this will be its perimeter, and then divide the resulting value by two. After this, it is necessary to subtract in turn the lengths of each of the three given sides of the triangle from the value of the semi-perimeter, that is, subtract a from n, then subtract b from n, and finally subtract c from n.

The resulting three differences should be multiplied among themselves and this product again multiplied by the value of the semi-perimeter. Having carried out all the above steps and obtained the result of multiplication, you need to extract the square root from this result. The number that comes out after taking the square root will be the area of ​​the given triangle. If we write it down briefly, the formula for the area of ​​a triangle will be: area (S) = square root of (n*(n-a) *(n-b) *(n-c)). As can be understood from the formula, the problem of finding a triangle with known side values ​​is very easy to solve.

For example, how to find the area of ​​a triangle if 3 sides are known: side a is 3 centimeters, side b is 4 centimeters and side c is 2 centimeters. The perimeter of this triangle will be equal to a + b + c = 3 centimeters + 4 centimeters + 2 centimeters = 9 cm. So the semi-perimeter is 9: 2 = 4.5 centimeters We get: S = square root of (4.5 centimeters * (4 .5 centimeters - 3 centimeters) * (4.5 centimeters - 4 centimeters) * (4.5 centimeters - 2 centimeters)) = 2.9 square centimeters

What if the values ​​of the sides are not only known, but it is also indicated that they are equal according to the conditions of the problem? In this case, how to find the area of ​​a triangle if all sides are known and they are also equal? You can, of course, also calculate it using Heron’s formula discussed above, but why unnecessary calculations if for such a triangle another formula has been derived, which is much simpler than Heron’s formula. Using this formula, you must first calculate the square root of the number 3, then raise the value of the length of the side of the triangle to the second power, multiply this value to the second power with the root of the number 3 and divide the resulting product by the number 4. You will get the area of ​​the given triangle . When written, this formula looks like this: S=(a^2*root(3)) /4

Let there be a triangle with equal side lengths equal to 3 centimeters. Using this formula, you can get the area of ​​such a triangle: S=(3^2*root(3)) /4=3.9 square centimeters. To check whether the area of ​​a particular triangle has been calculated correctly or not, you can carry out additional calculations using Heron’s formula and compare the results obtained.

Semi-perimeter (n) = (3+3+3) /2 = 4.5 centimeters. According to Heron's formula, we find: S = square root of (4.5 centimeters * (4.5 centimeters - 3 centimeters) * (4.5 centimeters - 3 centimeters) * (4.5 centimeters - 3 centimeters)) = 3 .9 square centimeters. Both area values, found using different formulas, coincide. This means that the area of ​​the triangle is determined correctly. When solving any other problems, you should take into account the data in the condition and use the formula corresponding to these data.

A triangle is one of the most common geometric shapes, which we become familiar with 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 in completely different ways, because in geometry there is more than one type of figure 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 of geometric figure is distinguished by the fact 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.