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.

For building 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 with 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 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.

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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 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 It's easier to calculate required amount 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 the 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 extract Square root from 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.

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 and 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 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).

Triangle is one of the most common geometric shapes, which we already get acquainted with in primary 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 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 them existing types 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 next.

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 the adjacent equal angles) 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 legs 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 equilateral triangle.

Heron's formula

Last option to find 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.

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 is 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 of a right triangle, when we multiply the length of side a by the sine of 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 also 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 the area of ​​an 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.

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 the length of one of the sides of the triangle and three angles are known (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 different 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 is 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.

Video on the topic

Sources:

• 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).

As you can 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 have completely forgotten 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 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. right triangle, in which all three sides are 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: