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

S = ½ * a * h,

where:
S is the 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. In this case, the area of ​​\u200b\u200bthe triangle will turn out in the corresponding "" units.

Example.
On one of the sides 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 you know the lengths of any two sides of a scalene triangle and the angle between them, 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 constructions 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 are the lengths of the sides of the triangle,
р – 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 is the radius of the inscribed circle (p is the semi-perimeter).

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

where: R is the radius of the circumscribed circle.

If the length of one of the sides of the triangle and three angles is 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 are the values ​​of the remaining two angles of the triangle.

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

Instruction

If from the condition we know the values ​​of the 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 the 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, further on the table we determine the angle itself.
Finding an angle? = 180°-?-?.
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 semiperimeter p = (a+b+c)/2

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

If we know the values ​​of the sides triangle a, b, c and the radius of the circumscribed near the given 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 semiperimeter, 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 is triangle.
If triangle ABC is a right triangle, 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.

Related videos

Sources:

• how to measure the area of ​​a triangle

Tip 3: How to find the area of ​​a triangle if you know the angle

Knowing only one parameter (the value of 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. A few of the most commonly used formulas are listed below.

Instruction

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

Area of ​​a triangle - formulas and examples of problem solving

Below are formulas for finding the area of ​​an arbitrary triangle which are suitable for finding the area of ​​any triangle, regardless of its properties, angles or dimensions. The formulas are presented in the form of a picture, here are explanations for the application or justification of their correctness. Also, a separate figure shows the correspondence of the letter symbols in the formulas and the graphic symbols in the drawing.

Note . If the triangle has special properties (isosceles, rectangular, equilateral), you can use the formulas below, as well as additionally special formulas that are true only for triangles with these properties:

• "Formulas for the area of ​​an equilateral triangle"

Triangle area formulas

Explanations for formulas:
a, b, c- the lengths of the sides of the triangle whose area we want to find
r- the radius of the circle inscribed in the triangle
R- the radius of the circumscribed circle around the triangle
h- the height of the triangle, lowered to the side
p- semiperimeter of a triangle, 1/2 the sum of its sides (perimeter)
α - the angle opposite side a of the triangle
β - the angle opposite side b of the triangle
γ - the angle opposite side c of the triangle
h a, h b , h c- the height of the triangle, lowered to the side a, b, c

Please note that the notation given corresponds to the figure above, so that when solving a real problem in geometry, it would be easier for you to visually substitute in right places formulas correct values.

• The area of ​​the triangle is half the product of the height of a triangle and the length of the side on which this height is lowered(Formula 1). The correctness of this formula can be understood logically. The height lowered to the base will split an arbitrary triangle into two rectangular ones. If we complete each of them to a rectangle with dimensions b and h, then, obviously, the area of ​​these triangles will be equal to exactly half the area of ​​the rectangle (Spr = bh)
• The area of ​​the triangle is half the product of its two sides and the sine of the angle between them(Formula 2) (see an example of solving a problem using this formula below). Despite the fact that it seems different from the previous one, it can easily be transformed into it. If we lower the height from angle B to side b, it turns out that the product of side a and the sine of angle γ, according to the properties of the sine in a right triangle, is equal to the height of the triangle drawn by us, which will give us the previous formula
• The area of ​​an arbitrary triangle can be found through work half the radius of a circle inscribed in it by the sum of the lengths of all its sides(Formula 3), in other words, you need to multiply the half-perimeter of the triangle by the radius of the inscribed circle (it's easier to remember this way)
• The area of ​​an arbitrary triangle can be found by dividing the product of all its sides by 4 radii of the circle circumscribed around it (Formula 4)
• Formula 5 is finding the area of ​​a triangle in terms of the lengths of its sides and its semi-perimeter (half the sum of all its sides)
• Heron's formula(6) is a representation of the same formula without using the concept of a semiperimeter, only through the lengths of the sides
• The area of ​​an arbitrary triangle is equal to the product of the square of the side of the triangle and the sines of the angles adjacent to this side divided by the double sine of the angle opposite to this side (Formula 7)
• The area of ​​an arbitrary triangle can be found as the product of two squares of a circle circumscribed around it and the sines of each of its angles. (Formula 8)
• If the length of one side and the magnitude of the two angles adjacent to it are known, then the area of ​​\u200b\u200bthe triangle can be found as the square of this side, divided by the double sum of the cotangents of these angles (Formula 9)
• If only the length of each of the heights of a triangle is known (Formula 10), then the area of ​​such a triangle is inversely proportional to the lengths of these heights, as by Heron's Formula
• Formula 11 allows you to calculate area of ​​a triangle according to the coordinates of its vertices, which are given as (x;y) values ​​for each of the vertices. Please note that the resulting value must be taken modulo, since the coordinates of individual (or even all) vertices can be in the area of ​​negative values

Note. The following are examples of solving problems in geometry to find the area of ​​a triangle. If you need to solve a problem in geometry, similar to which is not here - write about it in the forum. In solutions, instead of the symbol " Square root" the sqrt() function can be used, in which sqrt is the square root symbol, and the radical expression is indicated in brackets.Sometimes the symbol can be used for simple radical expressions √

A task. Find the area given two sides and the angle between them

The sides of the triangle are 5 and 6 cm. The angle between them is 60 degrees. Find the area of ​​a triangle.

Solution.

To solve this problem, we use formula number two from the theoretical part of the lesson.
The area of ​​a triangle can be found through the lengths of two sides and the sine of the angle between them and will be equal to
S=1/2 ab sin γ

Since we have all the necessary data for the solution (according to the formula), we can only substitute the values ​​from the problem statement into the formula:
S=1/2*5*6*sin60

In the table of values trigonometric functions find and substitute in the expression the value of the sine 60 degrees. It will be equal to the root of three by two.
S = 15 √3 / 2

Answer: 7.5 √3 (depending on the requirements of the teacher, it is probably possible to leave 15 √3/2)

A task. Find the area of ​​an equilateral triangle

Find the area of ​​an equilateral triangle with a side of 3 cm.

Solution .

The area of ​​a triangle can be found using Heron's formula:

S = 1/4 sqrt((a + b + c)(b + c - a)(a + c - b)(a + b -c))

Since a \u003d b \u003d c, the formula for the area of ​​​​an equilateral triangle will take the form:

S = √3 / 4 * a2

S = √3 / 4 * 3 2

Answer: 9 √3 / 4.

A task. Change in area when changing the length of the sides

How many times will the area of ​​a triangle increase if the sides are quadrupled?

Solution.

Since the dimensions of the sides of the triangle are unknown to us, to solve the problem we will assume that the lengths of the sides are respectively equal to arbitrary numbers a, b, c. Then, in order to answer the question of the problem, we find the area of ​​this triangle, and then we find the area of ​​a triangle whose sides are four times larger. The ratio of the areas of these triangles will give us the answer to the problem.

Next, we give a textual explanation of the solution of the problem in steps. However, at the very end, the same solution is presented in a graphical form that is more convenient for perception. Those who wish can immediately drop down the solution.

To solve, we use the Heron formula (see above in the theoretical part of the lesson). It looks like this:

S = 1/4 sqrt((a + b + c)(b + c - a)(a + c - b)(a + b -c))
(see the first line of the picture below)

The lengths of the sides of an arbitrary triangle are given by the variables a, b, c.
If the sides are increased by 4 times, then the area of ​​\u200b\u200bthe new triangle c will be:

S 2 = 1/4 sqrt((4a + 4b + 4c)(4b + 4c - 4a)(4a + 4c - 4b)(4a + 4b -4c))
(see the second line in the picture below)

As you can see, 4 is a common factor that can be taken out of brackets from all four expressions according to general rules mathematics.
Then

S 2 = 1/4 sqrt(4 * 4 * 4 * 4 (a + b + c)(b + c - a)(a + c - b)(a + b -c)) - on the third line of the picture
S 2 = 1/4 sqrt(256 (a + b + c)(b + c - a)(a + c - b)(a + b -c)) - fourth line

From the number 256, the square root is perfectly extracted, so we will take it out from under the root
S 2 = 16 * 1/4 sqrt((a + b + c)(b + c - a)(a + c - b)(a + b -c))
S 2 = 4 sqrt((a + b + c)(b + c - a)(a + c - b)(a + b -c))
(see the fifth line of the figure below)

To answer the question posed in the problem, it is enough for us to divide the area of ​​the resulting triangle by the area of ​​the original one.
We determine the area ratios by dividing the expressions into each other and reducing the resulting fraction.

Instruction

Parties and corners are considered basic elements a. A triangle is completely defined by any of the following basic elements: either three sides, or one side and two angles, or two sides and an angle between them. For existence triangle defined by three sides a, b, c, it is necessary and sufficient that 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 the point C of the segment CB=a how to draw a circle of radius b with a compass. Then, similarly, 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.

The area of ​​S constructed in this way triangle ABC with known parties a, b, c, is calculated by Heron's formula:
S=v(p(p-a)(p-b)(p-c)),
where a, b, c are sides triangle, p is the semiperimeter.
p = (a+b+c)/2

If the 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 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 the formula depending on what data is already known.

You will need

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

Instruction

If you know the value of one of the sides and the value of the height lowered to this side from the opposite corner, 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 way to determine the area of ​​a triangle if three of its sides are known. She is Heron's formula. To simplify its recording, an intermediate value is introduced - a semi-perimeter: p \u003d (a + b + c) / 2, where a, b, c - . Then Heron's formula is as follows: S = (p(p-a)(p-b)(p-c))^1, ^ exponentiation.

Suppose 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 side a, and α and γ are angles adjacent to the side.

Related videos

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 given three sides

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

You will need

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

Instruction

Heron's formula for the area of ​​a triangle is as follows: S = sqrt(p(p-a)(p-b)(p-c)). If you paint the semiperimeter p, then you 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 also derive a formula for the area of ​​a triangle from considerations, for example, by applying the cosine theorem.

By the law of cosines, AC^2 = (AB^2)+(BC^2)-2*AB*BC*cos(ABC). Using the introduced notation, these can also be 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 through two sides and the angle between them. The sine of angle ABC can be expressed in terms of it using the basic trigonometric identity: sin(ABC) = sqrt(1-((cos(ABC))^2) By substituting the sine into the formula for the area and painting it, we can arrive at the formula for the area of ​​the triangle ABC.

Related videos

For repair work may need to be measured square walls. It is easier to calculate the required amount of paint or wallpaper. For measurements, it is best to use a tape measure or centimeter tape. Measurements should be taken after walls have been aligned.

You will need

• -roulette;
• -ladder.

Instruction

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

Standard ceilings in typical - 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 margin 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 figures - this is square your walls. True, for painting work or for it is necessary to subtract square door and window openings. To do this, lay a centimeter along the opening. If a we are talking about the door that you are subsequently going to change, then carry out with the door frame removed, considering only square the opening itself. The window area is calculated along the perimeter of its frame. After square window and doorway calculated, subtract the result from the total room area obtained.

Please note that measurements of the length and width of the room are carried out together, it is 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.

Related videos

Finding the volume of a triangle is indeed 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 make the following assumption - the volume of a two-dimensional figure, this is its area. We are looking for the area of ​​the triangle.

You will need

• sheet of paper, pencil, ruler, calculator

Instruction

Draw on a sheet of paper with a ruler and pencil. By carefully examining the triangle, you can make sure that it really does not have, 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 that, restore the perpendicular to the measured side from the opposite vertex, such a perpendicular will be the height of the triangle. In the case shown in the figure, perpendicular "h" is restored to side "c" from vertex "A". Measure the resulting height with a ruler and record the result of the measurement.

It may happen that you find it difficult to restore the exact perpendicular. In this case, you should use a different formula. Measure all sides of the triangle with a ruler. After that, calculate the half-perimeter of the triangle "p" by adding the resulting lengths of the sides and dividing their sum in half. Having at your disposal the value of the semi-perimeter, you can use the Heron 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 desired 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 volume which is essentially a triangle in the 3D 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 and the area of ​​\u200b\u200bthe triangle we received.

note

Calculations will be more accurate the more carefully you take measurements.

Sources:

• All-to-All Calculator - Reference Portal
• 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 the one limited by its perimeter square. This can be done in several ways.

Instruction

Use Heron's formula to calculate area triangle. It involves the dimensions of the three sides of the figure, so start the 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 on 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 the calculations, enter an auxiliary variable - the semi-perimeter (P). From that this is half the sum of the lengths of all sides: P \u003d ½ * (AB + BC + AC) \u003d 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).

As you may remember from the school curriculum in geometry, a triangle is a figure formed from three segments connected by three points that do not lie on one straight line. The 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 corners, the answer will be just as true. Triangles are divided according to the number of equal sides and the size of the angles in the figures. So distinguish such triangles as isosceles, equilateral and scalene, as well as rectangular, acute-angled and obtuse-angled, respectively.

There are many formulas for calculating the area of ​​a triangle. Choose how to find the area of ​​a triangle, i.e. which formula to use, only you. But it is worth noting only some of the notation that is 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 come in handy if you have completely forgotten the course of geometry. The most understandable and not complicated options for calculating the unknown and mysterious area of ​​\u200b\u200bthe triangle will be given below. It is not difficult and will come in handy both for your household needs and for helping your children. Let's remember how to calculate the area of ​​a triangle as easy as shelling pears:

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 with one angle equal to 90 degrees (therefore called a right triangle). 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 the angles of any one triangle is 180 degrees. It turns out that 2 other angles should divide the remaining 90 degrees among themselves, for example, 70 and 20, 45 and 45, etc. So, you remembered the main thing, it remains to find out how to find the area right triangle. Imagine that we have such a right triangle in front of us, and we need to find its area S.

1. The easiest way to determine the area of ​​a right triangle is calculated using the following formula:

In our case, the area of ​​a right triangle is: S = 2.5 cm * 3 cm / 2 = 3.75 sq. cm.

In principle, it is no longer necessary to verify the area of ​​a triangle in other ways, since in everyday life it will come in handy and only this one will help. 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 areas of a right-angled triangle that you can still use:

We decided to use the first formula and with small blots (we drew in a notebook and used old ruler and a protractor), but we got the right calculation:

S \u003d (2.5 * 2.5) / (2 * 0.9) \u003d (3 * 3) / (2 * 1.2). We got such results 3.6=3.7, but taking into account the cell shift, we can forgive this nuance.

Isosceles triangle and its area.

If you are faced with the task of calculating the formula of an isosceles triangle, then the easiest way is to use the main one and, as is considered the classical formula for the area of ​​a triangle.

But first, before we find the area of ​​an isosceles triangle, we will find out what kind of figure it is. An isosceles triangle is a triangle whose two sides are the same length. These two sides are called the sides, the third side is called the base. Do not confuse an isosceles triangle with an equilateral one, i.e. an equilateral 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:

A triangle is a geometric figure that consists of three lines that meet at points that do not lie on the same line. The connection points of the lines are the vertices of the triangle, which are denoted with Latin letters(for example, A, B, C). The connecting straight lines of a triangle are called segments, which are also usually denoted in Latin letters. There are the following types of triangles:

• Rectangular.
• obtuse.
• Acute-angled.
• Versatile.
• Equilateral.
• Isosceles.

General formulas for calculating the area of ​​a triangle

Triangle area formula for length and height

S=a*h/2,
where a is the length of the side of the triangle whose area is to be found, h is the length of the height drawn to the base.

Heron's formula

S=√p*(p-a)*(p-b)*(p-c),
where √ is the square root, p is the semiperimeter of the triangle, a,b,c is the length of each side of the triangle. The semiperimeter of a triangle can be calculated using the formula p=(a+b+c)/2.

The formula for the area of ​​a triangle in terms of the angle and 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.

The formula for the area of ​​a triangle given the radius of the inscribed circle and three sides

S=p*r,
where p is the semiperimeter of the triangle whose area is to be found, r is the radius of the circle inscribed in this triangle.

The formula for the area of ​​a triangle given three sides and the radius of a 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 circumscribed circle around the triangle.

The formula for the area of ​​a triangle in Cartesian coordinates of points

The Cartesian coordinates of points are coordinates in the xOy system, where x is the abscissa and y is the ordinate. The Cartesian coordinate system xOy on the plane is called the mutually perpendicular numerical axes Ox and Oy with a common origin at the point O. If the coordinates of the 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 ​​a triangle using the following formula, which is obtained from the cross 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 that has one angle of 90 degrees. A triangle can have only one such angle.

The formula for the area of ​​a right triangle on two legs

S=a*b/2,
where a,b is the length of the legs. The legs are called the sides adjacent to the right angle.

The formula for the area of ​​a right triangle given 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.

The formula for the area of ​​a right triangle by leg and opposite angle

S = a*b/2*tg(β),
where a, b are the legs of the triangle, tg(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. You can use one of the following formulas to calculate the area of ​​an isosceles triangle.

The 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 on the lateral side and base

S=(c/2)* √(a*a – c*c/4),
where c is the base of the triangle, a is the value 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 an equilateral triangle.

The above formulas will allow you to calculate the required area of ​​\u200b\u200bthe triangle. It is important to remember that in order to calculate the spacing of triangles, you need to take into account the type of triangle and the available data that can be used for the calculation.