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 sizes. The formulas are presented in the form of a picture, with explanations for their application or justification for their correctness. Also, a separate figure shows the correspondence between 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 given below, as well as additional special formulas that are valid 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- radius of the circle inscribed in the triangle
R- radius of the circle circumscribed around the triangle
h- height of the triangle lowered to the side
p- semi-perimeter of a triangle, 1/2 the sum of its sides (perimeter)
α - angle opposite to side a of the triangle
β - angle opposite to side b of the triangle
γ - angle opposite to side c of the triangle
h a, h b , h c- height of the triangle lowered to side a, b, c

Please note that the given notations correspond to the figure above, so that when solving a real geometry problem, it will be visually easier for you to substitute the correct values in the right places in the formula.

The area of the triangle is half the product of the height of the triangle and the length of the side by 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 you build each of them into 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 we drew, which gives us the previous formula
The area of an arbitrary triangle can be found through work half the radius of the circle inscribed in it by the sum of the lengths of all its sides(Formula 3), simply put, you need to multiply the semi-perimeter of the triangle by the radius of the inscribed circle (this is easier to remember)
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 through 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 semi-perimeter, 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 the circle circumscribed around it by the sines of each of its angles. (Formula 8)
If the length of one side and the values of two adjacent angles are known, then the area of the 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 the triangle is known (Formula 10), then the area of such a triangle is inversely proportional to the lengths of these heights, as according to Heron’s Formula
Formula 11 allows you to calculate area of a triangle based on the coordinates of its vertices, which are specified 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 may be in the region of negative values

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

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 the 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 conditions into the formula:
S = 1/2 * 5 * 6 * sin 60

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

Answer: 7.5 √3 (depending on the teacher’s requirements, you can probably leave 15 √3/2)

Task. Find the area of an equilateral triangle

Find the area of an equilateral triangle with side 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 = b = c, the formula for the area of an equilateral triangle takes the form:

S = √3 / 4 * a 2

S = √3 / 4 * 3 2

Answer: 9 √3 / 4.

Task. Change in area when changing the length of the sides

How many times will the area of the triangle increase if the sides are increased by 4 times?

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 will find the area of the given triangle, and then we will find the area of the triangle whose sides are four times larger. The ratio of the areas of these triangles will give us the answer to the problem.

Below we provide a textual explanation of the solution to the problem step by step. However, at the very end, this same solution is presented in a more convenient graphical form. Those interested can immediately go down the solutions.

To solve, we use Heron’s 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 first line of picture below)

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

S 2 = 1/4 sqrt((4a + 4b + 4c)(4b + 4c - 4a)(4a + 4c - 4b)(4a + 4b -4c))
(see 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 the general rules of 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

The square root of the number 256 is perfectly extracted, so let’s 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 fifth line of the picture below)

To answer the question asked in the problem, we just need to divide the area of the resulting triangle by the area of the original one.
Let us determine the area ratios by dividing the expressions by each other and reducing the resulting fraction.

The triangle is a figure familiar to everyone. And this despite the rich variety of its forms. Rectangular, equilateral, acute, isosceles, obtuse. Each of them is different in some way. But for anyone you need to find out the area of a triangle.

Formulas common to all triangles that use the lengths of sides or heights

The designations adopted in them: sides - a, b, c; heights on the corresponding sides on a, n in, n with.

1. The area of a triangle is calculated as the product of ½, a side and the height subtracted from it. S = ½ * a * n a. The formulas for the other two sides should be written similarly.

2. Heron's formula, in which the semi-perimeter appears (it is usually denoted by the small letter p, in contrast to the full perimeter). The semi-perimeter must be calculated as follows: add up all the sides and divide them by 2. The formula for the semi-perimeter is: p = (a+b+c) / 2. Then the equality for the area of the figure looks like this: S = √ (p * (p - a) * ( р - в) * (р - с)).

3. If you don’t want to use a semi-perimeter, then a formula that contains only the lengths of the sides will be useful: S = ¼ * √ ((a + b + c) * (b + c - a) * (a + c - c) * (a + b - c)). It is slightly longer than the previous one, but it will help out if you have forgotten how to find the semi-perimeter.

General formulas involving the angles of a triangle

Notations required to read the formulas: α, β, γ - angles. They lie opposite sides a, b, c, respectively.

1. According to it, half the product of two sides and the sine of the angle between them is equal to the area of the triangle. That is: S = ½ a * b * sin γ. The formulas for the other two cases should be written in a similar way.

2. The area of a triangle can be calculated from one side and three known angles. S = (a 2 * sin β * sin γ) / (2 sin α).

3. There is also a formula with one known side and two adjacent angles. It looks like this: S = c 2 / (2 (ctg α + ctg β)).

The last two formulas are not the simplest. It's quite difficult to remember them.

General formulas for the situation when the radii of inscribed or circumscribed circles are known

Additional designations: r, R - radii. The first is used for the radius of the inscribed circle. The second is for the one described.

1. The first formula by which the area of a triangle is calculated is related to the semi-perimeter. S = r * r. Another way to write it is: S = ½ r * (a + b + c).

2. In the second case, you will need to multiply all the sides of the triangle and divide them by quadruple the radius of the circumscribed circle. In literal expression it looks like this: S = (a * b * c) / (4R).

3. The third situation allows you to do without knowing the sides, but you will need the values of all three angles. S = 2 R 2 * sin α * sin β * sin γ.

Special case: right triangle

This is the simplest situation, since only the length of both legs is required. They are designated by the Latin letters a and b. The area of a right triangle is equal to half the area of the rectangle added to it.

Mathematically it looks like this: S = ½ a * b. It is the easiest to remember. Because it looks like the formula for the area of a rectangle, only a fraction appears, indicating half.

Special case: isosceles triangle

Since it has two equal sides, some formulas for its area look somewhat simplified. For example, Heron's formula, which calculates the area of an isosceles triangle, takes the following form:

S = ½ in √((a + ½ in)*(a - ½ in)).

If you transform it, it will become shorter. In this case, Heron’s formula for an isosceles triangle is written as follows:

S = ¼ in √(4 * a 2 - b 2).

The area formula looks somewhat simpler than for an arbitrary triangle if the sides and the angle between them are known. S = ½ a 2 * sin β.

Special case: equilateral triangle

Usually in problems the side about it is known or it can be found out in some way. Then the formula for finding the area of such a triangle is as follows:

S = (a 2 √3) / 4.

Problems to find the area if the triangle is depicted on checkered paper

The simplest situation is when a right triangle is drawn so that its legs coincide with the lines of the paper. Then you just need to count the number of cells that fit into the legs. Then multiply them and divide by two.

When the triangle is acute or obtuse, it needs to be drawn to a rectangle. Then the resulting figure will have 3 triangles. One is the one given in the problem. And the other two are auxiliary and rectangular. The areas of the last two need to be determined using the method described above. Then calculate the area of the rectangle and subtract from it those calculated for the auxiliary ones. The area of the triangle is determined.

The situation in which none of the sides of the triangle coincides with the lines of the paper turns out to be much more complicated. Then it needs to be inscribed in a rectangle so that the vertices of the original figure lie on its sides. In this case, there will be three auxiliary right triangles.

Example of a problem using Heron's formula

Condition. Some triangle has known sides. They are equal to 3, 5 and 6 cm. You need to find out its area.

Now you can calculate the area of the triangle using the above formula. Under the square root is the product of four numbers: 7, 4, 2 and 1. That is, the area is √(4 * 14) = 2 √(14).

If greater accuracy is not required, then you can take the square root of 14. It is equal to 3.74. Then the area will be 7.48.

Answer. S = 2 √14 cm 2 or 7.48 cm 2.

Example problem with right triangle

Condition. One leg of a right triangle is 31 cm larger than the second. You need to find out their lengths if the area of the triangle is 180 cm 2.
Solution. We will have to solve a system of two equations. The first is related to area. The second is with the ratio of the legs, which is given in the problem.
180 = ½ a * b;

a = b + 31.
First, the value of “a” must be substituted into the first equation. It turns out: 180 = ½ (in + 31) * in. There is only one unknown quantity, so it is easy to solve. After opening the brackets, the quadratic equation is obtained: 2 + 31 360 = 0. This gives two values for "in": 9 and - 40. The second number is not suitable as an answer, since the length of the side of a triangle cannot be a negative value.

It remains to calculate the second leg: add 31 to the resulting number. It turns out 40. These are the quantities sought in the problem.

Answer. The legs of the triangle are 9 and 40 cm.

Problem of finding a side through the area, side and angle of a triangle

Condition. The area of a certain triangle is 60 cm 2. It is necessary to calculate one of its sides if the second side is 15 cm and the angle between them is 30º.

Solution. Based on the accepted notation, the desired side is “a”, the known side is “b”, the given angle is “γ”. Then the area formula can be rewritten as follows:

60 = ½ a * 15 * sin 30º. Here the sine of 30 degrees is 0.5.

After transformations, “a” turns out to be equal to 60 / (0.5 * 0.5 * 15). That is 16.

Answer. The required side is 16 cm.

Problem about a square inscribed in a right triangle

Condition. The vertex of a square with a side of 24 cm coincides with the right angle of the triangle. The other two lie on the sides. The third belongs to the hypotenuse. The length of one of the legs is 42 cm. What is the area of the right triangle?

Solution. Consider two right triangles. The first one is the one specified in the task. The second one is based on the known leg of the original triangle. They are similar because they have a common angle and are formed by parallel lines.

Then the ratios of their legs are equal. The legs of the smaller triangle are equal to 24 cm (side of the square) and 18 cm (given leg 42 cm subtract the side of the square 24 cm). The corresponding legs of a large triangle are 42 cm and x cm. It is this “x” that is needed in order to calculate the area of the triangle.

18/42 = 24/x, that is, x = 24 * 42 / 18 = 56 (cm).

Then the area is equal to the product of 56 and 42 divided by two, that is, 1176 cm 2.

Answer. The required area is 1176 cm 2.

You can find over 10 formulas for calculating the area of a triangle on the Internet. Many of them are used in problems with known sides and angles of a triangle. However, there are a number of complex examples where, according to the conditions of the assignment, only one side and angles of a triangle are known, or the radius of a circumscribed or inscribed circle and one more characteristic. In such cases, a simple formula cannot be applied.

The formulas below will solve 95 percent of problems in which you need to find the area of a triangle.
Let's move on to consider common area formulas.
Consider the triangle shown in the figure below

In the figure and below in the formulas, the classical designations of all its characteristics are introduced.
a,b,c – sides of the triangle,
R – radius of the circumscribed circle,
r – radius of the inscribed circle,
h[b],h[a],h[c] – heights drawn in accordance with sides a,b,c.
alpha, beta, hamma – angles near the vertices.

Basic formulas for the area of a triangle

1. The area is equal to half the product of the side of the triangle and the height lowered to this side. In the language of formulas, this definition can be written as follows

Thus, if the side and height are known, then every student will find the area.
By the way, from this formula one can derive one useful relationship between heights

2. If we take into account that the height of a triangle through the adjacent side is expressed by the dependence

Then the first area formula is followed by the second ones of the same type

Look carefully at the formulas - they are easy to remember, since the work involves two sides and the angle between them. If we correctly designate the sides and angles of the triangle (as in the figure above), we will get two sides a, b and the angle is connected to the third With (hamma).

3. For the angles of a triangle, the relation is true

The dependence allows you to use the following formulas for the area of a triangle in calculations:

Examples of this dependence are extremely rare, but you must remember that there is such a formula.

4. If the side and two adjacent angles are known, then the area is found by the formula

5. The formula for area in terms of side and cotangent of adjacent angles is as follows

By rearranging the indexes you can get dependencies for other parties.

6. The area formula below is used in problems when the vertices of a triangle are specified on the plane by coordinates. In this case, the area is equal to half the determinant taken modulo.

7. Heron's formula used in examples with known sides of a triangle.
First find the semi-perimeter of the triangle

And then determine the area using the formula

or

It is quite often used in the code of calculator programs.

8. If all the heights of the triangle are known, then the area is determined by the formula

It is difficult to calculate on a calculator, but in the MathCad, Mathematica, Maple packages the area is “time two”.

9. The following formulas use the known radii of inscribed and circumscribed circles.

In particular, if the radius and sides of the triangle, or its perimeter, are known, then the area is calculated according to the formula

10. In examples where the sides and the radius or diameter of the circumscribed circle are given, the area is found using the formula

11. The following formula determines the area of a triangle in terms of the side and angles of the triangle.

And finally - special cases:
Area of a right triangle with legs a and b equal to half their product

Formula for the area of an equilateral (regular) triangle=

= one-fourth of the product of the square of the side and the root of three.

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 and the resulting area of the triangle.

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).

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 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. 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: