Calculate the area of ​​a triangle online. How to calculate the area of ​​a triangle

Instructions

1. For two legs S = a * b/2, a, b – legs,

The second option for calculating area uses sines of known angles instead of cotangents. In this version square is equal to the square of the length of the known side, multiplied by the sines of each of the angles and divided by the double sine of these angles: S = A*A*sin(α)*sin(β)/(2*sin(α + β)). For example, for the same triangle with a known side of 15 cm, and adjacent to it corners at 40° and 60°, the area calculation will look like this: (15*15*sin(40)*sin(60))/(2*sin(40+60)) = 225*0.74511316*(-0.304810621)/( 2*(-0.506365641)) = -51.1016411/-1.01273128 = 50.4592305 square centimeters.

The version of calculating the area of ​​a triangle involves angles. The area will be equal to the square of the length of the known side, multiplied by the tangents of each of the angles and divided by double the sum of the tangents of these angles: S = A*A*tg(α)*tg(β)/2(tg(α)+tg(β) ). For example, for the triangle used in the previous steps with a side of 15 cm and adjacent corners at 40° and 60°, the area calculation will look like this: (15*15*tg(40)*tg(60))/(2*(tg(40)+tg(60)) = (225*(-1.11721493 )*0.320040389)/(2*(-1.11721493+0.320040389)) = -80.4496277/-1.59434908 = 50.4592305 square centimeters.

A triangle is the simplest polygon having three vertices and three sides. A triangle, one of whose angles is right, is called a right triangle. For right triangles, all formulas for general triangles are applicable. However, they can be modified, taking into account the properties of a right angle.

Instructions

Basic for finding area triangle through the base as follows: S = 1/2 * b * h, where b is the side triangle, and h – triangle. Height triangle is a perpendicular drawn from the vertex triangle to the line containing the opposite. For rectangular triangle height k b coincides with leg a. This way you will get the formula to calculate the area triangle with angle: S = 1/2 * a * b.

Consider. Let in a rectangular a = 3, b = 4. Then S = 1/2 * 3 * 4 = 6. Calculate square the same triangle, but now let only one side be known, b = 4. And the angle α, tan α = 3/4 is also known. Then, from the expression for the trigonometric function tangent α, express leg a: tg α = a/b => a = b * tan α. Substitute this value into the formula to calculate the area of ​​a rectangular triangle and we get: S = 1/2 * a * b = 1/2 *b^2 * tan α = 1/2 * 16 * 3/4 ​​= 6.

Consider as a special case the calculation of the area of ​​an isosceles rectangular triangle. An isosceles triangle is a triangle in which two sides are equal to each other. In the case of a rectangular triangle it turns out a = b. Write down the Pythagorean theorem for this case: c^2 = a^2 + b^2 = 2 * a^2. Next, substitute this value into the formula for calculating the area as follows: S = 1/2 * a * b = 1/2 * a^2 = 1/2 * (c^2 / 2) = c^2 / 4.

If the radii of the inscribed circle r and the circumcircle R are known, then square rectangular triangle is calculated by the formula S = r^2 + 2 * r * R. Let the radius of the inscribed circle in the triangle be r = 1, the radius of the circumscribed circle triangle circle R = 5/2. Then S = 1 + 2 * 1 * 5 / 2 = 6.

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Useful advice

The radius of a circle circumscribed around a right triangle is equal to half the hypotenuse: R = c / 2. The radius of a circle inscribed in a right triangle is found by the formula r = (a + b – c) / 2.

This is one of the simplest geometric figures, in which three segments connecting three points in pairs limit a part of the plane. Knowledge of some of the parameters of a triangle (lengths of sides, angles, radii of the inscribed or circumscribed circle, height, etc.) in various combinations allows us to calculate the area of ​​this limited section of the plane.

Instructions

If the lengths of the two sides of a triangle (A and B) and the magnitude of their angle (γ) are known, then the area (S) of the triangle will be equal to half the product of the lengths of the sides and the sine of the known angle: S=A∗B∗sin(γ)/2.

If the lengths of all three sides (A, B and C) in an arbitrary triangle are known, then to calculate its area (S) it is more convenient to introduce an additional variable - the semi-perimeter (p). This variable is calculated in half the sum of the lengths of all sides: p=(A+B+C)/2. Using this variable can be defined as the square root of the product of the semi-perimeter on this variable and the length of the sides: S=√(p∗(p-A)∗(p-B)∗(p-C)).

If, in addition to the lengths of all sides (A, B and C), the length of the radius (R) of a circle circumscribed near an arbitrary triangle is also known, then you can do without a semi-perimeter - the area (S) will be equal to the ratio of the product of the lengths of all sides to the quadruple radius of the circle: S=A ∗B∗C/(4∗R).

If the values ​​of all angles of a triangle (α, β and γ) and the length of one of its sides (A) are known, then the area (S) will be equal to the ratio of the product of the square of the length of the known side by the sines of two angles adjacent to it to the double sine of the opposite one angle: S=A²∗sin(β)∗sin(γ)/(2∗sin(α)).

If the values ​​of all angles of an arbitrary triangle (α, β and γ) and the radius (R) of the circumscribed circle are known, then the area (S) will be equal to twice the square of the radius and the sines of all angles: S=2∗R²∗sin(α)∗ sin(β)∗sin(γ).

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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
• volume of triangle

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 the 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 given below will allow you to 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.

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

• Rectangular.
• Obtuse.
• Acute angular.
• Versatile.
• Equilateral.
• Isosceles.

General formulas for calculating the area of ​​a triangle

Formula for the area of ​​a triangle based on length and height

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

Heron's formula

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

Formula for the area of ​​a triangle based on the angle and the length of the segment

S = (a*b*sin(α))/2,
where b,c is the length of the sides of the triangle, sin(α) is the sine of the angle between the two sides.

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

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

Formula for the area of ​​a triangle based on three sides and the radius of the circle circumscribed around it

S= (a*b*c)/4*R,
where a,b,c is the length of each side of the triangle, R is the radius of the circle circumscribed around the triangle.

Formula for the area of ​​a triangle using the Cartesian coordinates of points

Cartesian coordinates of points are coordinates in the xOy system, where x is the abscissa, y is the ordinate. The Cartesian coordinate system xOy on a plane is the mutually perpendicular numerical axes Ox and Oy with a common origin at point O. If the coordinates of points on this plane are given in the form A(x1, y1), B(x2, y2) and C(x3, y3 ), then you can calculate the area of ​​the triangle using the following formula, which is obtained from the vector product of two vectors.
S = |(x1 – x3) (y2 – y3) – (x2 – x3) (y1 – y3)|/2,
where || stands for module.

How to find the area of ​​a right triangle

A right triangle is a triangle with one angle measuring 90 degrees. A triangle can have only one such angle.

Formula for the area of ​​a right triangle on two sides

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

Formula for the area of ​​a right triangle based on the hypotenuse and acute angle

S = a*b*sin(α)/ 2,
where a, b are the legs of the triangle, and sin(α) is the sine of the angle at which the lines a, b intersect.

Formula for the area of ​​a right triangle based on the side and the opposite angle

S = a*b/2*tg(β),
where a, b are the legs of the triangle, tan(β) is the tangent of the angle at which the legs a, b are connected.

How to calculate the area of ​​an isosceles triangle

An isosceles triangle is one that has two equal sides. These sides are called the sides, and the other side is the base. To calculate the area of ​​an isosceles triangle, you can use one of the following formulas.

Basic formula for calculating the area of ​​an isosceles triangle

S=h*c/2,
where c is the base of the triangle, h is the height of the triangle lowered to the base.

Formula of an isosceles triangle based on side and base

S=(c/2)* √(a*a – c*c/4),
where c is the base of the triangle, a is the size of one of the sides of the isosceles triangle.

How to find the area of ​​an equilateral triangle

An equilateral triangle is a triangle in which all sides are equal. To calculate the area of ​​an equilateral triangle, you can use the following formula:
S = (√3*a*a)/4,
where a is the length of the side of the equilateral triangle.

The above formulas will allow you to calculate the required area of ​​the triangle. It is important to remember that to calculate the area of ​​triangles, you need to consider the type of triangle and the available data that can be used for the calculation.

As you may remember from your school geometry curriculum, a triangle is a figure formed from three segments connected by three points that do not lie on the same straight line. A triangle forms three angles, hence the name of the figure. The definition may be different. A triangle can also be called a polygon with three angles, the answer will also be correct. Triangles are divided according to the number of equal sides and the size of the angles in the figures. Thus, triangles are distinguished as isosceles, equilateral and scalene, as well as rectangular, acute and obtuse, respectively.

There are a lot of formulas for calculating the area of ​​a triangle. Choose how to find the area of ​​a triangle, i.e. Which formula to use is up to you. But it is worth noting only some of the notations that are used in many formulas for calculating the area of ​​a triangle. So, remember:

S is the area of ​​the triangle,

a, b, c are the sides of the triangle,

h is the height of the triangle,

R is the radius of the circumscribed circle,

p is the semi-perimeter.

Here are the basic notations that may be useful to you if you 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 this is. An isosceles triangle is a triangle in which two sides have the same length. These two sides are called lateral, the third side is called the base. Do not confuse an isosceles triangle with an equilateral triangle, i.e. a regular triangle with all three sides equal. In such a triangle there are no special tendencies to the angles, or rather to their size. However, the angles at the base in an isosceles triangle are equal, but different from the angle between equal sides. So, you already know the first and main formula; it remains to find out what other formulas for determining the area of ​​an isosceles triangle are known.

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. It has 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.