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 connection 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.

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 situations where 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 one 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 parentheses, 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.

Instructions

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

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

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

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

If the triangle is right-angled, that is, one of its angles is equal to 90°, and the sides forming it are legs, the third side is the hypotenuse. In 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

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.

Video on the topic

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.

Video on the topic

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 area of ​​the triangle we have obtained.

note

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

Sources:

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

The three points that uniquely define a triangle in the Cartesian coordinate system are its vertices. Knowing their position relative to each of the coordinate axes, you can calculate any parameters of this flat figure, including 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 - semiperimeter (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).

To determine the area of ​​a triangle, you can use different formulas. Of all the methods, the easiest and most frequently used is to multiply the height by the length of the base and then divide the result by two. However, this method is far from the only one. Below you can read how to find the area of ​​a triangle using different formulas.

Separately, we will look at ways to calculate the area of ​​specific types of triangles - rectangular, isosceles and equilateral. We accompany each formula with a short explanation that will help you understand its essence.

Universal methods for finding the area of ​​a triangle

The formulas below use special notation. We will decipher each of them:

• a, b, c – the lengths of the three sides of the figure we are considering;
• r is the radius of the circle that can be inscribed in our triangle;
• R is the radius of the circle that can be described around it;
• α is the magnitude of the angle formed by sides b and c;
• β is the magnitude of the angle between a and c;
• γ is the magnitude of the angle formed by sides a and b;
• h is the height of our triangle, lowered from angle α to side a;
• p – half the sum of sides a, b and c.

It is logically clear why you can find the area of ​​a triangle in this way. The triangle can easily be completed into a parallelogram, in which one side of the triangle will act as a diagonal. The area of ​​a parallelogram is found by multiplying the length of one of its sides by the value of the height drawn to it. The diagonal divides this conditional parallelogram into 2 identical triangles. Therefore, it is quite obvious that the area of ​​our original triangle must be equal to half the area of ​​this auxiliary parallelogram.

S=½ a b sin γ

According to this formula, the area of ​​a triangle is found by multiplying the lengths of its two sides, that is, a and b, by the sine of the angle formed by them. This formula is logically derived from the previous one. If we lower the height from angle β to side b, then, according to the properties of a right triangle, when we multiply the length of side a by the sine of angle γ, we obtain the height of the triangle, that is, h.

The area of ​​the figure in question is found by multiplying half the radius of the circle that can be inscribed in it by its perimeter. In other words, we find the product of the semi-perimeter and the radius of the mentioned circle.

S= a b c/4R

According to this formula, the value we need can be found by dividing the product of the sides of the figure by 4 radii of the circle described around it.

These formulas are universal, as they make it possible to determine the area of ​​any triangle (scalene, isosceles, equilateral, rectangular). This can be done using more complex calculations, which we will not dwell on in detail.

Areas of triangles with specific properties

How to find the area of ​​a right triangle? The peculiarity of this figure is that its two sides are simultaneously its heights. If a and b are legs, and c becomes the hypotenuse, then we find the area like this:

How to find the area of ​​an isosceles triangle? It has two sides with length a and one side with length b. Consequently, its area can be determined by dividing by 2 the product of the square of side a by the sine of angle γ.

How to find the area of ​​an equilateral triangle? In it, the length of all sides is equal to a, and the magnitude of all angles is α. Its height is equal to half the product of the length of side a and the square root of 3. To find the area of ​​a regular triangle, you need to multiply the square of side a by the square root of 3 and divide by 4.

Concept of area

The concept of the area of ​​any geometric figure, in particular a triangle, will be associated with a figure such as a square. For the unit area of ​​any geometric figure we will take the area of ​​a square whose side is equal to one. For completeness, let us recall two basic properties for the concept of areas of geometric figures.

Property 1: If geometric figures are equal, then their areas are also equal.

Property 2: Any figure can be divided into several figures. Moreover, the area of ​​the original figure is equal to the sum of the areas of all its constituent figures.

Let's look at an example.

Example 1

Obviously, one of the sides of the triangle is a diagonal of a rectangle, one side of which has a length of $5$ (since there are $5$ cells), and the other is $6$ (since there are $6$ cells). Therefore, the area of ​​this triangle will be equal to half of such a rectangle. The area of ​​the rectangle is

Then the area of ​​the triangle is equal to

Answer: $15$.

Next, we will consider several methods for finding the areas of triangles, namely using the height and base, using Heron’s formula and the area of ​​an equilateral triangle.

How to find the area of ​​a triangle using its height and base

Theorem 1

The area of ​​a triangle can be found as half the product of the length of a side and the height to that side.

Mathematically it looks like this

$S=\frac(1)(2)αh$

where $a$ is the length of the side, $h$ is the height drawn to it.

Proof.

Consider a triangle $ABC$ in which $AC=α$. The height $BH$ is drawn to this side, which is equal to $h$. Let's build it up to the square $AXYC$ as in Figure 2.

The area of ​​rectangle $AXBH$ is $h\cdot AH$, and the area of ​​rectangle $HBYC$ is $h\cdot HC$. Then

$S_ABH=\frac(1)(2)h\cdot AH$, $S_CBH=\frac(1)(2)h\cdot HC$

Therefore, the required area of ​​the triangle, by property 2, is equal to

$S=S_ABH+S_CBH=\frac(1)(2)h\cdot AH+\frac(1)(2)h\cdot HC=\frac(1)(2)h\cdot (AH+HC)=\ frac(1)(2)αh$

The theorem has been proven.

Example 2

Find the area of ​​the triangle in the figure below if the cell has an area equal to one

The base of this triangle is equal to $9$ (since $9$ is $9$ squares). The height is also $9$. Then, by Theorem 1, we get

$S=\frac(1)(2)\cdot 9\cdot 9=40.5$

Answer: $40.5$.

Heron's formula

Theorem 2

If we are given three sides of a triangle $α$, $β$ and $γ$, then its area can be found as follows

$S=\sqrt(ρ(ρ-α)(ρ-β)(ρ-γ))$

here $ρ$ means the semi-perimeter of this triangle.

Proof.

Consider the following figure:

By the Pythagorean theorem, from the triangle $ABH$ we obtain

From the triangle $CBH$, according to the Pythagorean theorem, we have

$h^2=α^2-(β-x)^2$

$h^2=α^2-β^2+2βx-x^2$

From these two relations we obtain the equality

$γ^2-x^2=α^2-β^2+2βx-x^2$

$x=\frac(γ^2-α^2+β^2)(2β)$

$h^2=γ^2-(\frac(γ^2-α^2+β^2)(2β))^2$

$h^2=\frac((α^2-(γ-β)^2)((γ+β)^2-α^2))(4β^2)$

$h^2=\frac((α-γ+β)(α+γ-β)(γ+β-α)(γ+β+α))(4β^2)$

Since $ρ=\frac(α+β+γ)(2)$, then $α+β+γ=2ρ$, which means

$h^2=\frac(2ρ(2ρ-2γ)(2ρ-2β)(2ρ-2α))(4β^2)$

$h^2=\frac(4ρ(ρ-α)(ρ-β)(ρ-γ))(β^2 )$

$h=\sqrt(\frac(4ρ(ρ-α)(ρ-β)(ρ-γ))(β^2))$

$h=\frac(2)(β)\sqrt(ρ(ρ-α)(ρ-β)(ρ-γ))$

By Theorem 1, we get

$S=\frac(1)(2) βh=\frac(β)(2)\cdot \frac(2)(β) \sqrt(ρ(ρ-α)(ρ-β)(ρ-γ) )=\sqrt(ρ(ρ-α)(ρ-β)(ρ-γ))$