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

S = ½ * a * h,

Where:
S – area of ​​the triangle,
a is the length of its side,
h is the height lowered to this side.

Side length and height must be presented in the same units of measurement. In this case, the area of ​​the triangle will be obtained in the corresponding “ ” units.

Example.
On one side of a scalene triangle 20 cm long, a perpendicular from the opposite vertex 10 cm long is lowered.
The area of ​​the triangle is required.
Solution.
S = ½ * 20 * 10 = 100 (cm²).

If the lengths of any two sides of a scalene triangle and the angle between them are known, then use the formula:

S = ½ * a * b * sinγ,

where: a, b are the lengths of two arbitrary sides, and γ is the angle between them.

In practice, for example, when measuring land plots, the use of the above formulas is sometimes difficult, since it requires additional construction and measurement of angles.

If you know the lengths of all three sides of a scalene triangle, then use Heron's formula:

S = √(p(p-a)(p-b)(p-c)),

a, b, c – lengths of the sides of the triangle,
p – semi-perimeter: p = (a+b+c)/2.

If, in addition to the lengths of all sides, the radius of the circle inscribed in the triangle is known, then use the following compact formula:

where: r – radius of the inscribed circle (р – semi-perimeter).

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

where: R – radius of the circumscribed circle.

If the length of one of the sides of the triangle and three angles are known (in principle, two are enough - the value of the third is calculated from the equality of the sum of the three angles of the triangle - 180º), then use the formula:

S = (a² * sinβ * sinγ)/2sinα,

where α is the value of the angle opposite to side a;
β, γ – values ​​of the remaining two angles of the triangle.

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

Instructions

If from the condition we know the values ​​of two sides b, c and the angle formed by them?, then the area triangle ABC is found by the formula:
S = (bcsin?)/2.

If from the condition we know the values ​​of two sides a, b and the angle not formed by them?, then the area triangle ABC is found as follows:
Finding the angle?, sin? = bsin?/a, then use the table to determine the angle itself.
Finding the angle?, ? = 180°-?-?.
We find the area itself S = (absin?)/2.

If from the condition we know the values ​​of only three sides triangle a, b and c, then the area triangle ABC is found by the formula:
S = v(p(p-a)(p-b)(p-c)), where p is the semi-perimeter p = (a+b+c)/2

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

If we know the meanings of the sides triangle a, b, c and the radius described about this triangle R, then the area of ​​this triangle ABC is determined by the formula:
S = abc/4R.
If three sides a, b, c and the radius of the inscribed in are known, then the area triangle ABC is found by the formula:
S = pr, where p is the semi-perimeter, p = (a+b+c)/2.

If ABC is equilateral, then the area is found by the formula:
S = (a^2v3)/4.
If triangle ABC is isosceles, then the area is determined by the formula:
S = (cv(4a^2-c^2))/4, where c – triangle.
If triangle ABC is right-angled, then the area is determined by the formula:
S = ab/2, where a and b are legs triangle.
If triangle ABC is a right isosceles triangle, then the area is determined by the formula:
S = c^2/4 = a^2/2, where c is the hypotenuse triangle, a=b – leg.

Video on the topic

Sources:

• how to measure the area of ​​a triangle

Tip 3: How to find the area of ​​a triangle if the angle is known

Knowing just one parameter (the angle) is not enough to find the area tre square . If there are any additional dimensions, then to determine the area you can choose one of the formulas in which the angle value is used as one of the known variables. Several of the most frequently used formulas are given below.

Instructions

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

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 area 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 is 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

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

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

You will need

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

Instructions

Heron's formula for the area of ​​a triangle is as follows: S = sqrt(p(p-a)(p-b)(p-c)). If we write the semi-perimeter p, we get: S = sqrt(((a+b+c)/2)((b+c-a)/2)((a+c-b)/2)((a+b-c)/2) ) = (sqrt((a+b+c)(a+b-c)(a+c-b)(b+c-a)))/4.

You can derive a formula for the area of ​​a triangle from considerations, for example, by applying the cosine theorem.

By the cosine theorem, AC^2 = (AB^2)+(BC^2)-2*AB*BC*cos(ABC). Using the introduced notations, these can also be written in the form: b^2 = (a^2)+(c^2)-2a*c*cos(ABC). Hence, cos(ABC) = ((a^2)+(c^2)-(b^2))/(2*a*c)

The area of ​​a triangle is also found by the formula S = a*c*sin(ABC)/2 using two sides and the angle between them. The sine of angle ABC can be expressed through it using the basic trigonometric identity: sin(ABC) = sqrt(1-((cos(ABC))^2). By substituting the sine into the formula for the area and writing it out, you can arrive at the formula for the area of ​​the triangle ABC.

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To carry out repair work, it may be necessary to measure square walls It's easier to calculate required quantity 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 it; such a perpendicular will be the height of the triangle. In the case shown in the figure, the perpendicular "h" is restored to side "c" from vertex "A". Measure the resulting height with a ruler and write down the measurement result.

It may be difficult for you to restore the exact perpendicular. In this case, you should use a different formula. Measure all sides of the triangle with a ruler. After this, calculate the semi-perimeter of the triangle “p” by adding the resulting lengths of the sides and dividing their sum in half. Having the value of the semi-perimeter at your disposal, you can use Heron's formula. To do this you need to extract square root from the following: p(p-a)(p-b)(p-c).

You have obtained the required area of ​​the triangle. The problem of finding the volume of a triangle has not been solved, but as mentioned above, the volume is not . You can find a volume that is essentially a triangle in the three-dimensional world. If we imagine that our original triangle has become a three-dimensional pyramid, then the volume of such a pyramid will be the product of the length of its base by the area of ​​the triangle we have obtained.

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 and limited by its perimeter square. This can be done in several ways.

Instructions

Use Heron's formula to calculate area triangle. It involves the dimensions of the three sides of the figure, so start your calculations with . The length of each side must be equal to the root of the sum of the squares of the lengths of its projections onto coordinate axes. If we denote the coordinates A(X₁,Y₁,Z₁), B(X₂,Y₂,Z₂) and C(X₃,Y₃,Z₃), the lengths of their sides can be expressed as follows: AB = √((X₁-X₂)² + (Y₁ -Y₂)² + (Z₁-Z₂)²), BC = √((X₂-X₃)² + (Y₂-Y₃)² + (Z₂-Z₃)²), AC = √((X₁-X₃)² + (Y₁-Y₃)² + (Z₁-Z₃)²).

To simplify calculations, introduce an auxiliary variable - 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).

A triangle is like this geometric figure, which consists of three lines connecting at points that do not lie on the same 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 common beginning reference 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 vector product 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 only have 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 a triangle 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.

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 right triangle, when multiplying the length of side a by the sine of the 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 regular triangle, you need to multiply the square of side a by the square root of 3 and divide by 4.