A triangle is the simplest geometric figure, which consists of three sides and three vertices. Due to its simplicity, the triangle has been used since ancient times to take various measurements, and today the figure can be useful for solving practical and everyday problems.

Features of a triangle

The figure has been used for calculations since ancient times, for example, land surveyors and astronomers operate with the properties of triangles to calculate areas and distances. It is easy to express the area of ​​any n-gon through the area of ​​this figure, and this property was used by ancient scientists to derive formulas for the areas of polygons. Constant work with triangles, especially the right triangle, became the basis for an entire branch of mathematics - trigonometry.

Triangle geometry

The properties of the geometric figure have been studied since ancient times: the earliest information about the triangle was found in Egyptian papyri from 4,000 years ago. Then the figure was studied in Ancient Greece and the greatest contributions to the geometry of the triangle were made by Euclid, Pythagoras and Heron. The study of the triangle never ceased, and in the 18th century, Leonhard Euler introduced the concept of the orthocenter of a figure and the Euler circle. At the turn of the 19th and 20th centuries, when it seemed that absolutely everything was known about the triangle, Frank Morley formulated the theorem on angle trisectors, and Waclaw Sierpinski proposed the fractal triangle.

There are several types of flat triangles that are familiar to us from school geometry courses:

• acute - all the corners of the figure are acute;
• obtuse - the figure has one obtuse angle (more than 90 degrees);
• rectangular - the figure contains one right angle equal to 90 degrees;
• isosceles - a triangle with two equal sides;
• equilateral - a triangle with all equal sides.
• There are all kinds of triangles in real life, and in some cases we may need to calculate the area of ​​a geometric figure.

Area of ​​a triangle

Area is an estimate of how much of the plane a figure encloses. The area of ​​a triangle can be found in six ways, using the sides, height, angles, radius of the inscribed or circumscribed circle, as well as using Heron's formula or calculating the double integral along the lines bounding the plane. The simplest formula for calculating the area of ​​a triangle is:

where a is the side of the triangle, h is its height.

However, in practice it is not always convenient for us to find the height of a geometric figure. The algorithm of our calculator allows you to calculate the area knowing:

• three sides;
• two sides and the angle between them;
• one side and two corners.

To determine the area through three sides, we use Heron's formula:

S = sqrt (p × (p-a) × (p-b) × (p-c)),

where p is the semi-perimeter of the triangle.

The area on two sides and an angle is calculated using the classic formula:

S = a × b × sin(alfa),

where alfa is the angle between sides a and b.

To determine the area in terms of one side and two angles, we use the relationship that:

a / sin(alfa) = b / sin(beta) = c / sin(gamma)

Using a simple proportion, we determine the length of the second side, after which we calculate the area using the formula S = a × b × sin(alfa). This algorithm is fully automated and you only need to enter the specified variables and get the result. Let's look at a couple of examples.

Examples from life

Paving slabs

Let's say you want to pave the floor with triangular tiles, and to determine the amount of material needed, you need to know the area of ​​\u200b\u200bone tile and the area of ​​​​the floor. Suppose you need to process 6 square meters of surface using a tile whose dimensions are a = 20 cm, b = 21 cm, c = 29 cm. Obviously, to calculate the area of ​​a triangle, the calculator uses Heron’s formula and gives the result:

Thus, the area of ​​one tile element will be 0.021 square meters, and you will need 6/0.021 = 285 triangles for the floor improvement. The numbers 20, 21 and 29 form a Pythagorean triple - numbers that satisfy . And that's right, our calculator also calculated all the angles of the triangle, and the gamma angle is exactly 90 degrees.

School task

In a school problem, you need to find the area of ​​a triangle, knowing that side a = 5 cm, and angles alpha and beta are 30 and 50 degrees, respectively. To solve this problem manually, we would first find the value of side b using the proportion of the aspect ratio and the sines of the opposite angles, and then determine the area using the simple formula S = a × b × sin(alfa). Let's save time, enter the data into the calculator form and get an instant answer

When using the calculator, it is important to indicate the angles and sides correctly, otherwise the result will be incorrect.

Conclusion

The triangle is a unique figure that is found both in real life and in abstract calculations. Use our online calculator to determine the area of ​​triangles of any kind.

Area of ​​a geometric figure- a numerical characteristic of a geometric figure showing the size of this figure (part of the surface limited by the closed contour of this figure). The size of the area is expressed by the number of square units contained in it.

Triangle area formulas

1. Formula for the area of ​​a triangle by side and height
   Area of ​​a triangle equal to half the product of the length of a side of a triangle and the length of the altitude drawn to this side
   
2. Formula for the area of ​​a triangle based on three sides and the radius of the circumcircle
   
3. Formula for the area of ​​a triangle based on three sides and the radius of the inscribed circle
   Area of ​​a triangle is equal to the product of the semi-perimeter of the triangle and the radius of the inscribed circle.
   
where S is the area of ​​the triangle,
- lengths of the sides of the triangle,
- height of the triangle,
- the angle between the sides and,
- radius of the inscribed circle,
R - radius of the circumscribed circle,

Square area formulas

1. Formula for the area of ​​a square by side length
   Square area equal to the square of the length of its side.
2. Formula for the area of ​​a square along the diagonal length
   Square area equal to half the square of the length of its diagonal.
   

   S=	1	2
   	2

where S is the area of ​​the square,
- length of the side of the square,
- length of the diagonal of the square.

Rectangle area formula

Area of ​​a rectangle equal to the product of the lengths of its two adjacent sides

where S is the area of ​​the rectangle,
- lengths of the sides of the rectangle.

Parallelogram area formulas

1. Formula for the area of ​​a parallelogram based on side length and height
   Area of ​​a parallelogram
2. Formula for the area of ​​a parallelogram based on two sides and the angle between them
   Area of ​​a parallelogram is equal to the product of the lengths of its sides multiplied by the sine of the angle between them.
   

   a b sin α

where S is the area of ​​the parallelogram,
- lengths of the sides of the parallelogram,
- length of parallelogram height,
- the angle between the sides of the parallelogram.

Formulas for the area of ​​a rhombus

1. Formula for the area of ​​a rhombus based on side length and height
   Area of ​​a rhombus is equal to the product of the length of its side and the length of the height lowered to this side.
2. Formula for the area of ​​a rhombus based on side length and angle
   Area of ​​a rhombus is equal to the product of the square of the length of its side and the sine of the angle between the sides of the rhombus.
3. Formula for the area of ​​a rhombus based on the lengths of its diagonals
   Area of ​​a rhombus equal to half the product of the lengths of its diagonals.
where S is the area of ​​the rhombus,
- length of the side of the rhombus,
- length of the height of the rhombus,
- the angle between the sides of the rhombus,
1, 2 - lengths of diagonals.

Trapezoid area formulas

1. Heron's formula for trapezoid

   Where S is the area of ​​the trapezoid,
   - lengths of the bases of the trapezoid,
   - lengths of the sides of the trapezoid,
   

Sometimes in life there are situations when you have to delve into your memory in search of long-forgotten school knowledge. For example, you need to determine the area of ​​a triangular-shaped plot of land, or the time has come for another renovation in an apartment or private house, and you need to calculate how much material will be needed for a surface with a triangular shape. There was a time when you could solve such a problem in a couple of minutes, but now you are desperately trying to remember how to determine the area of ​​a triangle?

Don't worry about it! After all, it is quite normal when a person’s brain decides to transfer long-unused knowledge somewhere to a remote corner, from which sometimes it is not so easy to extract it. So that you don’t have to struggle with searching for forgotten school knowledge to solve such a problem, this article contains various methods that make it easy to find the required area of ​​a triangle.

It is well known that a triangle is a type of polygon that is limited to the minimum possible number of sides. In principle, any polygon can be divided into several triangles by connecting its vertices with segments that do not intersect its sides. Therefore, knowing the triangle, you can calculate the area of ​​almost any figure.

Among all the possible triangles that occur in life, the following particular types can be distinguished: and rectangular.

The easiest way to calculate the area of ​​a triangle is when one of its angles is right, that is, in the case of a right triangle. It is easy to see that it is half a rectangle. Therefore, its area is equal to half the product of the sides that form a right angle with each other.

If we know the height of a triangle, lowered from one of its vertices to the opposite side, and the length of this side, which is called the base, then the area is calculated as half the product of the height and the base. This is written using the following formula:

S = 1/2*b*h, in which

S is the required area of ​​the triangle;

b, h - respectively, the height and base of the triangle.

It is so easy to calculate the area of ​​an isosceles triangle because the height will bisect the opposite side and can be easily measured. If the area is determined, then it is convenient to take the length of one of the sides forming a right angle as the height.

All this is of course good, but how to determine whether one of the angles of a triangle is right or not? If the size of our figure is small, then we can use a construction angle, a drawing triangle, a postcard or another object with a rectangular shape.

But what if we have a triangular plot of land? In this case, proceed as follows: count from the top of the supposed right angle on one side a distance multiple of 3 (30 cm, 90 cm, 3 m), and on the other side measure a distance multiple of 4 in the same proportion (40 cm, 160 cm, 4 m). Now you need to measure the distance between the end points of these two segments. If the result is a multiple of 5 (50 cm, 250 cm, 5 m), then we can say that the angle is right.

If the length of each of the three sides of our figure is known, then the area of ​​the triangle can be determined using Heron's formula. In order for it to have a simpler form, a new value is used, which is called semi-perimeter. This is the sum of all the sides of our triangle, divided in half. After the semi-perimeter has been calculated, you can begin to determine the area using the formula:

S = sqrt(p(p-a)(p-b)(p-c)), where

sqrt - square root;

p - semi-perimeter value (p = (a+b+c)/2);

a, b, c - edges (sides) of the triangle.

But what if the triangle has an irregular shape? There are two possible ways here. The first of them is to try to divide such a figure into two right triangles, the sum of the areas of which is calculated separately, and then added. Or, if the angle between two sides and the size of these sides are known, then apply the formula:

S = 0.5 * ab * sinC, where

a,b - sides of the triangle;

c is the size of the angle between these sides.

The latter case is rare in practice, but nevertheless, everything is possible in life, so the above formula will not be superfluous. Good luck with your calculations!

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.

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 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).

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.