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 triangle - formulas and examples of problem solving

Below are formulas for finding the area of an arbitrary triangle which are suitable for finding the area of any triangle, regardless of its properties, angles or sizes. The formulas are presented in the form of a picture, with explanations for their application or justification for their correctness. Also, a separate figure shows the correspondence between the letter symbols in the formulas and the graphic symbols in the drawing.

Note . If the triangle has special properties (isosceles, rectangular, equilateral), you can use the formulas given below, as well as additional special formulas that are valid only for triangles with these properties:

"Formula for the area of an equilateral triangle"

Triangle area formulas

Explanations for formulas:
a, b, c- the lengths of the sides of the triangle whose area we want to find
r- radius of the circle inscribed in the triangle
R- radius of the circle circumscribed around the triangle
h- height of the triangle lowered to the side
p- semi-perimeter of a triangle, 1/2 the sum of its sides (perimeter)
α - angle opposite to side a of the triangle
β - angle opposite to side b of the triangle
γ - angle opposite to side c of the triangle
h a, h b , h c- height of the triangle lowered to sides a, b, c

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

The area of the triangle is half the product of the height of the triangle and the length of the side by which this height is lowered(Formula 1). The correctness of this formula can be understood logically. The height lowered to the base will split an arbitrary triangle into two rectangular ones. If you build each of them into a rectangle with dimensions b and h, then obviously the area of these triangles will be equal to exactly half the area of the rectangle (Spr = bh)
The area of the triangle is half the product of its two sides and the sine of the angle between them(Formula 2) (see an example of solving a problem using this formula below). Despite the fact that it seems different from the previous one, it can easily be transformed into it. If we lower the height from angle B to side b, it turns out that the product of side a and the sine of angle γ, according to the properties of the sine in a right triangle, is equal to the height of the triangle we drew, which gives us the previous formula
The area of an arbitrary triangle can be found through work half the radius of the circle inscribed in it by the sum of the lengths of all its sides(Formula 3), simply put, you need to multiply the semi-perimeter of the triangle by the radius of the inscribed circle (this is easier to remember)
The area of an arbitrary triangle can be found by dividing the product of all its sides by 4 radii of the circle circumscribed around it (Formula 4)
Formula 5 is finding the area of a triangle through the lengths of its sides and its semi-perimeter (half the sum of all its sides)
Heron's formula(6) is a representation of the same formula without using the concept of semi-perimeter, only through the lengths of the sides
The area of an arbitrary triangle is equal to the product of the square of the side of the triangle and the sines of the angles adjacent to this side divided by the double sine of the angle opposite to this side (Formula 7)
The area of an arbitrary triangle can be found as the product of two squares of the circle circumscribed around it by the sines of each of its angles. (Formula 8)
If the length of one side and the values of two adjacent angles are known, then the area of the triangle can be found as the square of this side divided by the double sum of the cotangents of these angles (Formula 9)
If only the length of each of the heights of the triangle is known (Formula 10), then the area of such a triangle is inversely proportional to the lengths of these heights, as according to Heron’s Formula
Formula 11 allows you to calculate area of a triangle based on the coordinates of its vertices, which are specified as (x;y) values for each of the vertices. Please note that the resulting value must be taken modulo, since the coordinates of individual (or even all) vertices may be in the region of negative values

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

Task. Find the area given two sides and the angle between them

The sides of the triangle are 5 and 6 cm. The angle between them is 60 degrees. Find the area of the triangle.

Solution.

To solve this problem, we use formula number two from the theoretical part of the lesson.
The area of a triangle can be found through the lengths of two sides and the sine of the angle between them and will be equal to
S=1/2 ab sin γ

Since we have all the necessary data for the solution (according to the formula), we can only substitute the values from the problem conditions into the formula:
S = 1/2 * 5 * 6 * sin 60

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

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

Task. Find the area of an equilateral triangle

Find the area of an equilateral triangle with side 3 cm.

Solution .

The area of a triangle can be found using Heron's formula:

S = 1/4 sqrt((a + b + c)(b + c - a)(a + c - b)(a + b -c))

Since a = b = c, the formula for the area of an equilateral triangle takes the form:

S = √3 / 4 * a 2

S = √3 / 4 * 3 2

Answer: 9 √3 / 4.

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

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

Solution.

Since the dimensions of the sides of the triangle are unknown to us, to solve the problem we will assume that the lengths of the sides are respectively equal to arbitrary numbers a, b, c. Then, in order to answer the question of the problem, we will find the area of the given triangle, and then we will find the area of the triangle whose sides are four times larger. The ratio of the areas of these triangles will give us the answer to the problem.

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

To solve, we use Heron’s formula (see above in the theoretical part of the lesson). It looks like this:

S = 1/4 sqrt((a + b + c)(b + c - a)(a + c - b)(a + b -c))
(see first line of picture below)

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

S 2 = 1/4 sqrt((4a + 4b + 4c)(4b + 4c - 4a)(4a + 4c - 4b)(4a + 4b -4c))
(see second line in the picture below)

As you can see, 4 is a common factor that can be taken out of brackets from all four expressions according to the general rules of mathematics.
Then

S 2 = 1/4 sqrt(4 * 4 * 4 * 4 (a + b + c)(b + c - a)(a + c - b)(a + b -c)) - on the third line of the picture
S 2 = 1/4 sqrt(256 (a + b + c)(b + c - a)(a + c - b)(a + b -c)) - fourth line

The square root of the number 256 is perfectly extracted, so let’s take it out from under the root
S 2 = 16 * 1/4 sqrt((a + b + c)(b + c - a)(a + c - b)(a + b -c))
S 2 = 4 sqrt((a + b + c)(b + c - a)(a + c - b)(a + b -c))
(see fifth line of the picture below)

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

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 completely forgot 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:

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).

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.