The area of a triangle based on its three sides. How to calculate the area of a triangle. Task. Change in area when changing the length of the sides

06.03.2022 | Diagnostics

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 it 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.

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. In many geometry problems involving the calculation of areas, formulas for the area of a triangle are used. There are several of them, here we will look at the main ones.Listing these formulas would be too simple and of no use. We will analyze the origin of the basic formulas, those that are used most often.

Before you read the derivation of the formulas, be sure to look at the article about.After studying the material, you can easily restore the formulas in your memory (if they suddenly “fly out” at the moment you need).

First formula

The diagonal of a parallelogram divides it into two triangles of equal area:

Therefore, the area of the triangle will be equal to half the area of the parallelogram:

Area of triangle formula

*That is, if we know any side of the triangle and the height lowered to this side, then we can always calculate the area of this triangle.

Formula two

As already stated in the article on the area of a parallelogram, the formula looks like:

The area of a triangle is equal to half its area, which means:

*That is, if any two sides in a triangle and the angle between them are known, we can always calculate the area of such a triangle.

Heron's formula (third)

This formula is difficult to derive and it is of no use to you. Look how beautiful she is, you can say that she herself is memorable.

*If three sides of a triangle are given, then using this formula we can always calculate its area.

Formula four

Where r– radius of the inscribed circle

*If the three sides of a triangle and the radius of the circle inscribed in it are known, then we can always find the area of this triangle.

Formula five

Where R– radius of the circumscribed circle.

*If the three sides of a triangle and the radius of the circle circumscribed around it are known, then we can always find the area of such a triangle.

The question arises: if three sides of a triangle are known, then isn’t it easier to find its area using Heron’s formula!

Yes, it can be easier, but not always, sometimes complexity arises. This involves extracting the root. In addition, these formulas are very convenient to use in problems where the area of a triangle and its sides are given and you need to find the radius of the inscribed or circumscribed circle. Such tasks are available as part of the Unified State Examination.

Let's look at the formula separately:

It is a special case of the formula for the area of a polygon into which a circle is inscribed:

Let's consider it using the example of a pentagon:

Let us connect the center of the circle with the vertices of this pentagon and lower perpendiculars from the center to its sides. We get five triangles, with the dropped perpendiculars being the radii of the inscribed circle:

The area of the pentagon is:

Now it is clear that if we are talking about a triangle, then this formula takes the form:

Formula six

Can be found by knowing the base and height. The whole simplicity of the diagram lies in the fact that the height divides the base a into two parts a 1 and a 2, and the triangle itself into two right triangles, the area of which is and. Then the area of the entire triangle will be the sum of the two indicated areas, and if we take one second of the height out of the bracket, then in the sum we get back the base:

A more difficult method for calculations is Heron’s formula, for which you need to know all three sides. For this formula, you first need to calculate the semi-perimeter of the triangle: Heron's formula itself implies the square root of the semi-perimeter, multiplied in turn by its difference on each side.

The following method, also relevant for any triangle, allows you to find the area of the triangle through two sides and the angle between them. The proof of this comes from the formula with height - we draw the height on any of the known sides and through the sine of the angle α we obtain that h=a⋅sinα. To calculate the area, multiply half the height by the second side.

Another way is to find the area of a triangle, knowing 2 angles and the side between them. The proof of this formula is quite simple and can be clearly seen from the diagram.

We lower the height from the vertex of the third angle to the known side and call the resulting segments x accordingly. From right triangles it can be seen that the first segment x is equal to the product

Area formula is necessary to determine the area of a figure, which is a real-valued function defined on a certain class of figures of the Euclidean plane and satisfying 4 conditions:

Positivity - Area cannot be less than zero;
Normalization - a square with side unit has area 1;
Congruence - congruent figures have equal area;
Additivity - the area of the union of 2 figures without common internal points is equal to the sum of the areas of these figures.

Formulas for the area of geometric figures.

Geometric figure	Formula	Drawing
The result of adding the distances between the midpoints of opposite sides of a convex quadrilateral will be equal to its semi-perimeter.
Circle sector. The area of a sector of a circle is equal to the product of its arc and half its radius.
Circle segment. To obtain the area of segment ASB, it is enough to subtract the area of triangle AOB from the area of sector AOB.	S = 1 / 2 R(s - AC)
The area of the ellipse is equal to the product of the lengths of the major and minor semi-axes of the ellipse and the number pi.
Ellipse. Another option for calculating the area of an ellipse is through two of its radii.
Triangle. Through the base and height. Formula for the area of a circle using its radius and diameter.
Square . Through his side. The area of a square is equal to the square of the length of its side.
Square. Through its diagonals. The area of a square is equal to half the square of the length of its diagonal.
Regular polygon. To determine the area of a regular polygon, it is necessary to divide it into equal triangles that would have a common vertex at the center of the inscribed circle.	S= r p = 1/2 r n a