A triangle is one of the most common geometric shapes, which we become familiar with in elementary school. Every student faces the question of how to find the area of a triangle in geometry lessons. So, what features of finding the area of a given figure can be identified? In this article we will look at the basic formulas necessary to complete such a task, and also analyze the types of triangles.

Types of triangles

You can find the area of a triangle in completely different ways, because in geometry there is more than one type of figure containing three angles. These types include:

Obtuse.
Equilateral (correct).
Right triangle.
Isosceles.

Let's take a closer look at each of the existing types of triangles.

This geometric figure is considered the most common when solving geometric problems. When the need arises to draw an arbitrary triangle, this option comes to the rescue.

In an acute triangle, as the name suggests, all the angles are acute and add up to 180°.

This type of triangle is also very common, but is somewhat less common than the acute-angled one. For example, when solving triangles (that is, several of its sides and angles are known and you need to find the remaining elements), sometimes you need to determine whether the angle is obtuse or not. Cosine is a negative number.

B, the value of one of the angles exceeds 90°, so the remaining two angles can take small values (for example, 15° or even 3°).

To find the area of a triangle of this type, you need to know some nuances, which we will talk about later.

Regular and isosceles triangles

A regular polygon is a figure that includes n angles and all sides and angles are equal. This is what a regular triangle is. Since the sum of all the angles of a triangle is 180°, then each of the three angles is 60°.

A regular triangle, due to its property, is also called an equilateral figure.

It is also worth noting that only one circle can be inscribed in a regular triangle, and only one circle can be described around it, and their centers are located at the same point.

In addition to the equilateral type, one can also distinguish an isosceles triangle, which is slightly different from it. In such a triangle, two sides and two angles are equal to each other, and the third side (to which equal angles are adjacent) is the base.

The figure shows an isosceles triangle DEF whose angles D and F are equal and DF is the base.

Right triangle

A right triangle is so named because one of its angles is right, that is, equal to 90°. The other two angles add up to 90°.

The largest side of such a triangle, lying opposite the 90° angle, is the hypotenuse, while the remaining two sides are the legs. For this type of triangle, the Pythagorean theorem applies:

The sum of the squares of the lengths of the legs is equal to the square of the length of the hypotenuse.

The figure shows a right triangle BAC with hypotenuse AC and legs AB and BC.

To find the area of a triangle with a right angle, you need to know the numerical values of its legs.

Let's move on to the formulas for finding the area of a given figure.

Basic formulas for finding area

In geometry, there are two formulas that are suitable for finding the area of most types of triangles, namely for acute, obtuse, regular and isosceles triangles. Let's look at each of them.

By side and height

This formula is universal for finding the area of the figure we are considering. To do this, it is enough to know the length of the side and the length of the height drawn to it. The formula itself (half the product of the base and the height) is as follows:

where A is the side of a given triangle, and H is the height of the triangle.

For example, to find the area of an acute triangle ACB, you need to multiply its side AB by the height CD and divide the resulting value by two.

However, it is not always easy to find the area of a triangle this way. For example, to use this formula for an obtuse triangle, you need to extend one of its sides and only then draw an altitude to it.

In practice, this formula is used more often than others.

On both sides and corner

This formula, like the previous one, is suitable for most triangles and in its meaning is a consequence of the formula for finding the area by side and height of a triangle. That is, the formula in question can be easily derived from the previous one. Its formulation looks like this:

S = ½*sinO*A*B,

where A and B are the sides of the triangle, and O is the angle between sides A and B.

Let us recall that the sine of an angle can be viewed in a special table named after the outstanding Soviet mathematician V. M. Bradis.

Now let's move on to other formulas that are suitable only for exceptional types of triangles.

Area of a right triangle

In addition to the universal formula, which includes the need to find the altitude in a triangle, the area of a triangle containing a right angle can be found from its legs.

Thus, the area of a triangle containing a right angle is half the product of its legs, or:

where a and b are the legs of a right triangle.

Regular triangle

This type of geometric figure is different in that its area can be found with the indicated value of only one of its sides (since all sides of a regular triangle are equal). So, when faced with the task of “finding the area of a triangle when the sides are equal,” you need to use the following formula:

S = A 2 *√3 / 4,

where A is the side of the equilateral triangle.

Heron's formula

The last option for finding the area of a triangle is Heron's formula. In order to use it, you need to know the lengths of the three sides of the figure. Heron's formula looks like this:

S = √p·(p - a)·(p - b)·(p - c),

where a, b and c are the sides of a given triangle.

Sometimes the problem is given: “the area of a regular triangle is to find the length of its side.” In this case, we need to use the formula we already know for finding the area of a regular triangle and derive from it the value of the side (or its square):

A 2 = 4S / √3.

Examination tasks

There are many formulas in GIA problems in mathematics. In addition, quite often it is necessary to find the area of a triangle on checkered paper.

In this case, it is most convenient to draw the height to one of the sides of the figure, determine its length from the cells and use the universal formula for finding the area:

So, after studying the formulas presented in the article, you will not have any problems finding the area of a triangle 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 side 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). Even though 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 who wish can immediately go down the solution.

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.

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

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

Universal methods for finding the area of a triangle

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

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

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

S=½ a b sin γ

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

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

S= a b c/4R

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

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

Areas of triangles with specific properties

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

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

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

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 have completely forgotten 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 the other 2 angles should share 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:

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.