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 an acute triangle. 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 whose sides and angles are all 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.
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 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.
You can find over 10 formulas for calculating the area of a triangle on the Internet. Many of them are used in problems with known sides and angles of a triangle. However, there are a number of complex examples where, according to the conditions of the assignment, only one side and angles of a triangle are known, or the radius of a circumscribed or inscribed circle and one more characteristic. In such cases, a simple formula cannot be applied.
The formulas given below will allow you to solve 95 percent of problems in which you need to find the area of a triangle.
Let's move on to consider common area formulas.
Consider the triangle shown in the figure below
In the figure and below in the formulas, the classical designations of all its characteristics are introduced.
a,b,c – sides of the triangle,
R – radius of the circumscribed circle,
r – radius of the inscribed circle,
h[b],h[a],h[c] – heights drawn in accordance with sides a,b,c.
alpha, beta, hamma – angles near the vertices.
Basic formulas for the area of a triangle
1. The area is equal to half the product of the side of the triangle and the height lowered to this side. In the language of formulas, this definition can be written as follows
Thus, if the side and height are known, then every student will find the area.
By the way, from this formula one can derive one useful relationship between heights
2. If we take into account that the height of a triangle through the adjacent side is expressed by the dependence
Then the first area formula is followed by the second ones of the same type
Look carefully at the formulas - they are easy to remember, since the work involves two sides and the angle between them. If we correctly designate the sides and angles of the triangle (as in the figure above), we will get two sides a, b and the angle is connected to the third With (hamma).
3. For the angles of a triangle, the relation is true
The dependence allows you to use the following formulas for the area of a triangle in calculations:
Examples of this dependence are extremely rare, but you must remember that there is such a formula.
4. If the side and two adjacent angles are known, then the area is found by the formula
5. The formula for area in terms of side and cotangent of adjacent angles is as follows
By rearranging the indexes you can get dependencies for other parties.
6. The area formula below is used in problems when the vertices of a triangle are specified on the plane by coordinates. In this case, the area is equal to half the determinant taken modulo.
7. Heron's formula used in examples with known sides of a triangle.
First find the semi-perimeter of the triangle
And then determine the area using the formula
or
It is quite often used in the code of calculator programs.
8. If all the heights of the triangle are known, then the area is determined by the formula
It is difficult to calculate on a calculator, but in the MathCad, Mathematica, Maple packages the area is “time two”.
9. The following formulas use the known radii of inscribed and circumscribed circles. 
In particular, if the radius and sides of the triangle, or its perimeter, are known, then the area is calculated according to the formula
10. In examples where the sides and the radius or diameter of the circumscribed circle are given, the area is found using the formula
11. The following formula determines the area of a triangle in terms of the side and angles of the triangle.
And finally - special cases:
Area of a right triangle with legs a and b equal to half their product
Formula for the area of an equilateral (regular) triangle=
= one-fourth of the product of the square of the side and the root of three.
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.
The area of rectangle $AXBH$ is $h\cdot AH$, and the area of rectangle $HBYC$ is $h\cdot HC$. Then
$S_ABH=\frac(1)(2)h\cdot AH$, $S_CBH=\frac(1)(2)h\cdot HC$
Therefore, the required area of the triangle, by property 2, is equal to
$S=S_ABH+S_CBH=\frac(1)(2)h\cdot AH+\frac(1)(2)h\cdot HC=\frac(1)(2)h\cdot (AH+HC)=\ frac(1)(2)αh$
The theorem has been proven.
Example 2
Find the area of the triangle in the figure below if the cell has an area equal to one
The base of this triangle is equal to $9$ (since $9$ is $9$ squares). The height is also $9$. Then, by Theorem 1, we get
$S=\frac(1)(2)\cdot 9\cdot 9=40.5$
Answer: $40.5$.
Heron's formula
Theorem 2
If we are given three sides of a triangle $α$, $β$ and $γ$, then its area can be found as follows
$S=\sqrt(ρ(ρ-α)(ρ-β)(ρ-γ))$
here $ρ$ means the semi-perimeter of this triangle.
Proof.
Consider the following figure:
By the Pythagorean theorem, from the triangle $ABH$ we obtain
From the triangle $CBH$, according to the Pythagorean theorem, we have
$h^2=α^2-(β-x)^2$
$h^2=α^2-β^2+2βx-x^2$
From these two relations we obtain the equality
$γ^2-x^2=α^2-β^2+2βx-x^2$
$x=\frac(γ^2-α^2+β^2)(2β)$
$h^2=γ^2-(\frac(γ^2-α^2+β^2)(2β))^2$
$h^2=\frac((α^2-(γ-β)^2)((γ+β)^2-α^2))(4β^2)$
$h^2=\frac((α-γ+β)(α+γ-β)(γ+β-α)(γ+β+α))(4β^2)$
Since $ρ=\frac(α+β+γ)(2)$, then $α+β+γ=2ρ$, which means
$h^2=\frac(2ρ(2ρ-2γ)(2ρ-2β)(2ρ-2α))(4β^2)$
$h^2=\frac(4ρ(ρ-α)(ρ-β)(ρ-γ))(β^2 )$
$h=\sqrt(\frac(4ρ(ρ-α)(ρ-β)(ρ-γ))(β^2))$
$h=\frac(2)(β)\sqrt(ρ(ρ-α)(ρ-β)(ρ-γ))$
By Theorem 1, we get
$S=\frac(1)(2) βh=\frac(β)(2)\cdot \frac(2)(β) \sqrt(ρ(ρ-α)(ρ-β)(ρ-γ) )=\sqrt(ρ(ρ-α)(ρ-β)(ρ-γ))$