A triangle is such a geometric figure, which consists of three straight lines connecting at points that do not lie on one straight line. The connection points of the lines are the vertices of the triangle, which are denoted by Latin letters (for example, A, B, C). The connecting straight lines of a triangle are called segments, which are also usually denoted in Latin letters. Distinguish the following types triangles:
- Rectangular.
- obtuse.
- Acute-angled.
- Versatile.
- Equilateral.
- Isosceles.
General formulas for calculating the area of a triangle
Triangle area formula for length and height
S=a*h/2,
where a is the length of the side of the triangle whose area is to be found, h is the length of the height drawn to the base.
Heron's formula
S=√p*(p-a)*(p-b)*(p-c),
where √ is Square root, p is the half-perimeter of the triangle, a,b,c is the length of each side of the triangle. The semiperimeter of a triangle can be calculated using the formula p=(a+b+c)/2.
The formula for the area of a triangle in terms of the angle and 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.
The formula for the area of a triangle given the radius of the inscribed circle and three sides
S=p*r,
where p is the semiperimeter of the triangle whose area is to be found, r is the radius of the circle inscribed in this triangle.
The formula for the area of a triangle given three sides and the radius of a 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 circumscribed circle around the triangle.
The formula for the area of a triangle in Cartesian coordinates of points
The Cartesian coordinates of points are coordinates in the xOy system, where x is the abscissa and y is the ordinate. The Cartesian coordinate system xOy on the plane is called the mutually perpendicular numerical axes Ox and Oy with a common reference point 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 a triangle using the following formula, which is obtained from vector product 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 that has one angle of 90 degrees. A triangle can have only one such angle.
The formula for the area of a right triangle on two legs
S=a*b/2,
where a,b is the length of the legs. The legs are called the sides adjacent to the right angle.
The formula for the area of a right triangle given 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.
The formula for the area of a right triangle by leg and opposite angle
S = a*b/2*tg(β),
where a, b are the legs of the triangle, tg(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 isosceles triangle one of the following formulas can be used.
The 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 on the lateral side and base
S=(c/2)* √(a*a – c*c/4),
where c is the base of the triangle, a is the value 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 an equilateral triangle.
The above formulas will allow you to calculate the required area of \u200b\u200bthe triangle. It is important to remember that in order to calculate the spacing of triangles, you need to take into account the type of triangle and the available data that can be used for the calculation.
The triangle is one of the most common geometric shapes, which we are familiar with already in primary school. The question of how to find the area of a triangle is faced by every student in geometry lessons. So, what are the features of finding the area of \u200b\u200ba given figure can be distinguished? In this article, we will consider the basic formulas necessary to complete such a task, and also analyze the types of triangles.
Types of triangles
You can absolutely find the area of a triangle 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 existing types triangles.
Such a geometric figure is considered the most common in solving geometric problems. When it becomes necessary to draw an arbitrary triangle, this option comes to the rescue.
In an acute triangle, as the name implies, all angles are acute and add up to 180°.
Such a triangle is also very common, but is somewhat less common than an acute-angled one. For example, when solving triangles (that is, you know several of its sides and angles and need to find the remaining elements), sometimes you need to determine whether the angle is obtuse or not. Cosine is a negative number.
In 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 of the nuances, which we will talk about later.
Regular and isosceles triangles
regular polygon A figure is called a figure that includes n angles, in which all sides and angles are equal. This is the right triangle. Since the sum of all the angles of a triangle is 180°, each of the three angles is 60°.
The right 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 circumscribed around it, and their centers are located at one point.
In addition to the equilateral type, one can also distinguish an isosceles triangle, which differs slightly from it. In such a triangle, two sides and two angles are equal to each other, and the third side (to which equal angles) is the base.
The figure shows an isosceles triangle DEF, the angles D and F of which are equal, and DF is the base.
Right triangle
A right triangle is so named because one of its angles is a right angle, i.e. equal to 90°. The other two angles add up to 90°.
The largest side of such a triangle, lying opposite an angle of 90 °, is the hypotenuse, while the other two of its sides are the legs. For this type of triangles, the Pythagorean theorem is applicable:
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 \u200b\u200ba given figure.
Basic formulas for finding the area
In geometry, two formulas can be distinguished that are suitable for finding the area of most types of triangles, namely for acute-angled, obtuse-angled, regular and isosceles triangles. Let's analyze 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 the given triangle and H is the height of the triangle.
For example, to find the area of an acute-angled 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 in this way. For example, to use this formula for an obtuse-angled triangle, you need to continue one of its sides and only then draw a height to it.
In practice, this formula is used more often than others.
Two sides and a 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 the side and height of a triangle. That is, the formula under consideration can be easily deduced from the previous one. Its wording 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.
Recall that the sine of an angle can be viewed in a special table named after the outstanding Soviet mathematician V. M. Bradis.
And 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 draw a height in a triangle, the area of \u200b\u200ba triangle containing a right angle can be found from its legs.
So, 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.
right triangle
This type geometric figures differs in that its area can be found with the specified value of only one of its sides (since all sides of a regular triangle are equal). So, having met with the task of “find 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 an 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 the given triangle.
Sometimes the task is given: "the area of \u200b\u200ba regular triangle is to find the length of its side." AT this case you need to use the formula already known to us for finding the area of \u200b\u200ban regular triangle and derive from it the value of the side (or its square):
A 2 \u003d 4S / √3.
Exam problems
There are many formulas in the tasks of the GIA 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 by cells and use the universal formula for finding the area:
So, after studying the formulas presented in the article, you will not have problems finding the area of a triangle of any kind.
The concept of area
The concept of the area of any geometric figure, in particular a triangle, will be associated with such a figure as a square. For a unit area of any geometric figure, we will take the area of a square, the side of which is equal to one. For completeness, we recall two basic properties for the concept of areas of geometric shapes.
Property 1: If a geometric figures are equal, 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 values of the areas of all the figures that make it up.
Consider an example.
Example 1
It is obvious that one of the sides of the triangle is the diagonal of the rectangle , which has one side of length $5$ (since $5$ cells) and the other $6$ (since $6$ 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
Answer: $15$.
Next, consider several methods for finding the areas of triangles, namely using the height and base, using the Heron formula and the area of an equilateral triangle.
How to find the area of a triangle using the height and base
Theorem 1
The area of a triangle can be found as half the product of the length of a side times the height drawn 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 triangle $ABC$ where $AC=α$. The height $BH$ is drawn to this side and equals $h$. Let's build it up to the square $AXYC$ as in Figure 2.
The area of rectangle $AXBH$ is $h\cdot AH$, and that 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 desired area of the triangle, according to 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 $9$ (since $9$ is $9$ cells). The height is also $9$. Then, by Theorem 1, we obtain
$S=\frac(1)(2)\cdot 9\cdot 9=40.5$
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 half-perimeter of this triangle.
Proof.
Consider the following figure:
By the Pythagorean theorem, from the triangle $ABH$ we obtain
From the triangle $CBH$, by 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 $α+β+γ=2ρ$, hence
$h^2=\frac(2ρ(2ρ-2γ)(2ρ-2β)(2ρ-2α))(4β^2)$
$h^2=\frac(4ρ(ρ-α)(ρ-β)(ρ-γ))(β^2 )$
$h=\sqrt(\frac(4ρ(ρ-α)(ρ-β)(ρ-γ))(β^2))$
$h=\frac(2)(β)\sqrt(ρ(ρ-α)(ρ-β)(ρ-γ))$
By Theorem 1, we get
$S=\frac(1)(2) βh=\frac(β)(2)\cdot \frac(2)(β) \sqrt(ρ(ρ-α)(ρ-β)(ρ-γ) )=\sqrt(ρ(ρ-α)(ρ-β)(ρ-γ))$
From the opposite vertex) and divide the resulting product by two. In form it looks like this:
S = ½ * a * h,
where:
S is the area of the triangle,
a is the length of its side,
h is the height lowered to this side.
Side length and height must be presented in the same units. In this case, the area of \u200b\u200bthe triangle will turn out in the corresponding "" units.
Example.
On one of the sides of a scalene triangle 20 cm long, a perpendicular from the opposite vertex 10 cm long is lowered.
The area of the triangle is required.
Decision.
S = ½ * 20 * 10 = 100 (cm²).
If you know the lengths of any two sides of a scalene triangle and the angle between them, then use the formula:
S = ½ * a * b * sinγ,
where: a, b are the lengths of two arbitrary sides, and γ is the angle between them.
In practice, for example, when measuring land, the use of the above formulas is sometimes difficult, since it requires additional constructions and measurement of angles.
If you know the lengths of all three sides of a scalene triangle, then use Heron's formula:
S = √(p(p-a)(p-b)(p-c)),
a, b, c are the lengths of the sides of the triangle,
р – semi-perimeter: p = (a+b+c)/2.
If, in addition to the lengths of all sides, the radius of the circle inscribed in the triangle is known, then use the following compact formula:
where: r is the radius of the inscribed circle (p is the semi-perimeter).
To calculate the area of a scalene triangle of the circumscribed circle and the length of its sides, use the formula:
where: R is the radius of the circumscribed circle.
If the length of one of the sides of the triangle and three angles is known (in principle, two are enough - the value of the third is calculated from the equality of the sum of the three angles of the triangle - 180º), then use the formula:
S = (a² * sinβ * sinγ)/2sinα,
where α is the value of the angle opposite to side a;
β, γ are the values of the remaining two angles of the triangle.
The need to find various elements, including area triangle, appeared many centuries before our era among astronomers Ancient Greece. Square triangle can be calculated different ways using different formulas. The calculation method depends on which elements triangle known.
Instruction
If from the condition we know the values of the two sides b, c and the angle formed by them?, then the area triangle ABC is found by the formula:
S = (bcsin?)/2.
If from the condition we know the values of the two sides a, b and the angle not formed by them?, then the area triangle ABC is found as follows:
Finding the angle?, sin? = bsin? / a, further on the table we determine the angle itself.
Finding an angle? = 180°-?-?.
Find the area itself S = (absin?)/2.
If from the condition we know the values of only three sides triangle a, b and c, then the area triangle ABC is found by the formula:
S = v(p(p-a)(p-b)(p-c)) , where p is the semiperimeter p = (a+b+c)/2
If from the condition of the problem we know the height triangle h and the side to which this height is lowered, then the area triangle ABC by formula:
S = ah(a)/2 = bh(b)/2 = ch(c)/2.
If we know the values of the sides triangle a, b, c and the radius of the circumscribed near the given triangle R, then the area of this triangle ABC is determined by the formula:
S = abc/4R.
If three sides a, b, c and the radius of the inscribed in are known, then the area triangle ABC is found by the formula:
S = pr, where p is the semiperimeter, p = (a+b+c)/2.
If ABC is equilateral, then the area is found by the formula:
S = (a^2v3)/4.
If triangle ABC is isosceles, then the area is determined by the formula:
S = (cv(4a^2-c^2))/4, where c is triangle.
If triangle ABC is a right triangle, then the area is determined by the formula:
S = ab/2, where a and b are legs triangle.
If triangle ABC is a right isosceles triangle, then the area is determined by the formula:
S = c^2/4 = a^2/2, where c is the hypotenuse triangle, a=b - leg.
Related videos
Sources:
- how to measure the area of a triangle
Tip 3: How to find the area of a triangle if you know the angle
Knowing only one parameter (the value of the angle) is not enough to find the area tre square . If there are any additional dimensions, then to determine the area, you can choose one of the formulas in which the angle value is also used as one of the known variables. A few of the most commonly used formulas are listed below.
Instruction
If, in addition to the angle (γ) formed by the two sides tre square , the lengths of these sides (A and B) are also known, then square(S) figures can be defined as half the product of the lengths of the sides and the sine of this known angle: S=½×A×B×sin(γ).
Sometimes in life there are situations when you have to delve into your memory in search of long-forgotten school knowledge. For example, you need to determine the area of \u200b\u200bthe land plot of a triangular shape, or the turn of the next repair in an apartment or private house has come, and you need to calculate how much material it will take for a surface with a triangular shape. There was a time when you could solve such a problem in a couple of minutes, and now you are desperately trying to remember how to determine the area of a triangle?
You don't have to worry about this! After all, it is quite normal when the human brain decides to shift long-unused knowledge somewhere in a remote corner, from which it is sometimes not so easy to extract it. So that you do not have to suffer with the search for forgotten school knowledge to solve such a problem, this article contains various methods, which make it easy to find the desired area of the triangle.
It is well known that a triangle is a type of polygon that is limited to the minimum possible number sides. In principle, any polygon can be divided into several triangles by connecting its vertices with segments that do not intersect its sides. Therefore, knowing the triangle, you can calculate the area of almost any figure.
Among all the possible triangles that occur in life, the following particular types can be distinguished: and rectangular.
The easiest way to calculate the area of a triangle is when one of its corners is right, that is, in the case of a right triangle. It is easy to see that it is half a rectangle. Therefore, its area is equal to half the product of the sides that form a right angle between them.
If we know the height of a triangle dropped from one of its vertices to opposite side, and the length of this side, which is called the base, then the area is calculated as half the product of the height and the base. This is written using the following formula:
S = 1/2*b*h, in which
S is the desired area of the triangle;
b, h - respectively, the height and base of the triangle.
It is so easy to calculate the area of an isosceles triangle, since the height will bisect the opposite side, and it can be easily measured. If the area is determined, then it is convenient to take the length of one of the sides forming a right angle as the height.
All this is of course good, but how to determine whether one of the corners of a triangle is right or not? If the size of our figure is small, then you can use a building angle, a drawing triangle, a postcard or other object with rectangular shape.
But what if we have a triangular land plot? In this case, proceed as follows: count from the top of the proposed right angle on one side, a distance multiple of 3 (30 cm, 90 cm, 3 m), and on the other side, a distance multiple of 4 (40 cm, 160 cm, 4 m) is measured in the same proportion. Now you need to measure the distance between the end points of these two segments. If the value is a multiple of 5 (50 cm, 250 cm, 5 m), then it can be argued that the angle is right.
If the value of the length of each of the three sides of our figure is known, then the area of \u200b\u200bthe triangle can be determined using Heron's formula. In order for it to have a simpler form, a new value is used, which is called the semi-perimeter. This is the sum of all the sides of our triangle, divided in half. After the semi-perimeter is calculated, you can begin to determine the area using the formula:
S = sqrt(p(p-a)(p-b)(p-c)), where
sqrt - square root;
p is the value of the semi-perimeter (p =(a+b+c)/2);
a, b, c - edges (sides) of the triangle.
But what if the triangle has irregular shape? There are two possible ways here. The first of these is to try to divide such a figure into two right-angled triangles, the sum of the areas of which is calculated separately, and then added. Or, if the angle between the two sides and the size of these sides are known, then apply the formula:
S = 0.5 * ab * sinC, where
a,b - sides of the triangle;
c is the angle between these sides.
The latter case is rare in practice, but nevertheless, everything is possible in life, so the above formula will not be superfluous. Good luck with your calculations!
