P=a+b+c How to find the perimeter of a triangle: Everyone knows that the perimeter is easy to find - you just need to add up all three sides of the triangle. However, there are several other ways to find the sum of the lengths of the sides of a triangle. Step 1 Given the radius of the circle inscribed in the triangle and its area, find the perimeter using the formula P=2S/r. 
Step 2 If you know two angles, for example, α and β, adjacent to the side, and the length of this side, then to find the perimeter, use the formula a+sinα∙а/(sin(180°-α-β)) + sinβ∙а /(sin(180°-α-β)). 
Step 3 If the condition specifies adjacent sides and the angle β between them, consider the cosine theorem when finding the perimeter. Then P=a+b+√(a^2+b^2-2∙a∙b∙cosβ), where a^2 and b^2 are the squares of the lengths of adjacent sides. The expression under the root is the length of the third unknown party, expressed through the cosine theorem. 
4 step for isosceles triangle the perimeter formula takes the form P=2a+b, where a are the sides and b is its base. Step 5 Calculate the perimeter of a regular triangle using the formula P=3a. Step 6 Find the perimeter using the radii of the circles inscribed in the triangle or circumscribed around it. Yes, for equilateral triangle remember and use the formula P=6r√3=3R√3, where r is the radius of the inscribed circle and R is the radius of the circumscribed circle. Step 7 For an isosceles triangle, apply the formula P=2R(2sinα+sinβ), where α is the angle at the base and β is the angle opposite the base.
How to find the perimeter of a triangle? Each of us asked this question while studying at school. Let's try to remember everything we know about this amazing figure, and also answer question asked.
The answer to the question of how to find the perimeter of a triangle is usually quite simple - you just need to perform the procedure of adding the lengths of all its sides. However, there are several more simple methods desired value.
Adviсe
In the event that the radius (r) of the circle that is inscribed in the triangle and its area (S) are known, then answering the question of how to find the perimeter of the triangle is quite simple. To do this, you need to use the usual formula:
If two angles are known, say, α and β, which are adjacent to the side, and the length of the side itself, then the perimeter can be found using a very, very popular formula, which looks like:
sinβ∙a/(sin(180° - β - α)) + sinα∙a/(sin(180° - β - α)) + a
If you know the lengths of adjacent sides and the angle β between them, then in order to find the perimeter, you need to use the Perimeter is calculated by the formula:
P = b + a + √(b2 + a2 - 2∙b∙а∙cosβ),
where b2 and a2 are the squares of the lengths of the adjacent sides. The radical expression is the length of the third side, which is unknown, expressed by means of the cosine theorem.
If you do not know how to find the perimeter, then there is, in fact, nothing difficult. Calculate it using the formula:
where b is the base of the triangle and a are its sides.
To find the perimeter of a regular triangle, use the simplest formula:
where a is the length of the side.
How to find the perimeter of a triangle if only the radii of the circles that are described around it or inscribed in it are known? If the triangle is equilateral, then the formula should be applied:
P = 3R√3 = 6r√3,
where R and r are the radii of the circumscribed and inscribed circles, respectively.
If the triangle is isosceles, then the formula applies to it:
P=2R (sinβ + 2sinα),
where α is the angle that lies at the base and β is the angle that is opposite the base.
Often, the solution of mathematical problems requires the deepest analysis and specific skill find and derive the required formulas, and this, as many people know, is a rather difficult job. Although some problems can be solved with just one single formula.
Let's look at the formulas that are basic for answering the question of how to find the perimeter of a triangle, in relation to the most diverse types of triangles.
Of course, the main rule for finding the perimeter of a triangle is this statement: to find the perimeter of a triangle, you need to add the lengths of all its sides using the appropriate formula:
where b, a and c are the lengths of the sides of the triangle and P is the perimeter of the triangle.
There are several special cases of this formula. Let's say your problem is formulated as follows: "how to find the perimeter right triangle? In this case, you should use the following formula:
P = b + a + √(b2 + a2)
In this formula, b and a are the direct lengths of the legs of a right triangle. It is easy to guess that instead of the c side (hypotenuse), the expression obtained by the theorem of the great scientist of antiquity, Pythagoras, is used.
If you want to solve a problem where the triangles are similar, then it would be logical to use this statement: the ratio of the perimeters corresponds to the similarity coefficient. Let's say you have two similar triangles - ∆ABC and ∆A1B1C1. Then, to find the similarity coefficient, it is necessary to divide the perimeter ΔABC by the perimeter ΔA1B1C1.
In conclusion, it can be noted that the perimeter of a triangle can be found using a variety of methods, depending on the initial data that you have. It should be added that there are some special cases for right triangles.
How to find the perimeter of a triangle
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The perimeter is the sum of the lengths of all sides of any polygon. Therefore, without thinking about what kind of geometric figure is in front of you, feel free to measure the length of all sides with a ruler and summarize. Here is the perimeter.
If we are talking about the basics of geometry, then the perimeter is the sum of all sides of a triangle: P \u003d a + b + c.
However, if we are talking about more complex geometric and trigonometric problems, when we are given certain data, then there are several other formulas for calculating the perimeter of a triangle:
If the radius of the circle inscribed in the triangle and its area are known, then the perimeter is calculated by the formula: P=2S/r.
If two angles are known, for example amp;#945; and amp;#946;, adjacent to one side, and the length of this side, then the formula for the perimeter is as follows: P \u003d a + sinamp; amp;#946;)) + sinamp;#946;amp;#8729;a/(sin(180-amp;#945;-amp;#946;)).
If there are lengths of adjacent sides and angle amp;#946; between them, then the perimeter is calculated using the formula of the cosine theorem: where a2 and b2 are the squares of the lengths of the adjacent sides. The expression under the root is the length of the third unknown side, expressed through the cosine theorem.
The perimeter of an isosceles triangle has the following form P=2a+b, where a are the sides and b is its base.
Perimeter of a regular triangle: P=3a.
The perimeter formula for an equilateral triangle, if the radius of the circle inscribed in it is P=6ramp;#8730;3, or the radius of the circle circumscribed about it is P=3Ramp;#8730;3, where r and R, respectively, are the radii of the inscribed or circumscribed circle.
For an isosceles triangle there is a formula: P=2R(2sinamp;#945;+sinamp;#946;), where amp;#945; base angle, amp;#946; angle opposite the base.
Looking at what you know from the problem statement.
The simplest option is to add the lengths of all sides.
In an equilateral triangle, the length of a side is multiplied by three.
According to the formula P=2S/r, if S is the area and r is the radius of the inscribed circle.
There are also formulas for finding the area of a triangle if its angles are known.
If the triangle is equilateral, then to find its perimeter, you need to multiply the length of one side by three. And if the triangle is scalene, then to find its perimeter, you need to add the lengths of all its sides.
To find the perimeter of an equilateral triangle, multiply the length of one side by three.
To find the perimeter of an isosceles triangle, you need to take the length of one of the sides equal in length, multiply by two and add the length of the base.
Pick up a ruler, measure each side of the triangle (if it is equilateral, then only one can be measured) and add the lengths of its sides. In the case of an equilateral triangle, the length of its side is multiplied by 3.
In the mind, in a column, on a calculator - as you can, depending on mathematical ability and the presence or absence of a calculator.
Find the perimeter of a triangle, if the length of each of its sides is known, you just need to add the lengths of the sides and get the perimeter: (P=a+b+c).
Even easier to find perimeter of an equilateral triangle you just need to multiply the length of its side by 3: (P=3a).
But more often, the need to calculate the perimeter arises when the length of not all of its sides is known.
Therefore, if one side of the triangle c and the angles adjacent to it are known, then formula for calculating the perimeter will look like this:
The perimeter of a triangle is easy to find. Perimeter is the length of three sides of a triangle. It is necessary to add the first side, the second side and the third side - the total the length of three sides will be the perimeter of the triangle.
The perimeter is the sum of the lengths of the sides. We need to sum the lengths of all sides of the triangle. Or have I misunderstood something? What are the initial data of the task?
To find the perimeter of a triangle, you need to add the lengths of all three of its sides. If the triangle is isosceles, then you can multiply the length of one edge by 2 and add the length of the base, so you get the perimeter of an isosceles triangle.
One of the basic geometric shapes is a triangle. It is formed when three line segments intersect. These line segments form the sides of the figure, and the points of their intersection are called vertices. Every student studying a geometry course must be able to find the perimeter of this figure. The acquired skill will be useful for many and in adult life, for example, useful to a student, engineer, builder,
There are different ways to find the perimeter of a triangle. The choice of the formula you need depends on the available source data. To write this value in mathematical terminology, a special designation is used - P. Consider what the perimeter is, the main methods for calculating it for triangular figures of various types.
by the most in a simple way find the perimeter of a figure if all sides are given. In this case, the following formula is used:
The letter "P" denotes the value of the perimeter itself. In turn, "a", "b" and "c" are the lengths of the sides.
Knowing the size of the three quantities, it will be enough to get their sum, which is the perimeter.
Alternative option
AT mathematical problems all given lengths are rarely known. In such cases, it is recommended to use an alternative way to find the desired value. When the conditions specify the length of two straight lines, as well as the angle between them, the calculation is made through the search for the third one. To find this number, you need to get Square root according to the formula:
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Perimeter on both sides
To calculate the perimeter, it is not necessary to know all the data geometric figure. Consider the methods of calculation on two sides.
Isosceles triangle
A triangle is called isosceles if at least two of its sides have the same length. They are called lateral, and the third side is called the base. Equal lines form a vertex angle. A feature in an isosceles triangle is the presence of one axis of symmetry. Axis is a vertical line starting from the top corner and ending in the middle of the base. At its core, the axis of symmetry includes the following concepts:
- vertex angle bisector;
- median to base;
- the height of the triangle;
- median perpendicular.
To determine the perimeter of an isosceles triangular figure, use the formula.
In this case, you need to know only two quantities: the base and the length of one side. The designation "2a" implies multiplying the length of the side by 2. To the resulting figure, you need to add the value of the base - "b".
In the exceptional case, when the length of the base of an isosceles triangle is equal to its lateral line, a simpler method can be used. It is expressed in the following formula:
To get the result, it is enough to multiply this number by three. This formula is used to find the perimeter of a regular triangle.
Useful video: problems on the perimeter of a triangle
Triangle rectangular
The main difference between a right triangle and other geometric shapes of this category is the presence of an angle of 90 °. On this basis, the type of figure is determined. Before determining how to find the perimeter of a right triangle, it is worth noting that this value for any flat geometric figure is the sum of all sides. So in this case, the easiest way to find out the result is to sum the three values.
In scientific terminology, those sides that are adjacent to right angle, are called "legs", and the opposite to the angle of 90º is the hypotenuse. The features of this figure were studied by the ancient Greek scientist Pythagoras. According to the Pythagorean theorem, the square of the hypotenuse is equal to the sum squares of legs.
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Based on this theorem, another formula has been derived that explains how to find the perimeter of a triangle given two known sides. You can calculate the perimeter with the specified length of the legs using the following method.
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To find out the perimeter, having information about the size of one leg and the hypotenuse, you need to determine the length of the second hypotenuse. For this purpose, the following formulas are used:
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Also, the perimeter of the described type of figure is determined without data on the dimensions of the legs.
You will need to know the length of the hypotenuse as well as the angle adjacent to it. Knowing the length of one of the legs, if there is an angle adjacent to it, the perimeter of the figure is calculated by the formula:
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Preliminary information
The perimeter of any flat geometric figure in the plane is defined as the sum of the lengths of all its sides. The triangle is no exception to this. First, we give the concept of a triangle, as well as the types of triangles depending on the sides.
Definition 1
We will call a triangle a geometric figure, which is composed of three points connected by segments (Fig. 1).
Definition 2
The points within Definition 1 will be called the vertices of the triangle.
Definition 3
The segments within the framework of Definition 1 will be called the sides of the triangle.
Obviously any triangle will have 3 vertices as well as 3 sides.
Depending on the ratio of the sides to each other, triangles are divided into scalene, isosceles and equilateral.
Definition 4
A triangle is said to be scalene if none of its sides is equal to any other.
Definition 5
We will call a triangle isosceles if two of its sides are equal to each other, but not equal to the third side.
Definition 6
A triangle is called equilateral if all its sides are equal to each other.
You can see all types of these triangles in Figure 2.
How to find the perimeter of a scalene triangle?
Let us be given a scalene triangle with side lengths equal to $α$, $β$ and $γ$.
Conclusion: To find the perimeter of a scalene triangle, add all the lengths of its sides together.
Example 1
Find the perimeter of a scalene triangle equal to $34$ cm, $12$ cm and $11$ cm.
$P=34+12+11=57$ cm
Answer: $57 see.
Example 2
Find the perimeter of a right triangle whose legs are $6$ and $8$ cm.
First, we find the length of the hypotenuses of this triangle using the Pythagorean theorem. Denote it by $α$, then
$α=10$ According to the rule for calculating the perimeter of a scalene triangle, we get
$P=10+8+6=24$ cm
Answer: $24 see.
How to find the perimeter of an isosceles triangle?
Let us be given an isosceles triangle whose side lengths will be equal to $α$, and the length of the base will be equal to $β$.
By definition of the perimeter of a flat geometric figure, we get that
$P=α+α+β=2α+β$
Conclusion: To find the perimeter of an isosceles triangle, add twice the length of its sides to the length of its base.
Example 3
Find the perimeter of an isosceles triangle if its sides are $12$ cm and its base is $11$ cm.
From the example above, we see that
$P=2\cdot 12+11=35$ cm
Answer: $35 see.
Example 4
Find the perimeter of an isosceles triangle if its height drawn to the base is $8$ cm and the base is $12$ cm.
Consider the figure according to the condition of the problem:
Since the triangle is isosceles, $BD$ is also a median, hence $AD=6$ cm.
By the Pythagorean theorem, from the triangle $ADB$, we find the side. Denote it by $α$, then
According to the rule for calculating the perimeter of an isosceles triangle, we get
$P=2\cdot 10+12=32$ cm
Answer: $32 see.
How to find the perimeter of an equilateral triangle?
Let us be given an equilateral triangle with lengths of all sides equal to $α$.
By definition of the perimeter of a flat geometric figure, we get that
$P=α+α+α=3α$
Conclusion: To find the perimeter of an equilateral triangle, multiply the side length of the triangle by $3$.
Example 5
Find the perimeter of an equilateral triangle if its side is $12$ cm.
From the example above, we see that
$P=3\cdot 12=36$ cm
