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 $γ$.
Output: 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
$P=α+α+β=2α+β$
Output: 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
$P=α+α+α=3α$
Output: 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
P=a+b+c How to find the perimeter of a triangle: Everyone knows that finding the perimeter is easy - 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 side, expressed through the cosine theorem. 
Step 4 For an 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. So, for an 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.
The perimeter of a triangle, as in other things and any figure, is called the sum of the lengths of all sides. Quite often, this value helps to find the area or is used to calculate other parameters of the figure.
The formula for the perimeter of a triangle looks like this:
An example of calculating the perimeter of a triangle. Let a triangle be given with sides a = 4 cm, b = 6 cm, c = 7 cm. Substitute the data in the formula: cm
Formula for calculating the perimeter isosceles triangle will look like this:
Formula for calculating the perimeter equilateral triangle:
An example of calculating the perimeter of an equilateral triangle. When all the sides of the figure are equal, then they can simply be multiplied by three. Let's say a regular triangle with a side of 5 cm is given in this case: cm
In general, when all sides are given, finding the perimeter is fairly easy. In other situations, it is required to find the size of the missing side. In a right triangle, you can find the third side the Pythagorean theorem. For example, if the lengths of the legs are known, then you can find the hypotenuse using the formula: 
Consider an example of calculating the perimeter of an isosceles triangle, provided that we know the length of the legs in a right-angled isosceles triangle.
Given a triangle with legs a \u003d b \u003d 5 cm. Find the perimeter. First, let's find the missing side with . cm
Now let's calculate the perimeter: cm
The perimeter of a right isosceles triangle will be 17 cm.
In the case when the hypotenuse and the length of one leg are known, the missing one can be found using the formula: 
If the hypotenuse and one of the acute angles are known in a right triangle, then the missing side is found by the formula.
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, as well as answer the 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 some more simple methods of the 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 cosine theorem. 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 of an isosceles triangle, then there is, in fact, nothing complicated. 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, to solve mathematical problems, a deep analysis and a specific ability to find and derive the required formulas are required, 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 of a 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.
Content:
The perimeter is the total length of the boundaries of a 2D shape. If you want to find the perimeter of a triangle, then you must add the lengths of all its sides; if you don't know the length of at least one side of the triangle, you need to find it. This article will tell you (a) how to find the perimeter of a triangle given the three known sides; (b) how to find the perimeter of a right triangle when only two sides are known; (c) how to find the perimeter of any triangle when given two sides and the angle between them (using the law of cosines).
Steps
1 On three given sides
- 1 To find the perimeter, use the formula: P \u003d a + b + c, where a, b, c are the lengths of three sides, P is the perimeter.
- 2
Find the lengths of all three sides. In our example: a = 5, b = 5, c = 5.
- It is an equilateral triangle since all three sides are the same length. But the above formula applies to any triangle.
- 3
Add the lengths of all three sides to find the perimeter. In our example: 5 + 5 + 5 = 15, that is, P = 15.
- Another example: a = 4, b = 3, c = 5. P = 3 + 4 + 5 = 12.
- 4
Don't forget to include the unit of measurement in your answer. In our example, the sides are measured in centimeters, so your final answer must also include centimeters (or the units specified in the problem statement).
- In our example, each side is 5 cm, so the final answer is P = 15 cm.
2 Given two sides of a right triangle
- 1 Remember the Pythagorean theorem. This theorem describes the relationship between the sides of a right triangle and is one of the most famous and applied theorems in mathematics. The theorem says that in any right triangle the sides are connected by the following relationship: a 2 + b 2 \u003d c 2, where a, b are the legs, c is the hypotenuse.
- 2 Draw a triangle and label the sides as a, b, c. The longest side of a right triangle is the hypotenuse. It lies opposite the right angle. Label the hypotenuse as "c". The legs (sides adjacent to the right angle) are designated as "a" and "b".
- 3
Substitute the values of the known sides into the Pythagorean theorem (a 2 + b 2 = c 2). Instead of letters, substitute the numbers given in the condition of the problem.
- For example, a = 3 and b = 4. Substitute these values into the Pythagorean theorem: 3 2 + 4 2 = c 2 .
- Another example: a = 6 and c = 10. Then: 6 2 + b 2 = 10 2
- 4
Solve the resulting equation to find the unknown side. To do this, first square the known lengths of the sides (just multiply the number given to you by itself). If you are looking for the hypotenuse, add the squares of the two sides and take the square root of the resulting sum. If you are looking for a leg, subtract the square of the known leg from the square of the hypotenuse and take the square root of the resulting quotient.
- In the first example: 3 2 + 4 2 = c 2 ; 9 + 16 \u003d c 2; 25=c2; √25 = s. So c = 25.
- In the second example: 6 2 + b 2 = 10 2 ; 36 + b 2 \u003d 100. Transfer 36 to the right side of the equation and get: b 2 \u003d 64; b = √64. So b = 8.
- 5
- In our first example: P = 3 + 4 + 5 = 12.
- In our second example: P = 6 + 8 + 10 = 24.
3 According to two given sides and the angle between them
- 1 Any side of a triangle can be found using the law of cosines if you are given two sides and the angle between them. This theorem applies to any triangles and is a very useful formula. Cosine theorem: c 2 \u003d a 2 + b 2 - 2abcos (C), where a, b, c are the sides of the triangle, A, B, C are the angles opposite the corresponding sides of the triangle.
- 2 Draw a triangle and label the sides as a, b, c; label the angles opposite the corresponding sides as A, B, C (that is, the angle opposite the side "a", label it as "A", and so on).
- For example, given a triangle with sides 10 and 12 and an angle between them of 97°, that is, a = 10, b = 12, C = 97°.
- 3
Substitute the values given to you into the formula and find the unknown side "c". First, square the lengths of the known sides and add the resulting values. Then find the cosine of angle C (using a calculator or an online calculator). Multiply the lengths of the known sides by the cosine of the given angle and by 2 (2abcos(C)). Subtract the resulting value from the sum of the squares of the two sides (a 2 + b 2), and you get c 2 . Take the square root of this value to find the length of the unknown side "c". In our example:
- c 2 \u003d 10 2 + 12 2 - 2 × 10 × 12 × cos (97)
- c 2 \u003d 100 + 144 - (240 × -0.12187)
- c 2 \u003d 244 - (-29.25)
- c2 = 244 + 29.25
- c2 = 273.25
- c = 16.53
- 4
Add the lengths of the three sides to find the perimeter. Recall that the perimeter is calculated by the formula: P = a + b + c.
- In our example: P = 10 + 12 + 16.53 = 38.53.
