Definition of a triangle

Triangle is a geometric figure consisting of three points connected in series with each other.

A triangle has three sides and three angles.

There are many types of triangles, and they all have different properties. We list the main types of triangles:

1. Versatile(all sides of different lengths);
2. Isosceles(two sides are equal, two angles at the base are equal);
3. Equilateral(all sides and all angles are equal).

However, for all types of triangles, there is one universal formula for finding the perimeter of a triangle - this is the sum of the lengths of all sides of the triangle.

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Triangle Perimeter Formula

P = a + b + c P = a + b + c P=a +b +c

A , b , c a, b, c a, b, c are the lengths of the sides of the triangle.

Let's analyze the problem of finding the perimeter of a triangle.

Task

The triangle has sides: a = 28 cm, b = 46 cm, c = 51 cm. What is the perimeter of the triangle?

Solution
We use the formula for finding the perimeter of a triangle and substitute instead of a a a, bb b And c c c their numerical values:
P = a + b + c P = a + b + c P=a +b +c
P=28+46+51=125cm P=28+46+51=125\text(cm)P=2 8 + 4 6 + 5 1 = 1 2 5 cm

Answer:
P = 125 cm. P = 125 \text( cm.)P=1 2 5 cm .

Task

The triangle is equilateral with a side of 23 cm. What is the perimeter of the triangle?

Solution

P = a + b + c P = a + b + c P=a +b +c

But according to the condition, we have an equilateral triangle, that is, all its sides are equal. In this case, the formula will take the following form:

P = a + a + a = 3a P = a + a + a = 3aP=a +a +a =3a

Substitute the numerical value in the formula and find the perimeter of the triangle:

P = 3 ⋅ 23 = 69 cm P = 3\cdot23 = 69\text( cm)P=3 ⋅ 2 3 = 6 9 cm

Answer
P = 69 cm. P = 69 \text( cm.)P=6 9 cm .

Task

In an isosceles triangle, side b is 14 cm and base a is 9 cm. Find the perimeter of the triangle.

Solution
We use the formula for finding the perimeter of a triangle:

P = a + b + c P = a + b + c P=a +b +c

But by condition, we have an isosceles triangle, that is, its sides are equal. In this case, the formula will take the following form:

P = a + b + b = 2b + a P = a + b + b = 2b + aP=a +b +b=2b+a

We substitute numerical values ​​\u200b\u200binto the formula and find the perimeter of the triangle:

P = 2 ⋅ 14 + 9 = 28 + 9 = 37 cm P = 2 \cdot 14 + 9 = 28 + 9 = 37 \text( cm)P=2 ⋅ 1 4 + 9 = 2 8 + 9 = 3 7 cm

Answer
P = 37 cm. P = 37\text( cm.)P=3 7 cm .

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 in adulthood, for example, it will be 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.

The easiest way to find the perimeter of a shape is if you have data for all sides. 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

In 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 the square root using the formula:

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Perimeter on both sides

To calculate the perimeter, it is not necessary to know all the data of a 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 the right angle are called "legs", and the opposite to the 90º angle 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 of the squares of the 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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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 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.

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

A triangle is one of the fundamental geometric figures, which are three intersecting line segments. This figure was known even to the scientists of Ancient Egypt, Ancient Greece and Ancient China, who derived most of the formulas and patterns used by scientists, engineers and designers so far.

The main components of a triangle are:

Vertices - points of intersection of segments.

The sides are intersecting line segments.

Based on these components, they formulate such concepts as the perimeter of a triangle, its area, the inscribed and circumscribed circles. It has been known since school that the perimeter of a triangle is a numerical expression of the sum of all three of its sides. At the same time, there are a great many formulas for finding this value, depending on the initial data that the researcher has in this or that case.

1. The easiest way to find the perimeter of a triangle is used when the numerical values ​​​​of all three of its sides (x, y, z) are known, as a consequence:

2. The perimeter of an equilateral triangle can be found if we remember that for a given figure all sides, however, like all angles, are equal. Knowing the length of this side, the perimeter of an equilateral triangle can be determined by the formula:

3. In an isosceles triangle, unlike an equilateral one, only two sides have the same numerical value, so in this case, in general, the perimeter will be as follows:

4. The following methods are necessary in cases where the numerical values ​​of not all sides are known. For example, if the study has data on two sides, and the angle between them is known, then the perimeter of the triangle can be found using the definition of the third side and the known angle. In this case, this third party will be found by the formula:

z= 2x+2y-2xycosβ

Based on this, the perimeter of the triangle will be equal to:

P= x+y+2x+(2y-2xycos β)

5. In the case when the length of not more than one side of the triangle is initially given and the numerical values ​​\u200b\u200bof the two angles adjacent to it are known, then the perimeter of the triangle can be calculated based on the sine theorem:

P = x+sinβ x/(sin(180°-β)) + sinγ x/(sin(180°-γ))

6. There are cases when the known parameters of the circle inscribed in it are used to find the perimeter of a triangle. This formula is also known to most from the school bench:

P= 2S/r (S is the area of ​​the circle, while r is its radius).

From all of the above, it can be seen that the value of the perimeter of a triangle can be found in many ways, based on the data that the researcher owns. In addition, there are several more special cases of finding this value. So, the perimeter is one of the most important quantities and characteristics of a right triangle.

As you know, such a triangle is called a figure, the two sides of which form a right angle. The perimeter of a right triangle is found through the numerical expression of the sum of both legs and the hypotenuse. In the event that the researcher knows the data on only two sides, the rest can be calculated using the famous Pythagorean theorem: z \u003d (x2 + y2), if both legs are known, or x \u003d (z2 - y2), if the hypotenuse and leg are known.

In the event that the length of the hypotenuse and one of the angles adjacent to it are known, then the other two sides are found by the formulas: x \u003d z sinβ, y \u003d z cosβ. In this case, the perimeter will be:

P= z(cosβ + sinβ +1)

Also a special case is the calculation of the perimeter of a regular (or equilateral) triangle, that is, a figure in which all sides and all angles are equal. Calculating the perimeter of such a triangle from a known side is no problem, however, often the researcher knows some other data. So, if the radius of the inscribed circle is known, the perimeter of a regular triangle is found by the formula:

And if the value of the radius of the circumscribed circle is given, the perimeter of a regular triangle will be found as follows:

Formulas need to be memorized in order to be successfully applied in practice.