One of the basic geometric shapes is a triangle. It is formed at the intersection of three straight segments. These line segments form the sides of the figure, and their intersection points 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 adult life, 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 notation is used - P. Let's consider what the perimeter is, the main methods of calculating it for triangular figures of different types.

The easiest way to find the perimeter of a figure is if you have data on all sides. In this case, the following formula is used:

The letter “P” denotes 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 obtain 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 method of searching for the required value. When the conditions indicate the length of two straight lines, as well as the angle between them, the calculation is made by searching for the third. To find this number you need to find 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. Let's consider methods of calculation on both sides.

Isosceles triangle

An isosceles triangle is one in which at least two sides have the same length. They are called lateral, and the third side is called the base. Equal straight lines form a vertex angle. A special feature of an isosceles triangle is the presence of one axis of symmetry. The axis is a vertical line extending from the apical angle and ending in the middle of the base. At its core, the axis of symmetry includes the following concepts:

• bisector of the vertex angle;
• median to base;
• height of triangle;
• median perpendicular.

To determine the perimeter of an isosceles triangular figure, use the formula.

In this case, you only need to know 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, you can use a simpler method. It is expressed in the following formula:

To get the result, just multiply this number by three. This formula is used to find the perimeter of an equilateral triangle.

Useful video: problems on the perimeter of a triangle

Right triangle

The main difference between a right triangle and other geometric shapes in this category is the presence of an angle of 90°. Based on this feature, 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 quantities.

In scientific terminology, those sides that are adjacent to the right angle are called “legs,” and those opposite to the 90º angle are called 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 is derived that explains how to find the perimeter of a triangle using two known sides. You can calculate the perimeter for 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 using the formula:

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Calculation via height

You can calculate the perimeter of categories such as isosceles and right triangles using their midline indicator. As you know, the height of a triangle divides its base in half. Thus, it forms two rectangular shapes. Next, the desired indicator is calculated using the Pythagorean theorem. The formula will look like this:

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If you know the height and half of the base, using this method you will get the number you need without searching for the rest of the data about the figure.

Useful video: finding the perimeter of a triangle

A right triangle is one in which one of the angles is 90 degrees and the other two are acute angles. Calculation perimeter such triangle will depend on the number of known data about him.

You will need

• Depending on the case, skill 2 of the 3 sides of the triangle, as well as one of its acute angles.

Instructions

1. Method 1. If all three sides are famous triangle, then, regardless of whether the triangle is right-angled or not, its perimeter will be calculated as follows: P = a + b + c, where, possibly, c is the hypotenuse; a and b are the legs.

2. Method 2. If only 2 sides are known in a rectangle, then, using the Pythagorean theorem, the perimeter of this triangle can be calculated using the formula: P = v(a2 + b2) + a + b, or P = v(c2 – b2) + b + c.

3. Method 3. Let a hypotenuse c and an acute angle? be given in a right triangle, then it will be possible to find the perimeter in this way: P = (1 + sin? + cos?)*c.

4. Method 4. It is given that in a right triangle the length of one of the legs is equal to a, and opposite it lies an acute angle?. Then the calculation perimeter this triangle will be carried out according to the formula: P = a*(1/tg ? + 1/sin ? + 1)

5. Method 5. Let us enter leg a and the angle adjacent to it?, then the perimeter will be calculated as follows: P = a*(1/сtg ? + 1/cos ? + 1)

Video on the topic

Perimeter of a triangle, as with 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 into the formula: cm

Formula for calculating perimeter isosceles triangle will look like this:

Formula for calculating perimeter equilateral triangle:

An example of calculating the perimeter of an equilateral triangle. When all sides of a figure are equal, they can simply be multiplied by three. Suppose we are given a regular triangle with a side of 5 cm in this case: cm

In general, once all the sides are given, finding the perimeter is quite simple. In other situations, you need to find the size of the missing side. In a right triangle you can find the third side by Pythagorean theorem. For example, if the lengths of the legs are known, then you can find the hypotenuse using the formula:

Let's consider an example of calculating the perimeter of an isosceles triangle, provided that we know the length of the legs in a right isosceles triangle.
Given a triangle with legs a =b =5 cm. Find the perimeter. First, let's find the missing side c. 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, you can find the missing one using the formula:
If the hypotenuse and one of the acute angles are known in a right triangle, then the missing side is found using the formula.

The right triangle is a simple but extremely important figure for mathematics. Knowledge of its properties and the ability to operate with the basic parameters of a right triangle will allow you to cope with both school and real-life problems.

Geometry of a right triangle

Geometrically, a triangle is three points that do not lie on the same line, which are connected by segments. A right triangle is a figure whose two sides form a right angle. These sides are called the legs of the triangle, and the third, longest side is called the hypotenuse. The relationship between the squares of the legs and the hypotenuse is established by the Pythagorean theorem - one of the fundamental theorems of Euclidean geometry.

The relationships between the hypotenuse and legs also laid the basis for an entire branch of mathematics - trigonometry. Originally, sines and cosines were defined as functions of the angles of a right triangle, but in their modern meaning, trigonometric functions have been extended to the entire number line. Today, trigonometry is used in many areas of human activity: from astronomy and oceanography to financial market analysis and computer game development.

Right triangle in reality

The right triangle itself is found in reality at every corner, both literally and figuratively. The faces of tetrahedrons and prisms have the shape of a right triangle, which in reality turn into machine parts, ceramic tiles or roof slopes. A square is a drawing tool that a person first encounters in a geometry lesson; it has the shape of a right triangle and is used in design, construction and carpentry.

Perimeter of a triangle

Perimeter is a numerical estimate of the lengths of all sides of a flat geometric figure. The perimeter of an n-gon is found as the sum of the lengths of n sides. To determine the perimeter of a right triangle, use a simple formula:

a and b – legs, c – hypotenuse.

To calculate the perimeter of a triangle by hand, you would have to measure all three sides, perform additional trigonometric operations, or perform calculations using the Pythagorean theorem. Using an online calculator you just need to find out the following pairs of variables:

• two legs;
• leg and angle;
• hypotenuse and angle.

In school problems or in practice, you will be given initial data, so the calculator allows you to find the perimeter, knowing different pairs of parameters. In addition, the tool automatically calculates all other attributes of a right triangle, that is, the lengths of all sides and the magnitudes of all angles. Let's look at a couple of examples.

Examples from life

School task

Let's say in a school problem you are given a right triangle with a side length of 5 cm and an adjacent angle of 60 degrees. You need to find the perimeter of a geometric figure. The online calculator is accompanied by a drawing showing the sides and angles of a right triangle. We see that if leg a = 5 cm, then its adjacent angle is angle beta. This is an important point, because if you use the alpha angle for calculations, the result will be incorrect. We enter this data into the form and get a response like:

In addition to the perimeter itself, our program also determined the value of the opposite angle, as well as the length of the second leg and hypotenuse.

Flowerbed arrangement

Let's say you want to make a fence for a flower bed that has the shape of a right triangle. To do this, you need to know the perimeter of the figure. Of course, in reality you can simply measure all three sides, but it is easy to simplify your task and measure only two legs. Let them be 8 and 15 meters long. We enter this data into the calculator form and get the answer:

So, you will need to purchase materials to build 40 meters of fencing. Our calculator also calculated the length of the hypotenuse - 17 meters. The numbers 8, 15 and 17 form a Pythagorean triple - natural numbers that satisfy the conditions of the Pythagorean theorem.

Conclusion

Right triangles are widely used in everyday life, so determining the area or perimeter of a geometric figure will certainly be useful to you when solving school problems or everyday issues.

A right triangle is one in which one of the angles is 90 degrees and the other two are acute angles. Calculation of the perimeter of such triangle will depend on the amount of data known about it.

You will need

• Depending on the case, knowledge of two of the three sides of a triangle, as well as one of its acute angles.

Instructions

• Method 1. If all three sides are known triangle, then, regardless of whether the triangle is right-angled or not, its perimeter will be calculated as follows:
  P = a + b + c, where, let's say,
  c - hypotenuse;
  a and b are legs.
• Method 2. If only 2 sides are known in a rectangle, then, using the Pythagorean theorem, the perimeter of this triangle can be calculated using the formula:
  P = v(a2 + b2) + a + b, or
  P = v(c2 – b2) + b + c.
• Method 3. Let a hypotenuse c and an acute angle ? be given in a right triangle, then the perimeter can be found in this way:
  P = (1 + sin? + cos?)*s.
• Method 4. It is given that in a right triangle the length of one of the legs is equal to a, and opposite it lies an acute angle?. Then calculating the perimeter of this triangle will be carried out according to the formula:
  P = a*(1/tg ? + 1/sin ? + 1)
• Method 5. Let us know the side a and the angle adjacent to it?, then the perimeter will be calculated as follows:
  P = a*(1/сtg ? + 1/cos ? + 1)