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Finding the perimeter of a triangle in various ways. How to find the perimeter of a triangle if not all sides are known Formula for finding the perimeter and area of a right triangle

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:

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.

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.

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:

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:

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:

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

The perimeter of any triangle is the length of the line that bounds the figure. To calculate it, you need to find out the sum of all sides of this polygon.

Calculation from given side lengths

Once their meanings are known, this is easy to do. Denoting these parameters by the letters m, n, k, and the perimeter by the letter P, we obtain the formula for calculation: P = m+n+k. Assignment: It is known that a triangle has sides lengths of 13.5 decimeters, 12.1 decimeters and 4.2 decimeters. Find out the perimeter. We solve: If the sides of this polygon are a = 13.5 dm, b = 12.1 dm, c = 4.2 dm, then P = 29.8 dm. Answer: P = 29.8 dm.

Perimeter of a triangle that has two equal sides

Such a triangle is called isosceles. If these equal sides have a length of a centimeters, and the third side has a length of b centimeters, then the perimeter is easy to find out: P = b + 2a. Assignment: a triangle has two sides of 10 decimeters, a base of 12 decimeters. Find P. Solution: Let the side a = c = 10 dm, the base b = 12 dm. Sum of sides P = 10 dm + 12 dm + 10 dm = 32 dm. Answer: P = 32 decimeters.

Perimeter of an equilateral triangle

If all three sides of a triangle have an equal number of units of measurement, it is called equilateral. Another name is correct. The perimeter of a regular triangle is found using the formula: P = a+a+a = 3·a. Problem: We have an equilateral triangular plot of land. One side is 6 meters. Find the length of the fence that can be used to enclose this area. Solution: If the side of this polygon is a = 6 m, then the length of the fence is P = 3 6 = 18 (m). Answer: P = 18 m.

A triangle that has an angle of 90°

It is called rectangular. The presence of a right angle makes it possible to find unknown sides using the definition of trigonometric functions and the Pythagorean theorem. The longest side is called the hypotenuse and is designated c. There are two more sides, a and b. Following the theorem named after Pythagoras, we have c 2 = a 2 + b 2 . Legs a = √ (c 2 - b 2) and b = √ (c 2 - a 2). Knowing the length of two legs a and b, we calculate the hypotenuse. Then we find the sum of the sides of the figure by adding these values. Assignment: The legs of a right triangle have lengths of 8.3 centimeters and 6.2 centimeters. The perimeter of the triangle needs to be calculated. Solve: Let us denote the legs a = 8.3 cm, b = 6.2 cm. Following the Pythagorean theorem, the hypotenuse c = √ (8.3 2 + 6.2 2) = √ (68.89 + 38.44) = √107 .33 = 10.4 (cm). P = 24.9 (cm). Or P = 8.3 + 6.2 + √ (8.3 2 + 6.2 2) = 24.9 (cm). Answer: P = 24.9 cm. The values of the roots were taken with an accuracy of tenths. If we know the values of the hypotenuse and leg, then we obtain the value of P by calculating P = √ (c 2 - b 2) + b + c. Problem 2: A section of land lying opposite an angle of 90 degrees, 12 km, one of the legs is 8 km. How long will it take to walk around the entire area if you move at a speed of 4 kilometers per hour? Solution: if the largest segment is 12 km, the smaller one is b = 8 km, then the length of the entire path will be P = 8 + 12 + √ (12 2 - 8 2) = 20 + √80 = 20 + 8.9 = 28.9 ( km). We will find the time by dividing the path by the speed. 28.9:4 = 7.225 (h). Answer: you can get around it in 7.3 hours. We take the value of the square roots and the answer accurate to tenths. You can find the sum of the sides of a right triangle if one of the sides and the value of one of the acute angles are given. Knowing the length of the leg b and the value of the angle β opposite it, we find the unknown side a = b/ tan β. Find the hypotenuse c = a: sinα. We find the perimeter of such a figure by adding the resulting values. P = a + a/ sinα + a/ tan α, or P = a(1 / sin α+ 1+1 / tan α). Task: In a rectangular Δ ABC with right angle C, leg BC has a length of 10 m, angle A is 29 degrees. We need to find the sum of the sides Δ ABC. Solution: Let us denote the known side BC = a = 10 m, the angle opposite it, ∟A = α = 30°, then side AC = b = 10: 0.58 = 17.2 (m), hypotenuse AB = c = 10: 0.5 = 20 (m). P = 10 + 17.2 + 20 = 47.2 (m). Or P = 10 · (1 + 1.72 + 2) = 47.2 m. We have: P = 47.2 m. We take the value of trigonometric functions accurate to hundredths, round the length of the sides and perimeter to tenths. Having the value of the leg α and the adjacent angle β, we find out what the second leg is equal to: b = a tan β. The hypotenuse in this case will be equal to the leg divided by the cosine of the angle β. We find out the perimeter by the formula P = a + a tan β + a: cos β = (tg β + 1+1: cos β)·a. Assignment: The leg of a triangle with an angle of 90 degrees is 18 cm, the adjacent angle is 40 degrees. Find P. Solution: Let us denote the known side BC = 18 cm, ∟β = 40°. Then the unknown side AC = b = 18 · 0.83 = 14.9 (cm), hypotenuse AB = c = 18: 0.77 = 23.4 (cm). The sum of the sides of the figure is P = 56.3 (cm). Or P = (1 + 1.3 + 0.83) * 18 = 56.3 cm. Answer: P = 56.3 cm. If the length of the hypotenuse c and some angle α are known, then the legs will be equal to the product of the hypotenuse for the first - by the sine and for the second - by the cosine of this angle. The perimeter of this figure is P = (sin α + 1+ cos α)*c. Assignment: The hypotenuse of a right triangle AB = 9.1 centimeters and the angle is 50 degrees. Find the sum of the sides of this figure. Solution: Let us denote the hypotenuse: AB = c = 9.1 cm, ∟A= α = 50°, then one of the legs BC has a length a = 9.1 · 0.77 = 7 (cm), leg AC = b = 9 .1 · 0.64 = 5.8 (cm). This means the perimeter of this polygon is P = 9.1 + 7 + 5.8 = 21.9 (cm). Or P = 9.1·(1 + 0.77 + 0.64) = 21.9 (cm). Answer: P = 21.9 centimeters.

An arbitrary triangle, one of whose sides is unknown

If we have the values of two sides a and c, and the angle between these sides γ, we find the third by the cosine theorem: b 2 = c 2 + a 2 - 2 ac cos β, where β is the angle lying between sides a and c. Then we find the perimeter. Task: Δ ABC has a segment AB with a length of 15 dm and a segment AC with a length of 30.5 dm. The angle between these sides is 35 degrees. Calculate the sum of the sides Δ ABC. Solution: Using the cosine theorem, we calculate the length of the third side. BC 2 = 30.5 2 + 15 2 - 2 30.5 15 0.82 = 930.25 + 225 - 750.3 = 404.95. BC = 20.1 cm. P = 30.5 + 15 + 20.1 = 65.6 (dm). We have: P = 65.6 dm.

The sum of the sides of an arbitrary triangle in which the lengths of two sides are unknown

When we know the length of only one segment and the value of two angles, we can find out the length of two unknown sides using the sine theorem: “in a triangle, the sides are always proportional to the values of the sines of opposite angles.” Where does b = (a* sin β)/ sin a. Similarly c = (a sin γ): sin a. The perimeter in this case will be P = a + (a sin β)/ sin a + (a sin γ)/ sin a. Task: We have Δ ABC. In it, the length of side BC is 8.5 mm, the value of angle C is 47°, and angle B is 35 degrees. Find the sum of the sides of this figure. Solution: Let us denote the lengths of the sides BC = a = 8.5 mm, AC = b, AB = c, ∟ A = α= 47°, ∟B = β = 35°, ∟ C = γ = 180° - (47° + 35°) = 180° - 82° = 98°. From the relations obtained from the sine theorem, we find the legs AC = b = (8.5 0.57): 0.73 = 6.7 (mm), AB = c = (7 0.99): 0.73 = 9.5 (mm). Hence the sum of the sides of this polygon is P = 8.5 mm + 5.5 mm + 9.5 mm = 23.5 mm. Answer: P = 23.5 mm. In the case where there is only the length of one segment and the values of two adjacent angles, we first calculate the angle opposite to the known side. All angles of this figure add up to 180 degrees. Therefore ∟A = 180° - (∟B + ∟C). Next, we find the unknown segments using the sine theorem. Task: We have Δ ABC. It has a segment BC equal to 10 cm. The value of angle B is 48 degrees, angle C is 56 degrees. Find the sum of the sides Δ ABC. Solution: First, find the value of angle A opposite side BC. ∟A = 180° - (48° + 56°) = 76°. Now, using the theorem of sines, we calculate the length of the side AC = 10·0.74: 0.97 = 7.6 (cm). AB = BC* sin C/ sin A = 8.6. The perimeter of the triangle is P = 10 + 8.6 + 7.6 = 26.2 (cm). Result: P = 26.2 cm.

Calculating the perimeter of a triangle using the radius of the circle inscribed within it

Sometimes neither side of the problem is known. But there is a value for the area of the triangle and the radius of the circle inscribed in it. These quantities are related: S = r p. Knowing the area of the triangle and radius r, we can find the semi-perimeter p. We find p = S: r. Problem: The plot has an area of 24 m2, radius r is 3 m. Find the number of trees that need to be planted evenly along the line enclosing this plot, if there should be a distance of 2 meters between two neighboring ones. Solution: We find the sum of the sides of this figure as follows: P = 2 · 24: 3 = 16 (m). Then divide by two. 16:2= 8. Total: 8 trees.

Sum of the sides of a triangle in Cartesian coordinates

The vertices of Δ ABC have coordinates: A (x 1 ; y 1), B (x 2 ; y 2), C(x 3 ; y 3). Let's find the squares of each side AB 2 = (x 1 - x 2) 2 + (y 1 - y 2) 2 ; BC 2 = (x 2 - x 3) 2 + (y 2 - y 3) 2; AC 2 = (x 1 - x 3) 2 + (y 1 - y 3) 2. To find the perimeter, just add up all the segments. Assignment: Coordinates of vertices Δ ABC: B (3; 0), A (1; -3), C (2; 5). Find the sum of the sides of this figure. Solution: putting the values of the corresponding coordinates into the perimeter formula, we get P = √(4 + 9) + √(1 + 25) + √(1 + 64) = √13 + √26 + √65 = 3.6 + 5.1 + 8.0 = 16.6. We have: P = 16.6. If the figure is not on a plane, but in space, then each of the vertices has three coordinates. Therefore, the formula for the sum of the sides will have one more term.

Vector method

If a figure is given by the coordinates of its vertices, the perimeter can be calculated using the vector method. A vector is a segment that has a direction. Its module (length) is indicated by the symbol ǀᾱǀ. The distance between points is the length of the corresponding vector, or the absolute value of the vector. Consider a triangle lying on a plane. If the vertices have coordinates A (x 1; y 1), M(x 2; y 2), T (x 3; y 3), then the length of each side is found using the formulas: ǀAMǀ = √ ((x 1 - x 2 ) 2 + (y 1 - y 2) 2), ǀMTǀ = √ ((x 2 - x 3) 2 + (y 2 - y 3) 2), ǀATǀ = √ ((x 1 - x 3) 2 + ( y 1 - y 3) 2). We obtain the perimeter of the triangle by adding the lengths of the vectors. Similarly, find the sum of the sides of a triangle in space.

1) y = 2x + 5 2) y = 4 – 3x 3) y = 8x – 2 4) y = 5x 5) y = 0.1x + 8 6) X = 2 7) Y = x – 3, y = 2x + 3 y = -3x + 1 y = 4x – 2 y = 5x + 2 y = 3 y = -x y = -3 + x, 1) 0 2) 0 3) 1 4) 0 5) 1 6) 1 7) Infinite set. with flashcard tests. Card No. 1. A10. Correlate the functions given by the formulas with their graphs (Fig. 1).

?

A right triangle is a special type of arbitrary triangle. Like any other triangle, it has three sides, but one of its angles must be 90 degrees. Once you have determined that a given triangle is a right triangle, you can begin to find its basic dimensions. One of the characteristics of a right triangle is its perimeter. Many geometry problems are devoted to finding the perimeter of a right triangle.

Where P is the perimeter of the triangle;

A, b, c - sides of the triangle.

Based on the Pythagorean theorem, it became possible to determine the perimeter of a right triangle by its two sides of known length. If the lengths of the legs are known, then the perimeter of the triangle is determined by finding the value of the hypotenuse using the formula:

If only one of the legs and the length of the hypotenuse are known, then the perimeter of the triangle is determined by finding the value of the missing leg using the formula:

If in a right triangle only the length of the hypotenuse c and one of the acute angles α adjacent to it are known, then the perimeter of the triangle in this case can be determined by the formula:

In the case when the conditions of the problem specify the length of the leg a and the value of the acute angle α opposite it, then the perimeter of a right triangle in this case is calculated by the formula:

If a side a with an adjacent angle β is given, then the perimeter of the triangle can be calculated based on the expression:

How to find the perimeter of a right triangle

P = a + b + c, where, let's say,

P = v(a2 + b2) + a + b, or

P = v(c2 – b2) + b + c.

P = (1 + sin? + cos?)*s.

P = a*(1/tg? + 1/sin? + 1)

P = a*(1/сtg? + 1/cos? + 1)

Perimeter of a right triangle formula

How to find the perimeter of a right triangle

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

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

Posting sponsor P&G Articles on the topic “How to find the perimeter of a right triangle” How to find the surface area of a pyramid How to find the perimeter if the area is known How to find the perimeter of an equilateral triangle

Method 1. If all three sides of the triangle are known, 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,

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 the calculation of 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)

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 is a quantity that implies the length of all sides of a flat (two-dimensional) geometric figure. For different geometric shapes, there are different ways to find the perimeter.

In this article you will learn how to find the perimeter of a figure in different ways, depending on its known faces.

Possible methods:

all three sides of an isosceles or any other triangle are known;
how to find the perimeter of a right triangle given its two known faces;
two faces and the angle that is located between them (cosine formula) without a center line and height are known.

First method: all sides of the figure are known

How to find the perimeter of a triangle when all three faces are known, you must use the following formula: P = a + b + c, where a,b,c are the known lengths of all sides of the triangle, P is the perimeter of the figure.

For example, three sides of the figure are known: a = 24 cm, b = 24 cm, c = 24 cm. This is a regular isosceles figure; to calculate the perimeter we use the formula: P = 24 + 24 + 24 = 72 cm.

This formula applies to any triangle., you just need to know the lengths of all its sides. If at least one of them is unknown, you need to use other methods, which we will discuss below.

Another example: a = 15 cm, b = 13 cm, c = 17 cm. Calculate the perimeter: P = 15 + 13 + 17 = 45 cm.

It is very important to mark the unit of measurement in the response received. In our examples, the lengths of the sides are indicated in centimeters (cm), however, there are different tasks in which other units of measurement are present.

Second method: a right triangle and its two known sides

In the case when the task that needs to be solved is given a rectangular figure, the lengths of two faces of which are known, but the third is not, it is necessary to use the Pythagorean theorem.

Describes the relationship between the faces of a right triangle. The formula described by this theorem is one of the best known and most frequently used theorems in geometry. So, the theorem itself:

The sides of any right triangle are described by the following equation: a^2 + b^2 = c^2, where a and b are the legs of the figure, and c is the hypotenuse.

Hypotenuse. It is always located opposite the right angle (90 degrees), and is also the longest edge of the triangle. In mathematics, it is customary to denote the hypotenuse with the letter c.
Legs- these are the edges of a right triangle that belong to a right angle and are designated by the letters a and b. One of the legs is also the height of the figure.

Thus, if the conditions of the problem specify the lengths of two of the three faces of such a geometric figure, using the Pythagorean theorem it is necessary to find the dimension of the third face, and then use the formula from the first method.

For example, we know the length of 2 legs: a = 3 cm, b = 5 cm. Substitute the values into the theorem: 3^2 + 4^2 = c^2 => 9 + 16 = c^2 => 25 = c ^2 => c = 5 cm. So, the hypotenuse of such a triangle is 5 cm. By the way, this example is the most common and is called. In other words, if two legs of a figure are 3 cm and 4 cm, then the hypotenuse will be 5 cm, respectively.

If the length of one of the legs is unknown, it is necessary to transform the formula as follows: c^2 – a^2 = b^2. And vice versa for the other leg.

Let's continue with the example. Now you need to turn to the standard formula for finding the perimeter of a figure: P = a + b + c. In our case: P = 3 + 4 + 5 = 12 cm.

Third method: on two faces and the angle between them

In high school, as well as university, you most often have to turn to this method of finding the perimeter. If the conditions of the problem specify the lengths of two sides, as well as the dimension of the angle between them, then you need to use the cosine theorem.

This theorem applies to absolutely any triangle, which makes it one of the most useful in geometry. The theorem itself looks like this: c^2 = a^2 + b^2 – (2 * a * b * cos(C)), where a,b,c are the standard lengths of the faces, and A,B and C are angles that lie opposite the corresponding faces of the triangle. That is, A is the angle opposite to side a and so on.

Let's imagine that a triangle is described, sides a and b of which are 100 cm and 120 cm, respectively, and the angle lying between them is 97 degrees. That is, a = 100 cm, b = 120 cm, C = 97 degrees.

All you need to do in this case is to substitute all the known values into the cosine theorem. The lengths of the known faces are squared, after which the known sides are multiplied between each other and by two and multiplied by the cosine of the angle between them. Next, you need to add the squares of the faces and subtract the second value obtained from them. The square root is taken from the final value - this will be the third, previously unknown side.

After all three sides of the figure are known, it remains to use the standard formula for finding the perimeter of the described figure from the first method, which we already love.

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