How to find the perimeter of a triangle with an unknown side. How to find the perimeter of a triangle if not all sides are known

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

Preliminary information

The perimeter of any flat geometric figure on a plane is defined as the sum of the lengths of all its sides. The triangle is no exception to this. First, we present 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 that is made up of three points connected to each other by segments (Fig. 1).

Definition 2

Within the framework of Definition 1, we will call the points the vertices of the triangle.

Definition 3

Within the framework of Definition 1, the segments will be called sides of the triangle.

Obviously, any triangle will have 3 vertices, as well as three sides.

Depending on the relationship of the sides to each other, triangles are divided into scalene, isosceles and equilateral.

Definition 4

We will call a triangle scalene if none of its sides are 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

We will call a triangle 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 whose side lengths are equal to $α$, $β$ and $γ$.

Conclusion: To find the perimeter of a scalene triangle, you need to add all the lengths of its sides together.

Example 1

Find the perimeter of the scalene triangle equal to $34$ cm, $12$ cm and $11$ cm.

$P=34+12+11=57$ cm

Answer: $57$ cm.

Example 2

Find the perimeter of a right triangle whose legs are $6$ and $8$ cm.

First, let's find the length of the hypotenuses of this triangle using the Pythagorean theorem. Let us 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, the lengths of the sides will be equal to $α$, and the length of the base will be equal to $β$.

By determining the perimeter of a flat geometric figure, we obtain 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 discussed above, we see that

$P=2\cdot 12+11=35$ cm

Answer: $35$ cm.

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.

Let's look at the drawing according to the problem conditions:

Since the triangle is isosceles, $BD$ is also the median, therefore $AD=6$ cm.

Using the Pythagorean theorem, from the triangle $ADB$, we find the lateral side. Let us 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 whose lengths of all sides are equal to $α$.

By determining the perimeter of a flat geometric figure, we obtain that

$P=α+α+α=3α$

Conclusion: To find the perimeter of an equilateral triangle, multiply the length of the side of the triangle by $3$.

Example 5

Find the perimeter of an equilateral triangle if its side is $12$ cm.

From the example discussed above, we see that

$P=3\cdot 12=36$ cm

In this article we will show with examples, how to find the perimeter of a triangle. Let's consider all the main cases, how to find the perimeters of triangles, even when not all side values ​​are known.

Triangle is a simple geometric figure consisting of three straight lines intersecting each other. In which the points of intersection of lines are called vertices, and the straight lines connecting them are called sides.
Perimeter of a triangle is called the sum of the lengths of the sides of a triangle. It depends on how much initial data we have to calculate the perimeter of the triangle which option we will use to calculate it.
First option
If we know the lengths of the sides n, y and z of the triangle, then we can determine the perimeter using the following formula: in which P is the perimeter, n, y, z are the sides of the triangle

perimeter of a rectangle formula

P = n + y + z

Let's look at an example:
Given a triangle ksv whose sides are k = 10 cm, s = 10 cm, v = 8 cm. find its perimeter.
Using the formula we get 10 + 10 + 8 = 28.
Answer: P = 28cm.

For an equilateral triangle, we find the perimeter as follows: the length of one side multiplied by three. the formula looks like this:
P = 3n
Let's look at an example:
Given a triangle ksv whose sides are k = 10 cm, s = 10 cm, v = 10 cm. find its perimeter.
Using the formula we get 10 * 3 = 30
Answer: P = 30cm.

For an isosceles triangle, we find the perimeter like this: to the length of one side multiplied by two, add the side of the base
An isosceles triangle is the simplest polygon in which two sides are equal and the third side is called the base.

P = 2n + z

Let's look at an example:
Given a triangle ksv whose sides are k = 10 cm, s = 10 cm, v = 7 cm. find its perimeter.
Using the formula we get 2 * 10 + 7 = 27.
Answer: P = 27cm.
Second option
When we do not know the length of one side, but we know the lengths of the other two sides and the angle between them, and the perimeter of the triangle can only be found after we know the length of the third side. In this case, the unknown side will be equal to the square root of the expression b2 + c2 - 2 ∙ b ∙ c ∙ cosβ

P = n + y + √ (n2 + y2 - 2 ∙ n ∙ y ∙ cos α)
n, y - side lengths
α is the size of the angle between the sides known to us

Third option
When we do not know the sides n and y, but we know the length of the side z and the values ​​adjacent to it. In this case, we can find the perimeter of the triangle only when we find out the lengths of two sides unknown to us, we determine them using the theorem of sines, using the formula

P = z + sinα ∙ z / (sin (180°-α - β)) + sinβ ∙ z / (sin (180°-α - β))
z is the length of the side known to us
α, β - sizes of the angles known to us

Fourth option
You can also find the perimeter of a triangle by the radius inscribed in its circumference and the area of ​​the triangle. We determine the perimeter using the formula

P=2S/r
S - area of ​​the triangle
r is the radius of the circle inscribed in it

We have discussed four different options for finding the perimeter of a triangle.
Finding the perimeter of a triangle is not difficult in principle. If you have any questions or additions to the article, be sure to write them in the comments.

By the way, on referatplus.ru you can download abstracts on mathematics for free.

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.

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.