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
Content:
The perimeter is the total length of the boundaries of a two-dimensional 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 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 cosine theorem).
Steps
1 According to these three sides
- 1 To find the perimeter use the formula: P = a + b + c, where a, b, c are the lengths of the 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 because 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 indicate the unit of measurement in your answer. In our example, the sides are measured in centimeters, so your final answer should 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 For two given 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 states that in any right triangle the sides are connected by the following relationship: a 2 + b 2 = c 2, where a, b are 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 a right angle. Label the hypotenuse as "c". Label the legs (sides adjacent to the right angle) 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 problem statement.
- 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 (simply 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 = c 2 ; 25= c 2 ; √25 = s. So c = 25.
- In the second example: 6 2 + b 2 = 10 2 ; 36 + b 2 = 100. Move 36 to the right side of the equation and get: b 2 = 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 triangle and is a very useful formula. Cosine theorem: c 2 = 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 to the corresponding sides as A, B, C (that is, the angle opposite to side “a”, label it “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 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 = 10 2 + 12 2 - 2 × 10 × 12 × cos(97)
- c 2 = 100 + 144 – (240 × -0.12187)
- c 2 = 244 – (-29.25)
- c 2 = 244 + 29.25
- c 2 = 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.
You can find the perimeter of a triangle not only by summing the lengths of its sides. What, for example, should you do if you are given one side and the angles of a triangle or, for example, two sides and the angle between them?”
1. In case all three sides are known.
The perimeter of an arbitrary triangle is a+b+c.
If an equilateral (regular) triangle is given, then P=3a, that is, the length of the side multiplied by three.
If an isosceles triangle is given, then P=2a+c, where a is the side and c is the base.
2. Given two sides and the value of the angle between them.
To begin with, from the cosine theorem you can find out the third side lying opposite the angle "beta". This side (let's call it side c) will be equal to the square root of the expression a 2 + b 2 -2∙a∙b∙cosbeta;.
Therefore, the perimeter is equal to"a+b+radic;(a 2 +b 2 -2∙a∙b∙cosbeta;).
3. If a side and two adjacent angles are known.
In this case, to find the perimeter of the triangle, it is necessary to take into account the theorem of sines.
Then the formula for calculating the perimeter will take the form " а+sinalpha;∙а/(sin(180deg;-alpha;-beta;)) + sinbeta;∙а/(sin(180deg;-alpha;-beta;)).
4. If the area of the triangle and the radius of the circle inscribed in the triangle are known.
The perimeter of the triangle can then be found through the ratio of twice the area to the radius of the inscribed circle: "P=2S/r.
Special cases
(perimeter expressed in terms of the radii of inscribed and circumscribed circles).
1. For a regular triangle P=3Rradic;3=6rradic;3.
2. For an isosceles triangle P=2R(2sinalpha;+sinbeta;).
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.
