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 related by the following relation: a 2 + b 2 = c 2, where a, b are the 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 triangles 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 using the formula: P = a + b + c.
- In our example: P = 10 + 12 + 16.53 = 38.53.
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 answer you receive. 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.
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: $35$ 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.
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
How to find the perimeter of a triangle? Each of us asked this question while studying at school. Let's try to remember everything we know about this amazing figure, and also answer the question asked.
The answer to the question of how to find the perimeter of a triangle is usually quite simple - you just need to perform the procedure of adding the lengths of all its sides. However, there are several more simple methods for finding the desired value.
Adviсe
If the radius (r) of the circle inscribed in the triangle and its area (S) are known, then answering the question of how to find the perimeter of the triangle is quite simple. To do this you need to use the usual formula:
If two angles are known, say α and β, which are adjacent to the side, and the length of the side itself, then the perimeter can be found using a very, very popular formula, which looks like:
sinβ∙а/(sin(180° - β - α)) + sinα∙а/(sin(180° - β - α)) + а
If you know the lengths of adjacent sides and the angle β between them, then in order to find the perimeter, you need to use The perimeter is calculated using the formula:
P = b + a + √(b2 + a2 - 2∙b∙a∙cosβ),
where b2 and a2 are the squares of the lengths of adjacent sides. The radical expression is the length of the third side that is unknown, expressed using the cosine theorem.
If you don't know how to find the perimeter, then there's actually nothing complicated here. Calculate it using the formula:
where b is the base of the triangle, a is its sides.
To find the perimeter of a regular triangle, use the simplest formula:
where a is the length of the side.
How to find the perimeter of a triangle if only the radii of the circles that are circumscribed around it or inscribed in it are known? If the triangle is equilateral, then the formula should be applied:
P = 3R√3 = 6r√3,
where R and r are the radii of the circumcircle and inscribed circle, respectively.
If the triangle is isosceles, then the formula applies to it:
P=2R (sinβ + 2sinα),
where α is the angle that lies at the base, and β is the angle that is opposite to the base.
Often, solving mathematical problems requires in-depth analysis and a specific ability to find and derive the required formulas, and this, as many people know, is quite difficult work. Although some problems can be solved with just one formula.
Let's look at the formulas that are basic for answering the question of how to find the perimeter of a triangle, in relation to a wide variety of types of triangles.
Of course, the main rule for finding the perimeter of a triangle is this statement: to find the perimeter of a triangle, you need to add the lengths of all its sides using the appropriate formula:
where b, a and c are the lengths of the sides of the triangle, and P is the perimeter of the triangle.
There are several special cases of this formula. Let's say your problem is formulated as follows: “how to find the perimeter of a right triangle?” In this case, you should use the following formula:
P = b + a + √(b2 + a2)
In this formula, b and a are the immediate lengths of the legs of the right triangle. It is easy to guess that instead of the side with (hypotenuse), an expression is used, obtained from the theorem of the great scientist of antiquity - Pythagoras.
If you need to solve a problem where the triangles are similar, then it would be logical to use this statement: the ratio of the perimeters corresponds to the similarity coefficient. Let's say you have two similar triangles - ΔABC and ΔA1B1C1. Then, to find the similarity coefficient, it is necessary to divide the perimeter ΔABC by the perimeter ΔA1B1C1.
In conclusion, it can be noted that the perimeter of a triangle can be found using a variety of techniques, depending on the initial data that you have. It should be added that there are some special cases for right triangles.
Often mathematical problems require in-depth analysis, the ability to search for solutions and select the necessary statements and formulas. It's easy to get confused in this kind of work. And yet there are problems whose solution boils down to the application of one formula. Such problems include the question of how to find the perimeter of a triangle.
Let's consider the basic formulas for solving this problem in relation to different types of triangles.
- The basic rule for finding the perimeter of a triangle is the following statement: the perimeter of a triangle is equal to the sum of the lengths of all its sides. Formula P=a+b+c. Here a, b, c are the lengths of the sides of the triangle, P is its perimeter.
- There are special cases of this formula. For example:
- if the question is how to find the perimeter of a right triangle, then you can use both the classic formula (see point 1) and a formula that requires less data: P=a+b+ (a 2 +b 2). Here a, b are the lengths of the legs of a right triangle. It is easy to notice that the third side (hypotenuse) is replaced by an expression from the Pythagorean theorem.
- The perimeter of an isosceles triangle is found using the formula P=2*a+b. Here a is the length of the side of the triangle, b is the length of its base.
- to find the perimeter of an equilateral (or regular) triangle, we calculate the value of the expression P=3*a, where a is the length of the side of the triangle.
- To solve problems where similar triangles appear, it is useful to know the following statement: the ratio of the perimeters is equal to the similarity coefficient. It is convenient to use the formula
P(ABC)/P(A 1 B 1 C 1)=k, where ABC ~ A 1 B 1 C 1, and k is the similarity coefficient.
Given ABC with sides 6, 8, and 10 and A 1 B 1 C 1 with sides 9, 12. It is known that angle B is equal to angle B 1. Find the perimeter of triangle A 1 B 1 C 1.
- Let AB=6, BC=8, AC=10- A 1 B 1 =9- B 1 C 1 =12. Note that AB/ A 1 B 1 =BC/ B 1 C 1, because 6/9=8/12=2/3. Moreover, according to the condition B=B 1. These angles are contained between sides AB, BC and A 1 B 1, B 1 C 1, respectively. Conclusion - based on the 2nd criterion of similarity of triangles, ABC A 1 B 1 C 1. Similarity coefficient k=2/3.
- Let's find using the formula in step 1 P(ABC) = 6+8+10=24 (units). You can use the formula of item 2a, because The Pythagorean theorem proves that ABC is rectangular.
- From paragraph 2d it follows, P(ABC)/P(A 1 B 1 C 1)=2/3. Therefore P(A 1 B 1 C 1)=3*P(ABC)/2=3*24/2=36 (units).
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