Preliminary information
The perimeter of any flat geometric figure in the plane is defined as the sum of the lengths of all its sides. The triangle is no exception to this. First, we give 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, which is composed of three points connected by segments (Fig. 1).
Definition 2
The points within Definition 1 will be called the vertices of the triangle.
Definition 3
The segments within the framework of Definition 1 will be called the sides of the triangle.
Obviously any triangle will have 3 vertices as well as 3 sides.
Depending on the ratio of the sides to each other, triangles are divided into scalene, isosceles and equilateral.
Definition 4
A triangle is said to be scalene if none of its sides is 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
A triangle is called 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 with side lengths equal to $α$, $β$ and $γ$.
Conclusion: To find the perimeter of a scalene triangle, add all the lengths of its sides together.
Example 1
Find the perimeter of a scalene triangle equal to $34$ cm, $12$ cm and $11$ cm.
$P=34+12+11=57$ cm
Answer: $57 see.
Example 2
Find the perimeter of a right triangle whose legs are $6$ and $8$ cm.
First, we find the length of the hypotenuses of this triangle using the Pythagorean theorem. 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 whose side lengths will be equal to $α$, and the length of the base will be equal to $β$.
By definition of the perimeter of a flat geometric figure, we get 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 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.
Consider the figure according to the condition of the problem:
Since the triangle is isosceles, $BD$ is also a median, hence $AD=6$ cm.
By the Pythagorean theorem, from the triangle $ADB$, we find the side. 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 with lengths of all sides equal to $α$.
By definition of the perimeter of a flat geometric figure, we get that
$P=α+α+α=3α$
Conclusion: To find the perimeter of an equilateral triangle, multiply the side length of the triangle by $3$.
Example 5
Find the perimeter of an equilateral triangle if its side is $12$ cm.
From the example above, we see that
$P=3\cdot 12=36$ cm
Preliminary information
The perimeter of any flat geometric figure in the plane is defined as the sum of the lengths of all its sides. The triangle is no exception to this. First, we give 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, which is composed of three points connected by segments (Fig. 1).
Definition 2
The points within Definition 1 will be called the vertices of the triangle.
Definition 3
The segments within the framework of Definition 1 will be called the sides of the triangle.
Obviously any triangle will have 3 vertices as well as 3 sides.
Depending on the ratio of the sides to each other, triangles are divided into scalene, isosceles and equilateral.
Definition 4
A triangle is said to be scalene if none of its sides is 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
A triangle is called 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 with side lengths equal to $α$, $β$ and $γ$.
Conclusion: To find the perimeter of a scalene triangle, add all the lengths of its sides together.
Example 1
Find the perimeter of a scalene triangle equal to $34$ cm, $12$ cm and $11$ cm.
$P=34+12+11=57$ cm
Answer: $57 see.
Example 2
Find the perimeter of a right triangle whose legs are $6$ and $8$ cm.
First, we find the length of the hypotenuses of this triangle using the Pythagorean theorem. 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 whose side lengths will be equal to $α$, and the length of the base will be equal to $β$.
By definition of the perimeter of a flat geometric figure, we get 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 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.
Consider the figure according to the condition of the problem:
Since the triangle is isosceles, $BD$ is also a median, hence $AD=6$ cm.
By the Pythagorean theorem, from the triangle $ADB$, we find the side. 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 with lengths of all sides equal to $α$.
By definition of the perimeter of a flat geometric figure, we get that
$P=α+α+α=3α$
Conclusion: To find the perimeter of an equilateral triangle, multiply the side length of the triangle by $3$.
Example 5
Find the perimeter of an equilateral triangle if its side is $12$ cm.
From the example above, we see that
$P=3\cdot 12=36$ cm
The perimeter of a triangle, as in other things and 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 in the formula: cm
Formula for calculating the perimeter isosceles triangle will look like this:
Formula for calculating the perimeter equilateral triangle:
An example of calculating the perimeter of an equilateral triangle. When all the sides of the figure are equal, then they can simply be multiplied by three. Let's say a regular triangle with a side of 5 cm is given in this case: cm
In general, when all sides are given, finding the perimeter is fairly easy. In other situations, it is required to find the size of the missing side. In a right triangle, you can find the third side the Pythagorean theorem. For example, if the lengths of the legs are known, then you can find the hypotenuse using the formula: 
Consider an example of calculating the perimeter of an isosceles triangle, provided that we know the length of the legs in a right-angled isosceles triangle.
Given a triangle with legs a \u003d b \u003d 5 cm. Find the perimeter. First, let's find the missing side with . 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, the missing one can be found using the formula: 
If the hypotenuse and one of the acute angles are known in a right triangle, then the missing side is found by the formula.
Any triangle is equal to the sum of the lengths of its three sides. The general formula for finding the perimeter of triangles is:
P = a + b + c
where P is the perimeter of the triangle a, b and c- his sides.
It can be found by adding the lengths of its sides in series or by multiplying the length of the side by 2 and adding the length of the base to the product. The general formula for finding the perimeter of isosceles triangles will look like this:
P = 2a + b
where P is the perimeter of an isosceles triangle, a- any of the sides, b- base.
You can find it by adding the lengths of its sides in series or by multiplying the length of any of its sides by 3. The general formula for finding the perimeter of equilateral triangles will look like this:
P = 3a
where P is the perimeter of an equilateral triangle, a- any of its sides.
Square
To measure the area of a triangle, you can compare it with a parallelogram. Consider a triangle ABC:
If you take a triangle equal to it and attach it so that you get a parallelogram, you get a parallelogram with the same height and base as this triangle:
In this case, the common side of the triangles folded together is the diagonal of the formed parallelogram. From the properties of parallelograms, it is known that the diagonal always divides the parallelogram into two equal triangles, which means that the area of each triangle is equal to half the area of the parallelogram.
Since the area of a parallelogram is equal to the product of its base and its height, the area of a triangle will be equal to half of this product. So for Δ ABC area will be equal to
Now consider a right triangle:
Two equal right-angled triangles can be folded into a rectangle if they are leaned against each other by the hypotenuse. Since the area of a rectangle is equal to the product of its adjacent sides, the area of a given triangle is:
From this we can conclude that the area of any right triangle is equal to the product of the legs divided by 2.
From these examples, it can be concluded that the area of any triangle is equal to the product of the length of the base and the height dropped to the base, divided by 2. The general formula for finding the area of triangles will look like this:
| S = | ah a |
| 2 |
where S is the area of the triangle, a- its foundation h a- height lowered to the base a.
Perimeter is the sum of all sides of a figure. This characteristic, along with the area, is equally in demand for all figures. The formula for the perimeter of an isosceles triangle logically follows from its properties, but the formula is not as complicated as obtaining and consolidating practical skills.
Perimeter Formula
The sides of an isosceles triangle are equal to each other. This follows from the definition and is clearly visible even from the name of the figure. It is from this property that the perimeter formula follows:
P=2a+b, where b is the base of the triangle, a is the side value.
Rice. 1. Isosceles triangle
It can be seen from the formula that to find the perimeter, it is enough to know the size of the base and one of the sides. Consider several problems on finding the perimeter of an isosceles triangle. We will solve the problems as the complexity increases, this will allow us to better understand the way of thinking that needs to be followed to find the perimeter.
Task 1
- In an isosceles triangle, the base is 6, and the height drawn to this base is 4. You need to find the perimeter of the figure.
Rice. 2. Drawing for task 1
The height of an isosceles triangle drawn to the base is also the median and the height. This property is very often used in solving problems related to isosceles triangles.
Triangle ABC of height VM is divided into two right-angled triangles: ABM and BCM. In the triangle AVM, the leg VM is known, the leg AM is equal to half the base of the triangle ABC, since VM is the median of the bisector and height. Using the Pythagorean theorem, we find the value of the hypotenuse AB.
$$AB^2=AM^2+BM^2$$
$$AB=\sqrt(AM^2+BM^2)=\sqrt(3^2+4^2)=\sqrt(9+16)=\sqrt(25)=5$$
Find the perimeter: P=AC+AB*2=6+5*2=16
Task 2
- In an isosceles triangle, the height drawn to the base is 10, and the acute angle at the base is 30 degrees. you need to find the perimeter of the triangle.
Rice. 3. Drawing for task 2
This task is complicated by the lack of information about the sides of the triangle, but knowing the value of the height and angle, one can find the leg AH in the right triangle ABH, and then the solution will follow the same scenario as in problem 1.
Let's find AH through the value of the sine:
$$sin (ABH)=(BH\over AB)=(1\over2)$$ - the sine of 30 degrees is a table value.
Let's express the desired side:
$$AB=((BH\over (1\over 2))) =BH*2=10*2=20$$
Through the cotangent we find the value of AH:
$$ctg(BAH)=(AH\over BH)=(1\over\sqrt(3))$$
$$AH=(BH\over\sqrt(3))=10*\sqrt(3)=17.32$$ - round the resulting value to the nearest hundredth.
Let's find the base:
AC=AH*2=17.32*2=34.64
Now that all the required values have been found, let's define the perimeter:
P=AC+2*AB=34.64+2*20=74.64
Task 3
- An isosceles triangle ABC has an area equal to $$16\over\sqrt(3)$$ and an acute angle at the base of 30 degrees. Find the perimeter of the triangle.
The values in the condition are often given as the product of the root and the number. This is done in order to protect the subsequent decision from errors as much as possible. It is better to round the result at the end of calculations
With such a formulation of the problem, it may seem that there are no solutions, because it is difficult to express one of the sides or the height from the available data. Let's try to decide differently.
Let's denote the height and half of the base in Latin letters: BH=h and AH=a
Then the base will be: AC=AH+HC=AH*2=2a
Area: $$S=(1\over 2)*AC*BH=(1\over 2)*2a*h=ah$$
On the other hand, the value of h can be expressed from the triangle ABH in terms of the tangent of an acute angle. Why tangent? Because in the triangle ABH we have already marked two legs a and h. One has to be expressed in terms of the other. Two legs together connect the tangent and cotangent. Traditionally, the cotangent and cosine are only used when the tangent or sine does not fit. This is not a rule, you can decide how convenient it is, it's just accepted.
$$tg(BAH)=(h\over(a))=(1\over\sqrt(3))$$
$$h=(a\over\sqrt(3))$$
Substitute the resulting value into the area formula.
$$S=a*h=a*(a\over\sqrt(3))=((a^2)\over\sqrt(3))$$
Express a:
$$a=\sqrt(S*\sqrt(3))=\sqrt(16\over\sqrt(3)*\sqrt(3))=\sqrt(16)=4$$
Substitute the value of a in the area formula and determine the value of the height:
$$S=a*h=(16\over\sqrt(3))$$
$$h=(S\over(a))=((16\over\sqrt(3))\over(4))=(4\over\sqrt(3))=2.31$$- value received rounded up to hundredths.
Through the Pythagorean theorem, we find the side of the triangle:
$$AB^2=AH^2+BH^2$$
$$AB=\sqrt(AH^2+BH^2)=\sqrt(4^2+2.31^2)=4.62$$
Substitute the values into the perimeter formula:
P=AB*2+AH*2=4.62*2+4*2=17.24
What have we learned?
We figured out in detail all the intricacies of finding the perimeter of an isosceles triangle. We solved three problems of different levels of complexity, showing by example how typical problems are solved for solving an isosceles triangle.
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