Equilateral triangle

Definition of equilateral triangle

The equilateral triangle is a triangle that all its sides have the same length.

This also implies that all its angles are equal, that is, each angle measures $60°$ degrees (remember that the sum of the angles of a triangle is $180°$ degrees and, therefore, these $180°$ degrees are divided equally by the three angles).

Detailed explanation

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In an isosceles triangle, the angle between two ___ is called the "___ angle".

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Next, we will see some examples of equilateral triangles:

Equilateral triangle

Examples of equilateral triangles

Another name for the equilateral triangle

Recall that a regular polygon is a geometric figure that:

Has all its sides equal
Has all its angles equal

Therefore, the equilateral triangle is also known as the regular three-sided polygon.

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Test your knowledge

Question 1

In an isosceles triangle, the angle between ? and ? is the "base angle".

Question 2

In an isosceles triangle, the third side is called?

Question 3

In an isosceles triangle, what are each of the two equal sides called ?

Other characteristics of equilateral triangles

Recall that within a triangle there are what are called remarkable lines, which are the heights, medians, perpendicular bisectors and bisectors, these lines intersect at the so-called remarkable points (orthocenter, barycenter, circumcenter and incenter respectively).

In an equilateral triangle, the remarkable lines coincide.
In an equilateral triangle, the remarkable points coincide at the same point.

Recall that triangles can be classified according to the measure of their interior angles. Within this classification we find the acute triangles which are characterized by having all its acute angles (less than $90°$ degrees).

Since an equilateral triangle has all its interior angles equal to $60°$ degrees, it is also an acute triangle.

If you are interested in learning more about other triangle topics, you can enter one of the following articles:

On the Tutorela blog you will find a variety of articles about mathematics.

Equilateral Triangle Exercise

Exercise 1

Given the triangle $\triangle ACB$ equilateral

Task:

What is the value of angle $∢ACB$ ?

Solution:

Given the equilateral triangle $∆ ACB$

In an equilateral triangle all its angles are $60°$ .

Therefore the angle $∢ACB$ is equal to $60°$

Answer:

$60°$

Do you know what the answer is?

Question 1

In a right triangle, the side opposite the right angle is called....?

Question 2

In a right triangle, the two sides that form a right angle are called...?

Question 3

Is the triangle in the drawing a right triangle?

Exercise 2

Given the equilateral triangle:

Homework:

What is the perimeter?

Solution:

Since we are given an equilateral triangle, we will multiply the given side by 3 and get the perimeter of the triangle.

$3\cdot 5=15$

Answer: $3\cdot 5=15$

Exercise 3

Given the equilateral triangle:

Task:

The perimeter of the triangle is equal to $33 cm$ . What is the value of $X$ ?

Solution:

Since we are given an equilateral triangle whose perimeter is $33 cm$ , all we have to do is divide the circumference by 3 and we get the side measure $X$ .

$33:3=11$

Answer: $11$

Check your understanding

Question 1

Is the triangle in the drawing a right triangle?

Question 2

Is the triangle in the drawing a right triangle?

Question 3

Choose the appropriate triangle according to the following:

Angle B equals 90 degrees.

Exercise 4

In the figure we are given an equilateral triangle.

The length of each side is equal to $7 cm$

For each side there is a semicircle.

Exercise 4 In the figure we are given an equilateral triangle

Task:

What is the area of the whole figure? Replace a $\pi=3.14$

Solution:

$S=S1+3S_2$

When $S=$ the area of the whole figure

Area of the triangle $S_1=$

Area of the semicircle $S_2=$

In an equilateral triangle the height merges with the middle and so when $AD⊥BC$

$3.5=\frac{1}{2}\cdot7=DC=BD$

Right triangle: $∆\text{ADC}$

We perform pythagoras:

$AD²+DC²=AC²$

$AD²+3.5²=7²$

$AD²+DC²=AC²$

$AD²=\frac{7\sqrt{3}}{2}$

$S_1=\frac{AD\cdot BC}{2}=\frac{\frac{7\sqrt{3}}{2}\cdot7}{2}≈21.22\operatorname{cm}²$

$S_{2=}\frac{1}{2}(Diámetro del círculo=7 cm)=\frac{1}{2}(radio 3.5\operatorname{cm})$

$S_{2=}=\frac{1}{2}\cdot π\cdot 3.5²=\frac{1}{2}\cdot 3.14\cdot 3.5²$

$≈19.23$

$S=21.22+3\cdot 19.23=78.91$

Answer: $78.91$

Ejemplos y ejercicios con soluciones de triángulo equilátero

Exercise #1

Calculate the size of angle X given that the triangle is equilateral.

Video Solution

Step-by-Step Solution

Remember that the sum of angles in a triangle is equal to 180.

In an equilateral triangle, all sides and all angles are equal to each other.

Therefore, we will calculate as follows:

$x+x+x=180$

$3x=180$

We divide both sides by 3:

$x=60$

Answer

Exercise #2

What is the size of each angle in an equilateral triangle?

Video Solution

Step-by-Step Solution

To solve this problem, we'll follow these steps:

Step 1: Identify that an equilateral triangle has all sides of equal length, which implies its angles are also equal.
Step 2: Utilize the property that the sum of angles in any triangle is $180^\circ$ .
Step 3: Since each angle is equal in an equilateral triangle, divide the total sum of $180^\circ$ by 3.

Now, let's work through each step:
Step 1: In an equilateral triangle, all angles are equal in size.
Step 2: The sum of angles in any triangle is always $180^\circ$ .
Step 3: Divide $180^\circ$ by 3.

Calculating $180^\circ \div 3 = 60^\circ$ .

Therefore, the size of each angle in an equilateral triangle is $60^\circ$ .

Answer

Exercise #3

Which kind of triangle is given in the drawing?

Video Solution

Step-by-Step Solution

As we know that sides AB, BC, and CA are all equal to 6,

All are equal to each other and, therefore, the triangle is equilateral.

Answer

Equilateral triangle

Exercise #4

Given the size of the 3 sides of the triangle, is it an equilateral triangle?

Video Solution

Step-by-Step Solution

To determine if the triangle is equilateral, we need to check if all three sides of the triangle are equal.

The given side lengths are $2X$ , $12 - X$ , and $12 - X$ .

For the triangle to be equilateral, we must have the equality:

$2X = 12 - X$

Let's solve this equation:

$\begin{aligned} 2X &= 12 - X \\ 2X + X &= 12 \\ 3X &= 12 \\ X &= \frac{12}{3} \\ X &= 4 \end{aligned}$

Substitute $X = 4$ back into the expressions for the sides:

$2X = 2(4) = 8$
$12 - X = 12 - 4 = 8$
The third side, also $12 - X = 8$ .

The three calculated side lengths are $8$ , $8$ , and $8$ .

Since all three sides are equal, the triangle is an equilateral triangle.

Therefore, the answer is Yes, the triangle is equilateral.

Answer

Yes

Exercise #5

Is the triangle in the drawing an acute-angled triangle?

Video Solution

Step-by-Step Solution

To determine if the triangle is an acute-angled triangle, we need to understand the nature of its angles. In an acute-angled triangle, all three angles are less than $90^\circ$ . However, we do not have explicit angle measures or side lengths shown in the drawing. Instead, we assess the probable nature of the depicted triangle.

Given that an acute-angled triangle must have its largest angle smaller than $90^\circ$ , comparison property of triangle sides through Pythagorean type logic suggests that an acute triangle inequality $c^2 < a^2 + b^2$ (for sides $a$ , $b$ , and hypotenuse $c$ ) must hold.

In our problem, the depiction ultimately leads us to infer the implied relations among the triangle's angles. The given solution and analysis indicate it does not meet this criterion.

Hence, the triangle in the given drawing is not an acute-angled triangle, confirming the choice: No.

Answer

Questions on the subject

What is an equilateral triangle for children?

It is a geometric figure formed by three equal sides.

Why is the triangle equilateral?

Because its three sides have the same length.

What are the angles of an equilateral triangle?

In an equilateral triangle its interior angles are acute and these measure $60°$ degrees each.

What is an equilateral, isosceles and scalene triangle?

They are geometric figures with three sides, the first one is characterized by having all its sides equal, the second one by having two equal sides and the third one by not having any equal side.

What is another name for the equilateral triangle?

A regular three-sided polygon.

What are the types of triangles?

If the classification is made with respect to their sides we have three types of triangles: equilateral, isosceles and scalene.

If the classification is made with respect to their angles we also find three types of triangles: acute-angled, right-angled and obtuse-angled.

Do you think you will be able to solve it?

Question 1

Given the values of the sides of a triangle, is it a triangle with different sides?

Question 2

Is the triangle in the drawing an acute-angled triangle?

Question 3

Is the triangle in the drawing an acute-angled triangle?

Start practice

Equilateral triangle

Definition of equilateral triangle

Test yourself on types of triangles!

Equilateral triangle

Another name for the equilateral triangle

Other characteristics of equilateral triangles

Equilateral Triangle Exercise

Exercise 1

Exercise 2

Exercise 3

Exercise 4

Ejemplos y ejercicios con soluciones de triángulo equilátero

Exercise #1

Video Solution

Step-by-Step Solution

Answer

Exercise #2

Video Solution

Step-by-Step Solution

Answer

Exercise #3

Video Solution

Step-by-Step Solution

Answer

Exercise #4

Video Solution

Step-by-Step Solution

Answer

Exercise #5

Video Solution

Step-by-Step Solution

Answer

Questions on the subject

More Questions

Types of Triangles