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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Test yourself on types of triangles!

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

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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, what are each of the two equal sides called ?

Question 2

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

Question 3

Does the diagram show an obtuse triangle?

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

Does the diagram show an obtuse triangle?

Question 2

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

Question 3

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

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 an acute-angled triangle?

Question 2

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

Question 3

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

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

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

Step-by-Step Solution

The problem requires us to identify the side of a right triangle that is opposite to its right angle.
In right triangles, one of the most crucial elements to recognize is the presence of a right angle (90 degrees).
The side that is directly across or opposite the right angle is known as the hypotenuse. It is also the longest side of a right triangle.
Therefore, when asked for the side opposite the right angle in a right triangle, the correct term is the hypotenuse.

Selection from the given choices corroborates our analysis:

Choice 1: Leg - In the context of right triangles, the "legs" are the two sides that form the right angle, not the side opposite to it.
Choice 2: Hypotenuse - This is the correct identification for the side opposite the right angle.

Therefore, the correct answer is $\text{Hypotenuse}$ .

Answer

Hypotenuse

Exercise #2

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

Step-by-Step Solution

In an isosceles triangle, there are three sides: two sides of equal length and one distinct side. Our task is to identify what the equal sides are called.

To address this, let's review the basic properties of an isosceles triangle:

An isosceles triangle is defined as a triangle with at least two sides of equal length.
The side that is different in length from the other two is usually called the "base" of the triangle.
The two equal sides of an isosceles triangle are referred to as the "legs."

Therefore, each of the two equal sides in an isosceles triangle is called a "leg."

In our problem, we confirm that the correct terminology for these two equal sides is indeed "legs," distinguishing them from the "base," which is the unequal side. This aligns with both the typical definitions and properties of an isosceles triangle.

Thus, the equal sides in an isosceles triangle are known as legs.

Answer

Legs

Exercise #3

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

Step-by-Step Solution

In a right triangle, there are specific terms for the sides. The two sides that form the right angle are referred to as the legs of the triangle. To differentiate, the side opposite the right angle is called the hypotenuse, which is distinct due to being the longest side. Hence, in response to the problem, the sides forming the right angle are correctly identified as Legs.

Answer

Legs

Exercise #4

Does the diagram show an obtuse triangle?

Video Solution

Step-by-Step Solution

To determine if the triangle in the diagram is obtuse, we will visually assess the angles:

Step 1: Identify the angles in the diagram. The triangle has three angles, with one angle appearing between the horizontal base and the left slanted side.
Step 2: Evaluate the angle between the base and the left side. If it opens wider than a right angle, it's considered obtuse. This angle seems to be greater than $90^\circ$ , indicating obtuseness.
Step 3: Conclude based on visual inspection. Since this key angle is greater than $90^\circ$ , the triangle must be an obtuse triangle.

Therefore, the solution to the problem is Yes; the diagram does show an obtuse triangle.

Answer

Yes

Exercise #5

Does the diagram show an obtuse triangle?

Video Solution

Step-by-Step Solution

To determine if the triangle shown in the diagram is obtuse, we proceed as follows:

Step 1: Identify that the diagram is indeed a triangle by observing the confluence of three edges forming a closed shape.
Step 2: Appreciate the geometric arrangement of the triangle, focusing on the sides' lengths and angles visually.
Step 3: Noticeably, the longest side of the triangle represents a noticeable tilt indicating the presence of an obtuse angle.

Based on the observation above, notably from the triangle's longest side against the base, it's clear that one angle is larger than $90^\circ$ . Hence, the triangle in the diagram is indeed an obtuse triangle.

Therefore, the correct answer is Yes.

Answer

Yes

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

Is the triangle in the drawing a right triangle?

Question 2

Can a right triangle be equilateral?

Question 3

Does the diagram show an obtuse 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

Step-by-Step Solution

Answer

Exercise #2

Step-by-Step Solution

Answer

Exercise #3

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