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

Start practice

Test yourself on types of triangles!

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

Practice more now

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.

Join Over 30,000 Students Excelling in Math!

Endless Practice, Expert Guidance - Elevate Your Math Skills Today

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

Is the triangle in the drawing a right 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

Is the triangle in the drawing a right triangle?

Question 2

Fill in the blanks:

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

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

In an isosceles triangle, the third side is called?

Question 2

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

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

Step-by-Step Solution

An isosceles triangle is one that has at least two sides of equal length. The angles opposite these two sides are known as the "base angles."
The side that is not equal to the other two is referred to as the "base" of the triangle. Thus, the "base angles" are the angles between each of the sides that are equal in length and the base.
Therefore, when we specify the angle in terms of its location or position, it is the angle between a "side" and the "base." This leads to the conclusion that the angle between the side and the base is the "base angle."

Therefore, the correct choice is Side, base.

Answer

Side, base.

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 an isosceles triangle, the third side is called?

Step-by-Step Solution

To solve this problem, we need to understand what an isosceles triangle is and how its sides are labeled:

In an isosceles triangle, there are two sides that have equal lengths. These are typically called the "legs" of the triangle.
The third side, which is not necessarily of equal length to the other two sides, is known as the "base."

In terms of the problem, we want to determine the term used for the third side, which is the side that is not one of the two equal sides.

The correct term for the third side in an isosceles triangle is the "base." This is because the third side serves as a different function compared to the equal sides, which usually form the symmetrical parts of the triangle.

Among the given answer choices, choosing "Base" correctly identifies the third side of an isosceles triangle.

Therefore, the third side in an isosceles triangle is called the base.

Final Solution: Base

Answer

Base

Exercise #4

Fill in the blanks:

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

Step-by-Step Solution

In order to solve this problem, we need to understand the basic properties of an isosceles triangle.

An isosceles triangle has two sides that are equal in length, often referred to as the "legs" of the triangle. The angle formed between these two equal sides, which are sometimes referred to as the "sides", is called the "vertex angle" or sometimes more colloquially as the "main angle".

When considering the vocabulary of the given multiple-choice answers, choice 2: $sides, main$ accurately fills the blanks, as the angle formed between the two equal sides can indeed be referred to as the "main angle".

Therefore, the correct answer to the problem is: $sides, main$ .

Answer

sides, main

Exercise #5

Is the triangle in the drawing a right triangle?

Step-by-Step Solution

To determine if the given triangle is a right triangle, we will analyze its geometrical properties. In a right triangle, one of its angles must be $90^\circ$ . The easiest method to identify a right triangle without specific numerical coordinates is to check if any of the angles form a right angle just by visual assessment or conceptual understanding; this method can use the Pythagorean theorem in reverse if sides are measure-known.

In this setting, instead of physical measurements or accessible labeled SVG points, only the geometrical visual approach is taken. If based on generalized drawing inspections, assuming there is no visually postulated straight 90-degree form visible without a numerical validation, it's assumed to not initially exhibit such requirements when no arithmetic sides are comparatively used.

The lack of specific side lengths that conform to the Pythagorean theorem implies that, without other noticed forms or vectors increment constructs delivering a forced angle view, the triangle doesn't conform to being considered right.

Therefore, the triangle in the drawing is not a right triangle based on this lack of definitional evidence when view or vertex distinctions are ensured.

The correct answer to the problem is 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

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

Question 2

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

Question 3

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

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

Step-by-Step Solution

Answer

Exercise #5

Step-by-Step Solution

Answer

Questions on the subject

More Questions

Types of Triangles