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- Equilateral triangle - Definition with examples

wLyseOur mission is to make mathematics engaging and accessible to children. By introducing them to the beauty and meaning of the equilateral triangle, we encourage them to understand geometry and stimulate their curiosity to explore the world of shapes and patterns.

## What is an equilateral triangle?

An equilateral triangle is one of the simplest yet most fascinating concepts in geometry. It is a triangle, a triangular polygon, but not just any triangle. All three sides of an equilateral triangle are of equal length, hence the prefix "equi-" means equal. This perfect symmetry combines a number of special properties and characteristics, making the equilateral triangle a cornerstone of geometric research. Think of it as a perfect trinity, with each member contributing equally. it means an equilateral triangle in the world of forms.

## Equilateral triangular shape

The shape of an equilateral triangle is perfectly symmetrical. If you place it on a flat surface, you'll notice that it balances perfectly no matter which side it's on. This is due to equal sides and angles, a balance that can only be achieved when all three sides are of equal length. It's almost as if nature itself used a ruler to draw each side with absolute precision, creating a shape as symmetrical as possible. You'll find equilateral triangles in many aspects of everyday life, from your favorite brand's logo to the construction of a well-built bridge.

## Properties of an equilateral triangle

The properties of an equilateral triangle are quite interesting. All three sides are equal, as previously mentioned, but did you know that all three angles are also equal? Each angle is exactly 60 degrees, which adds up to the 180 degree rule for all triangles. Additionally, any height or altitude drawn from a vertex to the midpoint of the opposite side also bisects the vertex angle, creating two angles of 30 degrees each. The altitudes, medians, and bisectors of the angles coincide, making this triangle a line of symmetry. These properties make an equilateral triangle a perfect example of harmony and balance in mathematics.

## Compare: The scale, isosceles and equilateral triangles

By comparing the stairs, isosceles and equilateral triangles, we see that each has unique properties. An isosceles triangle has no equal sides or angles, an isosceles triangle has two equal sides and two equal angles, while an isosceles triangle exceeds both with all sides and all angles equal. It's like comparing different teams in a game, each with unique strategies and strengths, but an equilateral triangle is a team where all players are equally strong and coordinated.

## Types of equilateral triangles

Formulas play a vital role in understanding and calculating various aspects of an equilateral triangle. The area of an equilateral triangle can be calculated using the formula`Pole = (sqrt(3)/4) * Hi^2`

. The perimeter of an equilateral triangle can be found from the formula`Perimeter = 3 * side`

. There is also a special theorem known as the equilateral triangle theorem, which states that if a triangle is equilateral, then it must also be equilateral and vice versa.

## What are equilateral triangles NOT?

A triangle that does not have all three sides equal is NOT an equilateral triangle. This includes isosceles triangles with only two equal sides and scalar triangles with no equal sides. Similarly, right triangles, obtuse triangles, or acute triangles with odd sides are not equilateral triangles. Remember that an equilateral triangle is the epitome of symmetry and balance, and any deviation from this symmetry does not qualify as an equilateral triangle.

## Perimeter and area of an equilateral triangle

The perimeter and area of an equilateral triangle are fairly easy to calculate because of its symmetry. The perimeter is three times as long as one side, while the area is calculated with the formula`Pole = (sqrt(3)/4) * Hi^2`

. Understanding these formulas can help solve various geometric problems, such as finding the size of a plot or the area of a triangular park. Remember that math is not just about memorizing formulas, but also about understanding how to apply them in real life.

## The height of an equilateral triangle

The height of an equilateral triangle can be calculated using the formula`Height = (sqrt(3)/2) * Side`

. Notably, this height also intersects the base, dividing the equilateral triangle into two equal triangles, adding another layer of symmetry and balance to this fascinating shape.

## Center of an equilateral triangle

The midpoint of an equilateral triangle is the point where all three medians of the triangle intersect. Due to the symmetrical properties of an equilateral triangle, the centroid, the centroid (the center of the circle that surrounds the triangle), and the orthocenter (the point of intersection of the altitudes) coincide at the same point. It is like a point of contact between the different sides of a triangle, further enhancing its symmetry.

## Around the center

The circumcenter of an equilateral triangle is the center of the circumscribed circle that can be drawn around the triangle so that it touches all three vertices. Because of the symmetry of an equilateral triangle, the center of the circumcircle, the centroid, and the orthocenter lie at the same point, which is the center of the triangle.

## Practice equilateral triangle problems

Now that we've covered the theory, let's move on to the practical. Here are some equilateral triangle problems to better understand:

- Given that one side of an equilateral triangle has 10 units, find the area.
- If the perimeter of an equilateral triangle is 45 units, find the length of one side.
- Given that the area of an equilateral triangle is 15,588 square units, find the length of one side.

## Application

At Brighterly, we believe that learning maths should be an exciting adventure, full of discovery and wonder. In this exploration of equilateral triangles, we delved into their definitions, properties, comparisons, and types. This tour revealed the inherent beauty and significance of these special triangles, demonstrating their unparalleled symmetry and their role as the fundamental building blocks of geometry.

Equilateral triangles are widely used in a number of fields, proving their importance beyond class boundaries. They can be found in architectural designs, where their balanced proportions create visually appealing structures. For example, the height of a pyramid can be calculated using the properties of equilateral triangles. In addition, equilateral triangles often appear in logo designs, symbolizing unity, balance and strength. Understanding the principles behind these forms allows us to appreciate the thought and creativity involved in such visual representations.

## Frequently asked questions about equilateral triangle

### Is every equilateral triangle also an isosceles triangle?

Yes, every equilateral triangle is also an isosceles triangle. An isosceles triangle has at least two sides of equal length, and since an isosceles triangle has all three sides equal, it meets the criteria for classification as both an isosceles and isosceles triangle.

### Can an equilateral triangle have a right angle?

No, an equilateral triangle cannot be a right triangle. In a right triangle, one of the angles is 90 degrees, which is not possible in an equilateral triangle where all angles are 60 degrees. If you want to know more about different types of triangles.

### What is the relationship between the sides and angles of an equilateral triangle?

In an equilateral triangle, all three sides are equal in length and all three angles are equal, each measuring 60 degrees. The sides and angles of an equilateral triangle are interconnected and form a harmonious relationship. It is this unique balance that distinguishes equilateral triangles from other types of triangles.

### How to find the height of an equilateral triangle?

The height of an equilateral triangle can be calculated using the formula`Height = (sqrt(3)/2) * Side`

where "Side" is the length of any side of an equilateral triangle. Height is the vertical distance from one side to the opposite mountain. This formula allows you to determine the height of an equilateral triangle without measuring it directly. More information on properties and calculations of equilateral triangles can be found in various geometry textbooks such as "Geometry" by Ray C. Jurgensen and Richard G. Brown.

## Information sources:

## FAQs

### What is the definition and properties of equilateral triangle? ›

In geometry, an equilateral triangle is **a triangle that has all its sides equal in length**. Since the three sides are equal therefore the three angles, opposite to the equal sides, are equal in measure. Therefore, it is also called an equiangular triangle, where each angle measure 60 degrees.

**What are the 3 properties of equilateral triangle? ›**

**Properties**

- All three sides are equal; in ∆ABC, sides AB = BC = CA.
- All three angles are equal (equiangular), ...
- There are three lines of symmetry.
- The altitude, median, angular bisector, and the perpendicular line are all same and together called the perpendicular bisector, shown as AE.

**What is the definition of an equilateral triangle? ›**

: **a triangle in which all three sides are the same length**.

**What is an example of an equilateral triangle? ›**

An equilateral triangle is a type of triangle. It is a regular polygon and has special properties: all three sides are equal in length, and all three angles in the corners are 60º. Examples in real life can include **traffic signs and tortilla chips**.

**What are the 7 properties of triangle? ›**

**Let us discuss here some of the properties of triangles.**

- A triangle has three sides and three angles.
- The sum of the angles of a triangle is always 180 degrees.
- The exterior angles of a triangle always add up to 360 degrees.
- The sum of consecutive interior and exterior angle is supplementary.

**What are the properties and definitions of triangles? ›**

The properties of a triangle are: **A triangle has three sides, three angles, and three vertices**. The sum of all internal angles of a triangle is always equal to 180^{°}^{.} This is called the angle sum property of a triangle. The sum of the length of any two sides of a triangle is greater than the length of the third side.

**Are all 3 angles of an equilateral triangle equal? ›**

Sal proves that **the angles of an equilateral triangle are all congruent** (and therefore they all measure 60°), and conversely, that triangles with all congruent angles are equilateral. Created by Sal Khan.

**Does an equilateral triangle have 3 right angles? ›**

So according to the definition of an equilateral triangle, **an equilateral triangle can not be a right -angled triangle**. Hence, an equilateral triangle can never be a right- angled triangle.

**What are the properties of equilateral? ›**

The three properties of an equilateral triangle are: All three sides are equal. All three angles are congruent. The figure has three lines of symmetry.

**What are the properties of equilateral triangle in a circle? ›**

By symmetry, the center of the equilateral triangle coincides with the center of the circle, and the distance from the center of the equilateral triangle to any of its vertices is equal to the radius of the circle.

### What is an equilateral triangle with 3 sides? ›

An equilateral triangle is **a triangle in which all three sides are equal**. Equilateral triangles also called equiangular. That means, all three internal angles are equal to each other and the only value possible is 60° each. It is a regular polygon with 3 sides.

**What shapes are equilateral examples? ›**

Equilateral polygon definition: An equilateral polygon is one having sides of equal length. **An equilateral triangle**, for example, has three sides of the same length, while a square has four. The term equilateral is derived from the Latin words equi-, or same, and latus, or side.

**What are the four equilateral triangle? ›**

**A regular tetrahedron** is made of four equilateral triangles.

**What are the 3 main types of triangles? ›**

Let's start with the three types of triangles that are categorized by the measure of their largest angle. These are the **acute, right, and obtuse triangles**.

**What are the 12 types of triangles? ›**

Types of Triangle Based on Sides | Types of Triangles Based on Angles |
---|---|

Equilateral Triangle | Acute-Angled Triangle (Acute Triangle) |

Isosceles Triangle | Right-Angled Triangle (Right Triangle) |

Scalene Triangle | Obtuse-Angled Triangle (Obtuse Triangle) |

**What are the 4 types of triangle with their properties? ›**

Equilateral Triangle: All the sides are equal and all the three angles equal to 60°. Acute Angled Triangle: A triangle having all its angles less than 90°. Right Angled Triangle: A triangle having one of the three angles exactly 90°. Obtuse Angled Triangle: A triangle having one of the three angles more than 90°.

**What is the definition of a triangle? ›**

A triangle is **a closed, 2-dimensional shape with 3 sides, 3 angles, and 3 vertices**. A triangle is also a polygon. The above figure is a triangle denoted as △ABC.

**What do the 4 triangles mean? ›**

The Tetractys symbolizes the four classical elements—**air, fire, water, and earth**. The Tetractys represented the organization of space: the first row represented zero dimensions (a point)

**How do you measure equilateral? ›**

Since the sum of a triangle's angles is always 180 degrees, **each angle in an equilateral triangle must measure 60 degrees**. This is because we must divide 180 degrees evenly between the three angles: 180 / 3 = 60.

**How do you classify equilateral triangle? ›**

In an equilateral triangle, **all the lengths of the sides are equal**. In such a case, each of the interior angles will have a measure of 60 degrees. Since the angles of an equilateral triangle are same, it is also known as an equiangular triangle.

### Are all equilateral triangles scalene? ›

If none of the sides of a triangle are equal (of equal length), the triangle is scalene. If two or more of the triangles sides are equal, the triangle is isosceles. If all three of the sides of a triangle are equal, it is equilateral. **All equilateral triangles are also isosceles**, by definition.

**Is a rectangle a equilateral? ›**

**A rectangle is equilateral but not equiangular**. Convex polygons have no portions of their diagonals in their exteriors.

**What is the measure of one of the interior angles of a regular twenty sided polygon? ›**

∴ The measure of each interior angle =324020=**162∘** Q. Find the measure of each exterior angle of a regular polygons having 20 sides.

**How do you calculate the size of the interior angles of a regular quadrilateral square? ›**

The common property for all the above four-sided shapes is the sum of interior angles is always equal to 360 degrees. For a regular quadrilateral such as square, each interior angle will be equal to: **360/4 = 90 degrees**.

**Which is an acute scalene triangle? ›**

In an acute-angled scalene triangle, **each angle of the triangle is less than 90°**. In simple words, all angles are acute angles. In an obtuse angled scalene triangle, there is one obtuse angle (between 90° and 180°) and remaining two angles are acute.

**Can any shape be equilateral? ›**

**A shape is equilateral if all the sides are the same length**. In geometry class, people learn about many shapes, such as triangles and squares. A square is equilateral, because all of its sides are the same length. A rhombus is also equilateral — its sides are also the same length.

**Are all squares equilateral? ›**

**All sides of a square are the same length (equilateral)** and all its angles have the same measure (equiangular). This makes the square the only equilateral and equiangular quadrilateral. Of course, we can just call it a "regular quadrilateral."

**How many degrees is an equilateral triangle? ›**

These are acute angles because they measure less than 90 degrees. Ultimately, an equilateral triangle is a triangle with 3 equal sides and three angles, each of 60 degrees, ensuring that there are 180 degrees in such a triangle.

**Is an equilateral triangle acute? ›**

All the sides of an equilateral triangle are of equal lengths—each interior angle of this triangle measures 60°. So, **an equilateral triangle is always an acute triangle**.

**What is the base of a triangle? ›**

Summary. We can choose **any of the three sides of a triangle** to call the base. The term “base” refers to both the side and its length (the measurement). The corresponding height is the length of a perpendicular segment from the base to the vertex opposite of it.

### What is the property of equilateral triangle and isosceles triangle? ›

**An equilateral triangle is the one in which all the three sides are equal**. It is a special case of the isosceles triangle where the third side is also equal. In the triangle ABC, AB = BC = CA.

**What is the exterior angle property of equilateral triangle? ›**

Each angle of an equilateral triangle is of 60∘. Therefore, each exterior angle is equal to **180∘−60∘=120∘**

**What is an obtuse equilateral triangle? ›**

An obtuse – angled equilateral triangle means **any one angle of a triangle is obtuse and the side adjacent to the angle is equal**. A right – angled equilateral triangle means any one angle of a triangle is the right angle and the side adjacent to the angle is equal. Last updated date: 14th Jun 2023.

**How many angles does an equilateral triangle have? ›**

An equilateral triangle is a triangle that has three sides that are all the same length and **three angles** that are all the same size (60°).

**How many types of equilateral are there? ›**

angles. Central triangles that are equilateral include the circumnormal triangle, circumtangential triangle, first Morley triangle, inner Napoleon triangle, outer Napoleon triangle, second Morley triangle, Stammler triangle, and third Morley triangle.

**What is a right triangle in real life? ›**

One of the most common examples of a right triangle seen in real life is **a situation in which a shadow is cast by a tall object**. For example, if a 40 ft. tree casts a 20 ft. shadow, at what angle from vertical is the sun shining?

**Does a square have equilateral triangles? ›**

Four Equilateral Triangles inside a Square – GeoGebra.

**Is an equilateral triangle 360? ›**

Angles on a straight line equal 180 degrees. Adding the three exterior angles or multiplying 120 by three gives us 360 degrees. Therefore, **the sum of the measures of the exterior angles of an equilateral triangle are 360 degrees**. This fact is also true for any triangle.

**Are all squares with equal areas congruent? ›**

**If two squares have equal area, they are congruent**.

**What are the properties of an equiangular triangle? ›**

Equiangular triangles have **equal sides and equal angles**. The sum of all the interior angles of a triangle is equal to 180°, and each angle of an equiangular triangle is equal to 60°. An equilateral triangle has a predictable shape.

### What are the properties of an equilateral square? ›

All the angles of a square are 90° All sides of a square are equal and parallel to each other. Diagonals bisect each other perpendicularly.

**What are the properties of an equilateral triangle in a circle? ›**

By symmetry, the center of the equilateral triangle coincides with the center of the circle, and the distance from the center of the equilateral triangle to any of its vertices is equal to the radius of the circle.

**Does an equilateral triangle have all the properties of an isosceles triangle? ›**

Yes. For a triangle to be isosceles,any two sides should be equal in length. In equilateral triangle, all the three sides are equal in length. The minimum criterion of two equal sides is hence met.So, **all the equilateral triangles are isosceles triangle too.**

**Are equilateral triangles always similar? ›**

For the equilateral triangles since they always have 3 angles that are each 60∘, **any equilateral triangles will be similar**.

**What are the properties of the 3 types of triangles? ›**

Scalene Triangle: All the sides and angles are unequal. Isosceles Triangle: It has two equal sides. Also, the angles opposite these equal sides are equal. Equilateral Triangle: All the sides are equal and all the three angles equal to 60°.

**Are all triangles equilateral? ›**

Corollary. **Every triangle is equilateral**. Perhaps you object to my figure, because depending on the triangle, perhaps the angle bisector of passes on the other side of the midpoint of , which would make the point lie outside the triangle, as in the following figure.

**Can an equilateral triangle have a right angle? ›**

It is possible to have a right angled equilateral triangle.

**What are the rules for equilateral triangles? ›**

The three properties of an equilateral triangle are: **All three sides are equal.** **All three angles are congruent.** **The figure has three lines of symmetry.**