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A triangle is a 2D shape having three sides, three vertices, and three angles. Although the sum of three angles of a triangle is always $180^{\circ}$, there are various types of triangles classified according to their length of sides – equilateral, isosceles, or scalene, or measures of angles – acute, right or obtuse.

Let’s understand what is an equilateral triangle and its properties.

## What is an Equilateral Triangle? – Equilateral Triangle Definition

An equilateral triangle is a triangle in which all three sides are equal and the angles are also equal. The value of each angle of an equilateral triangle is $60^{\circ}$ therefore, it is also known as an equiangular triangle. The equilateral triangle is a regular polygon or a regular triangle as angles are equal and sides are also equal.

In the above figure, $\triangle \text{ABC}$ is an equilateral. The length of all its sides is equal, i.e., $\text{AB} = \text{BC} = \text{CA} = a$. Also, the measure of all three angles is equal, i.e., $\angle \text{A} = \angle \text{B} = \angle \text{C} = 60^{\circ}$.

## Shape of Equilateral Triangle

The shape of an equilateral triangle is regular. The word ‘Equilateral’ is formed by the combination of two words, i.e., ‘Equi’ meaning equal, and ‘Lateral’ meaning sides. An equilateral triangle is also called a regular polygon or regular triangle since all its sides are equal.

## Properties of Equilateral Triangle

The following are the important properties of an equilateral triangle that helps to distinguish it from other types of triangles.

- The sum of all the angles of an equilateral triangle is equal to $180^{\circ}$.
- In an equilateral all three sides are equal.
- All three angles are congruent(or qual) to $60^{\circ}$ in a equilateral triangle.
- An equilateral triangle is a regular polygon with three sides.
- The perpendicular drawn from the vertex of the equilateral triangle to the opposite side bisects it into equal halves.
- The perpendicular drawn from the vertex of the equilateral triangle divides the angle from where the perpendicular is drawn into two equal angles, i.e., $30^{\circ}$ each.
- The orthocentre, incentre, and centroid are at the same point.
- In an equilateral triangle, the median, the angle bisector, and the altitude for all sides are the same.
- The area of an equilateral triangle is $\frac{\sqrt{3}a^2}{4}$
- The perimeter of an equilateral triangle is $3a$.

## Each Angle of an Equilateral Triangle is $60^{\circ}$

Let $\triangle \text{ABC}$ be an equilateral triangle, then $\angle \text{A} = \angle \text{B} = \angle \text{C} = 180^{\circ}$.

Let’s prove the above statement.

In a triangle the sum of three angles is $180^{\circ}$ (Angle Sum Property of Triangle)

Therefore, $\angle \text{A} + \angle \text{B} + \angle \text{C} = 180^{\circ}$ ————– (1)

Also, $\angle \text{A} = \angle \text{B} = \angle \text{C}$ ($\triangle \text{ABC}$ is an equilateral triangle) ———— (2)

From (1) and (2), $\angle \text{A} + \angle \text{A} + \angle \text{A} = 180^{\circ}$

$=> 3\angle \text{A} = 180^{\circ} => \angle \text{A} = 60^{\circ}$

From (2) $\angle \text{A} = \angle \text{B} = \angle \text{C} = 60^{\circ}$.

## Perpendicular Drawn From the Vertex Divides an Equilateral Triangle into Two Equal Halves

Let $\triangle \text{ABC}$ be an equilateral triangle and a perpendicular $\text{AD}$ is drawn the vertex $\text{A}$ on the opposite side $\text{BC}$, then $\triangle \text{ABD} \cong \triangle \text{ADC}$.

Let’s prove the above statement.

In $\triangle \text{ABD}$ and $\triangle \text{ADC}$

$\text{AB} = \text{AC}$ (Sides of an equilateral triangle)

$\text{AD} = \text{AD}$ (Common Side)

$\angle \text{ADB} = \angle \text{ADC} = 90^{\circ}$ ($\text{AD} \perp \text{BC}$)

Therefore, $\triangle \text{ABD} \cong \triangle \text{ADC}$ (RHS Congruence Criterion)

Thus, $\triangle \text{ABD}$ and $\triangle \text{ADC}$ are equal.

## Perpendicular Drawn From the Vertex Divides The Angle into Two Equal Angles

Let $\triangle \text{ABC}$ be an equilateral triangle and a perpendicular $\text{AD}$ is drawn the vertex $\text{A}$ on the opposite side $\text{BC}$, then $\angle \text{BAD} = \angle \text{DAC}$.

Let’s prove the above statement.

In $\triangle \text{ABD}$ and $\triangle \text{ADC}$

$\text{AB} = \text{AC}$ (Sides of an equilateral triangle)

$\text{AD} = \text{AD}$ (Common Side)

$\angle \text{ADB} = \angle \text{ADC} = 90^{\circ}$ ($\text{AD} \perp \text{BC}$)

Therefore, $\triangle \text{ABD} \cong \triangle \text{ADC}$ (RHS Congruence Criterion)

Thus, $\angle \text{BAD} = \angle \text{DAC}$ (Corresponding Parts of Congruent Triangles)

## Equilateral Triangle Theorem

Let $\triangle \text{ABC}$ be an equilateral triangle and $\text{P}$ be a point on the arc $\widehat{\text{BC}}$ of the circumcircle of the $\triangle \text{ABC}$, then according to equilateral triangle theorem $\text{PA} = \text{PB} + \text{PC}$.

Let’s prove the above statement.

For a cyclic quadrilateral $\text{ABPC}$, we have $\text{PA}⋅\text{BC}=\text{PB}⋅\text{AC}+\text{PC}⋅\text{AB}$

Also, for an equilateral triangle ABC, $\text{AB} = \text{BC} = \text{AC}$

Therefore, $\text{PA}.\text{AB} = \text{PB}.\text{AB}+\text{PC}.\text{AB}$

Taking $\text{AB}$ as a common, we get $\text{PA}.\text{AB}=\text{AB} \left(\text{PB}+\text{PC} \right)$

$=>\text{PA} = \text{PB} + \text{PC}$.

## Difference Between Equilateral, Isosceles, and Scalene Triangle

The following are the main differences between equilateral, isosceles, and scalene Triangle

Equilateral Triangle | Isosceles Triangle | Scalene Triangle |

All three sides are equal | Any two sides are equal | All three sides are unequal |

All three interior angles are equal to $60^{\circ}$ | Angles opposite to equal sides are equal | All the angles are unequal in measure |

Medians, altitudes, and angle bisectors of an equilateral triangle are the same | Medians, altitudes, and angle bisectors of an isosceles triangle are different | Medians, altitudes, and angle bisectors of a scalene triangle are different |

The centroid, the orthocentre, and the incentre of an equilateral triangle are the same. | The centroid, the orthocentre, and the incentre of an isosceles triangle are different. | The centroid, the orthocentre, and the incentre of a scalene triangle are different. |

## Practice Problems

- What is meant by an equilateral triangle?
- Prove that the sum of each angle of an equilateral triangle is $60^{\circ}$
- Prove that a perpendicular drawn from the vertex of the equilateral triangle to the opposite side bisects it into equal halves.
- Prove that a perpendicular drawn from the vertex of the equilateral triangle divides the angle from where the perpendicular is drawn into two equal angles.

## FAQs

### What is meant by an equilateral triangle?

An equilateral triangle is a triangle in which all sides are equal and angles are also equal. The value of each angle of an equilateral triangle is $60^{\circ}$ therefore, it is also known as an equiangular triangle.

### What is an equilateral triangle and its formula?

An equilateral triangle is a triangle in which all sides are equal and angles are also equal. For an equilateral triangle, $\text{ABC}$ formulas are

a) $\angle \text{A} = \angle \text{B} = \angle \text{C}$

b) $\text{Perimeter} = 3a$ ($a$ is the length of each side)

c) $\text{Area} = \frac{\sqrt{3}}{4}a^2$ ($a$ is the length of each side)

### Why is it called an equilateral?

The word ‘Equilateral’ is formed by the combination of two words, i.e., ‘Equi’ meaning equal, and ‘Lateral’ meaning sides. An equilateral triangle is also called a regular polygon or regular triangle since all its sides are equal.

## Conclusion

An equilateral triangle is a triangle in which all three sides are equal and the angles are also equal. The value of each angle of an equilateral triangle is $60^{\circ}$ therefore, it is also known as an equiangular triangle. In an equilateral triangle, the medians, altitudes, and angle bisectors are the same lines and meet at the same point.

## Recommended Reading

- How to Construct a Rectangle Using Compass and Ruler – Methods, Steps & Examples
- How to Construct a Square Using Compass and Ruler – Methods, Steps & Examples
- What is a Scalene Triangle – Definition, Properties & Examples
- What is an Isosceles Triangle – Definition, Properties & Examples
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- How to Construct a Triangle(With Steps, Diagrams & Examples)
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- Congruence of Triangles Criteria – SSS, SAS, ASA, RHS
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