Equiangular Triangle Explained with Properties and Proof

Before going to know about equiangular triangles, one should know the basic details about a triangle and the types of triangles. A triangle is a type of polygon, which has three sides, and the two sides are joined end to end to form the vertex of the triangle. An angle is formed between two sides of a triangle.

In Geometry, we define a triangle as a three-sided polygon that consists of three edges and three vertices. One of the most important properties of a triangle is that the sum of the internal angles of a triangle is equal to 180 degrees. This property is called the angle sum property of a triangle.

Types of triangle 

Triangles can be classified according to the length of the sides and the interior angles.

1.  Classification on the Basis of Sides:

1. A triangle in which all three sides are equal is called an equilateral triangle.

2. A triangle in which any two sides are equal is called as an isosceles triangle.

3. A triangle in which all three sides are different lengths is called the scalene Triangle.

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2.   Classification on the Basis of Angles

1. A triangle in which all the angles are less than 90° is called an acute angle triangle.

2. A triangle in which one angle is a right angle (90°) is called as a right angle triangle.

3. A triangle in which one of the angles is more than 90° is called an obtuse angle triangle.

Introduction to Equiangular Triangle

A triangle whose all three sides and interior angles are equal is called an equiangular triangle. This is also called an equilateral triangle. For a triangle to be equiangular, the measure of all its three interior angles must be equal to 60 degrees.

An equiangular triangle has three equal sides, and it is similar to an equilateral triangle.

Equilateral Triangle Angles

There are three angles of an equilateral triangle. Each angle of an equilateral triangle is 60°.

Basic Properties of an Equilateral Triangle

• All three sides are equal.

• All three angles are congruent and equal. Each angle is equal to 60 degrees.

• It is a regular polygon having three sides.

• The perpendicular drawn from the vertex of the equilateral triangle to the opposite side bisects it into equal halves. The angle of the vertex from where the perpendicular is drawn divides it into two equal angles, i.e. 30 degrees each.

• The orthocentre and centroid of an equilateral triangle are at the same point.

• In an equilateral triangle, median, angle bisector, and altitude for all the sides are the same.

• The area of an equilateral triangle is \[\frac{\sqrt{3}}{4}a^{2}\], where a is the sides of the triangle.

• The perimeter of an equiangular triangle is 3a.

Area of Equilateral Triangle

The area of an equilateral triangle is defined as the region occupied by it in a two-dimensional plane. The formula of an equiangular triangle is A = \[\frac{\sqrt{3}}{4}a^{2}\].

Derivation of the formula is given here:

From the above figure, the area of a triangle is given by,

Area = 1/2 x base x height

Here a = base and h = height.

Therefore, Area = 1/2 x a x h --(i)

Now, in the above figure, the altitude h bisects the base into equal halves, such that each equal part is a/2. It also forms two equivalent right-angled triangles.

So, for a right triangle, applying Pythagoras theorem, we can write:

a2 = h2 + (a/2)2 

Or h2 = a2 − (a/2)2

h2 = 3a2/4

h = √3a/2

Now put h value in equation (i), we get;,

Area =1/2 x a x √3a/2

Area = √3a2/4

Hence, the area of the equilateral triangle is √3a2/4.

The Perimeter of the Equilateral Triangle

In geometry, we know that the perimeter of any polygon is equal to the length of its sides. So in an equilateral triangle, the perimeter will be the sum of all three sides.

Suppose, ABC is an equilateral triangle having the length of sides is a, then the perimeter of ∆ABC is the sum of its sides.

Perimeter = AB + BC + AC

P = a + a + a

P = 3a

Hence, the perimeter of an equilateral triangle is 3a.

How can we Find the Measure of Each Angle of an Equilateral Triangle?

We will apply here the angle sum property of the triangle.

If ∆ ABC is an equilateral triangle and the sides of a triangle are x.

As we know that in a triangle, the sum of all the angles is 180 degrees.

x + x + x = 180°

⇒ 3x = 180°

⇒ x = 60°.

Hence, the measure of each angle of an equilateral triangle is equal to 60°.

Are all Equiangular Triangles Similar?

As we know in an equilateral triangle, the lengths of all three sides are equal. So, each of the interior angles will have a measure of 60 degrees. Since the angles of an equilateral triangle are the same, it is also known as an equiangular triangle. So we can say, all equiangular triangles will have each interior angle of 60 degrees. Hence, by angle-angle-angle similarity, all equiangular triangles are similar.

Centroid of Equilateral Triangle

The centroid of the equilateral triangle lies at the centre of the triangle. Since all its sides are equal in length, hence it is easy to find its centroid.

To find the centroid, we have to draw perpendiculars from each vertex of the triangle to the opposite sides. These perpendiculars are all equal in length and intersect each other at a single point, that point is known as centroid. The centroid of the triangle is shown below:

Circumcentre

The circumcentre of an equilateral triangle is the point of intersection of the perpendicular bisectors of the sides. Hence, the circumcircle passes through all three vertices of the triangle.

If any of the incenter, orthocenter, or centroid coincides with the circumcenter of a triangle, then it is called an equilateral triangle.

Did you Know?

• The word “equiangular” means “equal angles”.

• An acute angle triangle is a triangle whose measure of all three interior angles is less than 90˚. Since the measure of all three interior angles of an equiangular triangle is 60˚,.hence, an equiangular triangle is always an acute-angled triangle.

Conclusion

An equilateral triangle is a triangle whose all three sides are equal. It is a special kind of isosceles triangle whose base is equal to each leg, and whose vertex angle is equal to its base angles. Therefore, since all three sides of an equilateral triangle are equal, so all three angles are also equal. Hence, every equilateral triangle is also known as equiangular.