Equiangular Triangle – Definition with Examples

Equiangular Triangle: Introduction

Equiangular triangle is a triangle in which angles are congruent and all sides are of equal length. It is often called an equilateral triangle. It is a regular polygon with three equal sides and the angles of the same measure. 

Equiangular Triangle: Definition 

A triangle with three equal interior angles is called an equiangular triangle. In an equiangular triangle, the measure of each of its interior angles is 60 ̊.

What Is an Equiangular Triangle?

In an equiangular triangle, all interior angles are of equal measure. Thus, all its three sides are also of equal length. It is the same as an equilateral triangle.

Non-examples of Equiangular Triangle

In the given non-examples, all the figures are of triangles, but none of them is an equiangular triangle because:

• In a right-angled triangle, one of the interior angles is $60^\circ$.
• In an isosceles triangle, only two interior angles are equal.
• In a scalene triangle, none of the interior angles is equal. 

For a triangle to be equiangular all its three interior angles must be equal, that is, each angle should measure $60^\circ$.

Properties of Equiangular Triangles

Triangles have different classifications depending on their properties, angles, and sides. We have different triangles, such as an isosceles triangle or a right triangle. An equiangular triangle has its own properties based on the angles and sides. The properties of an equiangular triangle are:

• An equiangular triangle has all the three sides congruent to each other.
• All three angles are equal to each other and measure at $60^\circ$ each.
• Since the angles of an equiangular triangle are always $60^\circ$, the triangle is always an acute-angled triangle.
• The radius of an incircle in an equiangular triangle is considered exactly half of the radius of a circumcircle.
• In an equiangular triangle, an orthocenter and centroid are all considered to be the same points.

How to Construct an Equiangular Triangle?

An equiangular triangle can be constructed just by using a compass and a ruler. Since the sides of the triangle are equal in length, construction becomes very simple. Here are the steps to draw an equiangular triangle:

Step 1: Draw a line segment AB, which will be considered as the length of the sides of the triangle.

Step 2: Mark a point X anywhere that will be one vertex of the triangle.

Step 3: Measure the length of line segment AB with a compass. With one point of the compass at A and the drawing part of the compass at B.

Step 4: With the obtained length, keeping the compass at point X marks two arcs from the drawing end without changing the length of the compass.

Step 5: These two arcs will be considered the other two vertices of the triangle.

Step 6: On the second arc, mark a point Y. It could be any of two arcs.

Step 7: Place the compass on point Y, and draw arcs crossing the third arc creating the point Z.

Step 8: Using a ruler, join all the points creating an equiangular triangle.

Perimeter of an Equiangular Triangle

Perimeter of an equiangular triangle is considered as the sum of all sides or three times of a side since all the sides are equal to each other. Hence, the formula to calculate the perimeter is given as:

Perimeter $= 3 \times a$, 

where a is the side.

Area of Equiangular Triangle

The area of an equiangular triangle is the space covered within the perimeter or three sides of the equiangular triangle and is expressed in square units. Some important units used to express the area are sq. in, sq. m, sq. cm, sq yd, etc. The formula to calculate the area of an equiangular triangle is given as:

Area $= \sqrt{\frac{3}{4}} (side)^2$  square units

Conclusion

In this article, we learned about equiangular triangles. We saw their properties and understood ways of construction. Let us now look at some examples and solve practice problems based on area and perimeters. 

Solved Example

1. What is the angle opposite to the side AB if ABC is an equiangular triangle?

Solution:

All the angles of an equiangular triangle measure the same, i.e., $60^\circ$.

2. What is the measure of exterior angle of an  equiangular triangle?

Solution:

The measure of exterior angle of an  equiangular triangle is:

We know that sum of angle on a strength line is $180^\circ$,

Now 

$Interior angle + Exterior angle = 180^\circ$

$60^\circ + Exterior angle = 180^\circ$

$Exterior angle = 180^\circ – 60^\circ$

$Exterior angle = 120^\circ$

3. What is the sum of angles of an equiangular triangle?

Solution:

The sum of angles of a triangle is always 180 degrees.

$\angle \text{A} + \angle \text{B} + \angle \text{C} = 60^\circ + 60^\circ + 60^\circ = 180^\circ$

4. In an isosceles triangle, the equal sides measure 4 inches and all the angles measure $60^\circ$. Find out the length of the other side.

Solution:

We observe that since all the angles measure $60^\circ$. So, the triangle is not isosceles but equiangular. 

Hence, the other side measures the same as given two equal sides, i.e., 4 inches.

5. What is the perimeter of an equiangular triangle of sides measuring 5 inches?

Solution:

Perimeter of equiangular triangle $= 3 \times side$

$= 3 \times 5$ inches

$= 15$ inches

Practice Problems

Frequently Asked Questions