What is 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. The term ‘equilateral’ is created by combining the words ‘equi’ and ‘lateral’, which means equal sides. If ABC is an equilateral triangle, then AB = BC = AC, where AB, BC, and AC are the sides of the triangle.
Consequently, ∠A = ∠B = ∠C = 60°
Properties of Equiangular Triangles
According to its angles and sides, an equiangular triangle has the following characteristics:
- All three sides of an equiangular triangle are equal in length.
- An equiangular triangle has three equal angles, each measuring 60 degrees.
- Since the angles of an equiangular triangle are always 60 degrees, they are always acute angle triangles.
Area of an Equiangular Triangle
The space enclosed by an equiangular triangle is known as its area. It is measured in square units like inch\(^2\), m\(^2\), cm\(^2\), yd\(^2\).
The area of an equiangular triangle is calculated using the following formula:
Area, A = \(\frac{\sqrt{~3~}}{4}\) × (side)\(^2\) square units.
Area, A = \(\frac{\sqrt{~3~}}{4}\) × a\(^2\) square units.
Here, a is side length.
Perimeter of an Equiangular Triangle
The perimeter of an equiangular triangle is equal to the sum of the length of its three sides.
If ABC is an equilateral triangle, its perimeter is Perimeter, P = AB + BC + AC
P = a + a + a
P = 3a
Where ‘a’ is the side length of the triangle.
Height of an Equiangular Triangle
The height (h) of an equiangular triangle is calculated using the formula:
Height, h = \(\frac{\sqrt{~3}a}{2}\)
Where ‘a’ denotes the side length of the triangle.
Rapid Recall
Area | \(\frac{\sqrt{~3}}{4}\) × (a)\(^2\) |
Perimeter | 3a |
Height | \(\frac{\sqrt{~3}a}{2}\) |
Solved Equiangular Triangle Examples
Example 1: Determine the height of an equiangular triangle with sides measuring 10 inches each.
Solution:
Side, a = 10 inches
Height, h = \(\frac{\sqrt{~3}a}{2}\) [Formula of height]
Substituting the value in the formula,
h = \(\frac{\sqrt{~3}~\times~(10)}{2}\)
h = 8.66 inches [Simplify]
Hence, the height of the triangle is 8.66 inches.
Example 2: Determine the perimeter of an equiangular triangle with sides measuring 19 inches each.
Solution:
Perimeter of an equiangular triangle, P = 3a
Side, a = 19 inches
Substituting the value in the formula,
P = 3 x 19 [Simplify]
P = 57 inches
Hence, the perimeter of the triangle is 57 inches.
Example 3: A traffic sign board has the shape of an equilateral triangle with sides that measure 40 centimeters in length. Find the area of the material needed to make that sign board.
Solution:
Length of the sides, a = 40 centimeters.
It is an equiangular triangle.
Area = \(\frac{\sqrt{~3~}}{4}\) × (a)\(^2\) [Area formula of an equiangular triangle]
Area = \(\frac{\sqrt{~3~}}{4}\) × (40)\(^2\) [Substitute the value of a]
Area = \(\frac{\sqrt{~3~}}{4}\) × 1600 [Simplify]
Area = 692.82
Therefore, the area of the traffic sign board is 692.82 square centimeters.
An equilateral triangle or polygon has congruent sides similar to a rhombus, whereas an equiangular triangle or polygon has congruent inner angles similar to a rectangle. This is the primary difference between the two triangles. But all equilateral angles are equiangular, and all equiangular triangles are equilateral.
An equiangular triangle is also known as an equilateral triangle.
Yes, three congruent angles and their opposite congruent sides make an equiangular triangle symmetric.