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Equiangular triangles are a type of triangles with the angles and sides equal to each other. We can say that an equiangular triangle is also called an equilateral triangle. It is a regular polygon with three equal sides with angles of the same measure. Let us see the definition, properties, formulas of an equiangular triangle.
|What are Equiangular Triangles?
|Properties of Equiangular Triangles
|Constructing Equiangular Triangles
|Area of an Equiangular Triangles
|Perimeter of an Equiangular Triangles
|FAQs on Equiangular Triangles
What are Equiangular Triangles?
An equiangular triangle is a triangle that has all three sides equal to each other along with all three angles measured the same at 60°. An equiangular triangle can also be called an equilateral triangle as the properties of both the triangles are the same. This triangle is considered a regular polygon with equal angles and equal sides. The word equiangular means equal angles.
Definition of Equiangular Triangle
A triangle with three interior angles equal to each other measured at 60° along with the sides of the same length is called an equiangular triangle. The interior angles of any triangle sum up to 180° and in an equiangular triangle, each angle is always the third of that, hence 60°. This type of triangle can also be called an equilateral triangle and is seen like the image below.
Properties of Equiangular Triangles
Triangles are of different kinds depending on their properties, angles, and sides such as an isosceles triangle or a right triangle. An equiangular triangle has its own properties based on the angles and sides and they are:
- An equiangular triangle has all three sides congruent to each other.
- The radius of an incircle in an equiangular triangle is considered exactly half the radius of a circumcircle.
- All three angles are equal to each other and measure at 60°.
- In an equiangular triangle, an orthocenter and centroid are considered as the same point.
- A right-angled triangle, isosceles triangle, and scalene triangle are examples of triangles that are not equiangular since each of the three interior angles are not equal to each other and not at 60°.
- Since the angles of an equiangular triangle are always 60°, the triangle is always an acute-angled triangle.
Constructing an Equiangular Triangles
An equiangular triangle can be constructed by using a compass and a ruler. Since the sides of the triangle are equal in length, constructing becomes very simple. Here are the steps:
- Draw a line segment AB which will be considered as the length of the sides of the triangle.
- Mark a point X anywhere that will be one vertex of the triangle.
- 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.
- With the obtained length, keeping the compass at point X mark two arcs from the drawing end without changing the length of the compass.
- These two arcs will be considered the other two vertices of the triangle.
- On the second arc, mark a point Y. It could be any of two arcs.
- Place the compass on point Y, and draw arcs crossing the third arc creating the point Z.
- Using a ruler, join all the points creating an equiangular triangle.
Area of an Equiangular Triangle
The area of an equiangular triangle is the space covered within the three sides of the equiangular triangle and is expressed in square units. Some important units used to express the area are in2, m2, cm2, yd2, etc. The formula to calculate the area of an equiangular triangle is:
Area = √3/4 × (side)2 square units.
Perimeter of an Equiangular Triangle
Perimeter of an equiangular triangle is considered 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:
Perimeter = 3a, where a is the side.
Also, when we need to find the height and the semi-perimeter of an equiangular triangle, we use the formulas mentioned below respectively.
- Height of an Equiangular Triangle = √3a/ 2
- Semi Perimeter of an Equiangular Triangle = 3a/2
Listed below are a few topics related to an equiangular triangle, they are:
Examples on Equiangular Triangle
Example 1: Elle is eating a large triangular-shaped pizza where the lengths of the sides are 16 in. Help Elle find the area of the pizza.
The length of the sides = 16 in.
Since the sides of the pizza are of equal length, we can say that it is an equiangular triangle. Hence, using the area formula we get,
Area = √3/4 × (side)2
Area = √3/4 × (16)2
Area = √3/4 × 256
Area = 110.85
Therefore, the area of the large pizza eaten by Elle is 110.85 in2.
Example 2: Calculate the perimeter of an equiangular triangle whose each side is 12 inches.
The formula to calculate the perimeter is:
Perimeter of equilateral triangle = 3a, where a is the side.
Given, a = 12 inches.
Thus, perimeter = 3 × 12 = 36 inches.
Therefore, the perimeter of the equiangular triangle is 36 inches.
FAQs on Equiangular Triangles
What is an Equiangular Triangle?
An equiangular triangle is a triangle with all three sides equal to each other along with the three angles equal in measure i.e. 60°. The triangle can be considered as a regular polygon as the angles and sides are equal to each other.
What is Another Name of Equiangular Triangle?
The other name of an equiangular triangle is an equilateral triangle. Since both the triangles have similar properties such as sides are equal and all three angles are measured at 60°.
What is the Difference Between Equilateral and Equiangular Triangle?
Both the triangles are considered as regular polygons. The main difference between the two triangles is that an equilateral triangle or polygon has congruent sides like a rhombus whereas an equiangular triangle or polygon has congruent interior angles like a rectangle. If a polygon is both equilateral and equiangular, it is considered a regular polygon.
Are Equiangular Triangles Congruent?
An equiangular triangle has three congruent angles with congruent sides opposite them making all three sides congruent. Hence, equiangular triangles are congruent.
What is the Formula to Find the Area of an Equiangular Triangle?
The area of an equiangular triangle can be found by lengths of the side and the formula is, Area of Equiangular Triangle = √3/4 × (side)2 square units.
What is the Formula to Find the Perimeter of an Equiangular Triangle?
The perimeter of an equiangular triangle can be found when the length of the sides is mentioned and the formula is, Perimeter of Equiangular Triangle = 3a, where a is the length of the sides.
How to Construct an Equiangular Triangle?
An equiangular triangle can be constructed by using a compass and a ruler and the steps are as follows:
- Consider the length of a line segment with the help of a compass.
- Mark any point near the line segment, using the compass draw two arcs keeping the point as one vertex of the triangle.
- Mark a second point on any of the two arcs.
- Without changing the length in the compass, cross another arc on the third arc by placing one end of the compass on the second point.
- Use a ruler to use all the points and constructing an equiangular triangle.