Equilateral Triangle

Triangles are fundamental geometric shapes that form the basis of many mathematical and real-world concepts. Among the various types of triangles, the equilateral triangle stands out for its unique properties and applications. With its perfect balance of sides and angles, equilateral triangles possess a unique charm in the world of shapes and geometry.

Definition of an Equilateral Triangle: An equilateral triangle is a type of triangle that has all three sides of equal length and three interior angles of equal measure. This regularity and symmetry make equilateral triangles distinct from other types of triangles, where side lengths and angle measures may vary.

In the figure above, all the three sides $\mathrm{LM}$, $\mathrm{MN}$, $\mathrm{LN}$ and angles $\mathrm{\angle }L$, $\mathrm{\angle }M$, $\mathrm{\angle }N$ of the $\mathrm{△}\mathrm{LMN}$ are equal, hence it is an equilateral triangle.

Properties of an Equilateral Triangle

• Equal Sides

  : All three sides of an equilateral triangle are congruent in length.

• Equal Angles

  : All angles in an equilateral triangle are equal and measure 60°. This is a consequence of the fact that the sum of all the three angles in any triangle is 180°, and in an equilateral triangle, all three angles are equal.

• Symmetry

  : Equilateral triangles exhibit symmetry with respect to their axes of rotation. They have three lines of symmetry, each bisecting the angles and passing through the midpoints of the sides.

• Concentric Circles

  : The center of the circle that can be inscribed within an equilateral triangle (incircle) is also the center of the triangle itself. Additionally, the center of the circle that circumscribes the triangle (circumcircle) coincides with the center of the triangle.

• Height and Median Relationships

  : In an equilateral triangle, the altitude (height) from any vertex is also a median and a bisector. It splits the base into two equal segments, and it also bisects the opposite angle.

• Regular Polygon

  : Equilateral triangles are a specific type of regular polygon with equal sides and angles.

These properties make equilateral triangles fundamental shapes in geometry, and their symmetry and simplicity make them useful in various fields, including architecture, engineering, and mathematics.

Equilateral Triangle Formulas

Perimeter of an Equilateral Triangle

Area of an Equilateral Triangle

Height/Altitude of an Equilateral Triangle

Radius of the Circle Inscribed within an Equilateral Triangle

Radius of the Circle Circumscribing an Equilateral Triangle

Equilateral triangles are important in geometry and have applications in various fields, such as architecture, engineering, and trigonometry. Understanding their properties and relationships can help solve problems involving these triangles and provide insights into broader mathematical concepts.

Solved Examples on Equilateral Triangle