Triangle

A triangle is a fundamental geometric shape in mathematics and geometry. It is a two-dimensional with the fewest number of sides and is considered one of the simplest geometric shapes.

Definition of triangle: Triangle can be defined as a polygon with three sides, three vertices (or corners), and three angles. The term "triangle" is derived from the Latin words "tri" meaning "three" and "angulus" meaning "angle".

A triangle is denoted by the symbol △. Triangles are widely studied and utilized in various fields, including mathematics, physics, engineering, and architecture. Hence it is highly important to learn and understand the concept of triangle.

Characteristics of a Triangle:

• Vertices: A triangle has three vertices, which are the points where the sides intersect. The vertices are typically labeled using uppercase letters.
• Sides: A triangle has three sides, each having specific length, which are line segments connecting the vertices of the triangle.
• Angles: A triangle has three interior angles formed by the intersection of its sides. The angles are usually denoted by capital letters corresponding to the vertices. The sum of the interior angles in any triangle is always 180 degrees.

Types of Triangles:

Types of Triangle Based on Sides:

• Scalene Triangle: A scalene triangle has three unequal side lengths and three different angles. None of the angles are equal to each other.

  In the figure above, all the three sides and angles of △PQR are unequal, hence it is scalene triangle.

• Isosceles Triangle: An isosceles triangle has at least two sides of equal length.

  In the figure above, the sides XY and XZ are equal and consequently, angles ∠Y and ∠Z have equal measure in △XYZ, hence it is an isosceles triangle.

• Equilateral Triangle: An equilateral triangle has three equal side lengths, and therefore, all three angles are equal. Each angle in an equilateral triangle measures 60 degrees.

  In the figure above, all the three sides and angles of the △LMN are equal, hence it is an equilateral triangle.

Types of Triangle Based on Angles:

• Right Triangle: A right triangle has one angle that measures 90 degrees, known as the right angle. The side opposite the right angle is called the hypotenuse, and the other two sides are known as the legs.

  In the figure above, ∠E is right angle in △DEF, hence △DEF is right triangle.

• Obtuse Triangle: An obtuse triangle has one angle that measures more than 90 degrees.

  In the figure above, ∠S measures more than 90 degrees, hence the △RST is an obtuse triangle.

• Acute Triangle: An acute triangle has all three angles measuring less than 90 degrees.

  In the figure above, all the three angles of the △JKL measure less than 90 degrees, hence it is an acute triangle.

Properties of Triangles:

• Triangle Inequality Theorem: The sum of the lengths of any two sides of a triangle is greater than the length of the third side. In mathematical terms, if $a$, $b$ and $c$ are the lengths of the sides of a triangle, then:
  $$a+b>c$$$$a+c>b$$$$b+c>a$$
• Angle Sum Property: The sum of the three interior angles of a triangle is always 180 degrees.
• Pythagorean Theorem: In a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides. This theorem is expressed as a² + b² = c², where c represents the length of the hypotenuse, and a and b represent the lengths of the other two sides.

Triangle Formulas

Perimeter of a Triangle

Area of a Triangle

Area of Triangle by Heron’s Formula

Solved Examples on Triangle

Applications of Triangles:

Triangles, as fundamental shapes in geometry, have numerous real-world applications across various fields. Their properties such as the relationships between angles, sides, and the ability to calculate areas and volumes make them invaluable tools in many practical areas. Below are some of the important real world applications of triangles.

• Trigonometry: Triangles form the foundation of trigonometry, which deals with the relationships between angles and the lengths of the sides of triangles. Trigonometric functions such as sine, cosine, and tangent are used extensively in various fields, including physics, engineering, and navigation.
• Geometry: Triangles are essential in geometry for various purposes, such as calculating areas and perimeters of shapes, and solving geometric problems.
• Engineering and Architecture: Triangles are used extensively in structural engineering and architecture to provide stability and support in various structures like buildings, bridges, and trusses.
• Computer Graphics and 3D Modeling: Triangles are often used to represent three-dimensional objects in computer graphics. They form the basis for mesh structures used in 3D modeling and rendering. The surfaces of complex objects in video games, animations, and simulations are broken down into triangular meshes to make rendering efficient.
• Navigation and Surveying: Triangulation, a technique that uses triangles to determine distances and locations, is commonly employed in navigation, surveying, and geodesy.