Triangle: Any two-dimensional figure that has three sides is known as a triangle. The three sides of a triangle are length, breadth, and height, denoted by symbols such as l, b, and h. The points of intersections where the sides of the triangle meet are called the vertices of the triangle. The angles produced at the vertices are called the angles of the triangle. 

 Parts of a triangle:

 In a triangle, there are 3 sides, 3 vertices, and 3 angles. 

  • In the triangle given above:

  • The three angles are ∠1, ∠2, and ∠3.

  • The three sides are side 1, side 2, and side 3.

  • The three vertices are Vertex 1 (P), Vertex 1 (Q) and Vertex 1 ®

Essential conditions of a triangle:

  • Condition on the sides

The sum of the lengths of any 2 sides of a triangle must be greater than or equal to the length of the 3rd side.

  • Conditions on the angles

(a) Each of the angles is positive.

(b) The sum of three angles is always 180°

The area of a triangle: It is the amount of space inside the triangle. The units are represented as  square units. These units can be:

  • Square millimetres (mm2)

  • Square inches (in2)

  • Square kilometres (km2)

  • Square yards

Area of different types of triangles:

Different types of triangles can be classified based on their sides and angles.

The general formula to find the area of a triangle is given by:

Area (A) = ½ (b h) square units

Where; 

A = Area, b is Base, and “h” represents height of the triangle. 

The triangles might be different, but this formula can be applied to all the triangles. 

  1. Acute angled triangle: Any triangle in which all the angles are less than 900.

A triangle that has at least two interior angles that are always acute, i.e less than 900. There are three possibilities in the case of the third angle:

  • Less than 90° – All three angles are acute and so the triangle is acute.

  • Exactly 90° – It is a right triangle

  • Greater than 90° (obtuse) – It is an obtuse triangle

  1. Obtuse angled triangle: Any triangle with one of the angles more than 900. 

obtuse triangle

A  triangle in which at least two interior angles are always acute or less than 90°. There are three possibilities in the case of the third angle:

  • Less than 90° – All three angles are acute and so the triangle is acute.

  • Exactly 90° – It is a right triangle

  • Greater than 90° (obtuse) – Then, a triangle is an obtuse triangle

  1. Equilateral triangle: A triangle in which all three sides (angles) are equal.

 

  • The area of an equilateral triangle is:

Area=a234

 

Where a = length of the sides.

 

  1. Right-angled triangle: A triangle that has at least one angle equal to 900 is called a right-angled triangle. 

https://vervetimes.com/wp-content/uploads/2021/06/What-are-the-Main-Properties-of-the-Right-Angled-Triangle.png

 

To find an area of a right-angled triangle, we use the Pythagoras theorem, which defines the relationship between the three sides of a right triangle.

 

hypotenuse2 = base2 + height2

 

In short;

H2=B2 + P2 

Where; H is “hypotenuse”
B is  “base” 

P represents “perpendicular”.

 

Types of right-angled triangles:

  1. Isosceles right triangle: A triangle in which any two sides and angles are equal and include the right angle between them. The third side formed is called hypotenuse.

  2. Scalene right triangle: A triangle in which all sides, as well as angles, are unequal and the angle opposite to the longest side forms the right angle (90 degrees).

  1. Isosceles triangle: A triangle in which any two sides and angles are equal.

 

  • The area of an isosceles triangle is:

           Area of a triangle = base x height2  or 12(bxh)

Where b is the base length and h is the height of the triangle.

(a) The following formula is usually used when the height of the isosceles triangle is not given:

Height =√x2-y2/4

Where; y = triangle’s base

x = two equal side lengths

So, the isosceles triangle area will be;

Area=½[(√x2-y2/4)y]

 

(b) Also, the area of an isosceles right triangle is given by:

Area=½*a2

Where; a = Side length of the two equal sides

(c) This formula is used to find an isosceles triangle area when it has length of 2 sides and angle between them or using 2 angles and length between them:

Area = 1/2 b a sin()

Where; b =triangle base

a =two equal side lengths

α =Angle 2 equal sides make with each other

Using the lengths and 2 angles between them:

Area=[a2×sin(θ/2)×sin(β)]

Where;

  • a = two equal sides length

  • θ, β = isosceles triangle angles

(d) A right isosceles triangle is an isosceles triangle that has one angle equal to 90°. The formula to calculate the area for an isosceles right triangle can be expressed as:

Area =12 a2

Area=1/2a2

             Where; a = Length of equal sides.

  1. Scalene triangle: A triangle in which all sides, as well as angles, are unequal. 

Scalene triangles

 

  • The area of a scalene triangle can be calculated using Heron’s formula.

Area=s(s-a)(s-b)(s-c)

S=a+b+c2

Where;

 S represents semi-perimeter 

And a, b, c are the three sides of a triangle.

Conclusion 

In this article, we have learned about different types of triangles formed in mathematics, their properties, and methods to find their areas. However, the triangle is a very wide topic. It governs the entire geometry. By knowing all this data, students will get a reasonably good idea about a triangle and how to use formulas to obtain an area of different types of triangles.