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How to Find the Area of Triangle with 3 Sides - Formula and Examples

 

To understand how to calculate the area of a triangle, you first need to know what is a triangle. Triangle, also known as a trigon, is a three-sided closed figure. The word is taken from the Greek word ‘tri’, which means three. Therefore, deriving the literal meaning of a three-angled figure is known as a triangle. Triangles are classified into the following based on two properties - the length of the sides and the degree of angles. Read below to find out more details about types of triangles:

According to the length of the sides:

  • Equilateral Triangle: A triangle having all three sides of equal length is known as an equilateral triangle. Click here to know how to find Area of Equilateral Triangle.
  • Isosceles Triangle: An Isosceles triangle is a type of triangle having any two sides of equal length and the angles opposite to the equal sides as equal.
  • Scalene Triangle: A triangle having all three sides of distinct sizes or lengths is known as a scalene triangle.

According to the degree of angles:

  • Acute Triangle: A triangle having all internal angles less than 90 degrees is known as an acute triangle.
  • Obtuse Triangle: A triangle having one of the interior angles greater than 90 degrees and lesser than 180 degrees is known as an Obtuse angle.
  • Right-angle Triangle: A triangle having one of the angles exactly equal to 90 degrees is known as a right-angled triangle. The remaining two angles inside the triangle are acute.

Area of Triangle Using Heron’s Formula:

Heron’s formula is often referred to as Hero’s formula, which was named after the great Hero of Alexandria. This formula is beneficial in calculating the area of any three-sided figure if all the lengths of that figure are well defined.

To apply Heron’s Formula, we must first determine the value of the semi-perimeter. It is denoted by the symbol ‘s’. The semi-perimeter is defined as the sum of all three lengths of the triangle divided by 2.

For example, let us consider a triangle QPR with lengths k, l, and m.

Therefore, s = (k + l + m)/2

Next, we substitute the value of ‘s’ in Heron’s main formula, which is stated as:

Area of a triangle = √ s (s-k) (s-l) (s-m)

We saw that a triangle format might be used to determine the area of specific triangles. But the computation of height would not be easy for a triangle with the sides specified. So we rely on Heron's Formula for the same reason to compute the triangular area with uniform lengths. Heron’s formula is very beneficial in higher studies, including engineering.

You may also want to know 'how to find perimeter of the triangle'. 

List of Areas of Different Triangles:

Type of Triangle Formula for Area
Standard triangle (with height) ½ x base x height
Standard triangle with three sides given (Scalene triangle) Using Heron’s Formula = √ s (s-k) (s-l) (s-m)
Isosceles triangle ½ x base (non-equal side) x height (perpendicular drawn from the vertex to the base)
Equilateral triangle (√3)/4 × s2, where s is the sides of the triangle

We can also determine the area of any triangle if any two sides and the angle between those particular two sides are given. Let us assume a triangle KLM with a length of the side k, l, and m, respectively. Let the angle associated be A, B, and C; therefore, an area of the triangle is calculated using one of the three formulas :

  • Area of triangle= ½ kl sin A
  • Area of triangle = ½ lm sin B
  • Area of triangle = ½ km sin C, according to the given sides and angle.

 

Click here to know more about Trigonometric functions (Sine, Cosine, Tangent, Secant, Cosecant and Cotangent).

 

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