How To Calculate the Area of a Triangle Using Heron's Formula

• Heron's Formula

• When you are given the lengths of the three sides of a triangle, you can apply a very useful formula for calculating the area of ​​the triangle, called Heron's Formula. Area of the triangle can be calculated as a product of factors:

• Area² = s × (s - a) × (s - b) × (s - c)
• Where:
• a, b and c are the lengths of the three sides of the triangle
• s is the semiperimeter of the triangle
• s = ^{(a + b + c)} / ₂

• » Click here to see a Proof of Heron's Formula

• Example 1.
  Calculate the area of a triangle using the Heron's Formula

• Calculate the area of a triangle with the sides' lengths a = 18, b = 10 and c = 13 units. The semiperimeter can be calculated right away as:
• s = ^{(a + b + c)} / ₂ = ^{(18 + 10 + 13)} / ₂ = ⁴¹/₂ = 20.5.

Area ≈ 63.529028797865 ≈ 63.53 square units

• Example 2.
  Calculate the area of a right triangle

• Given the lengths of the sides of a right triangle, we calculate its area using two methods, then compare the results.
• We calculate the area of ​​the triangle using the classic formula, as the product of a side and its corresponding height, then dividing the result by two, then using Heron's Formula.
• We will compare the results at the end.
• Note: Since in a right triangle one of the legs is the height corresponding to the other leg, the area of ​​a right triangle is calculated as the product of the two legs, divided by two.
• The lengths of the sides of the right triangle are: a = 3 (leg), b = 4 (leg) and c = 5 (hypothenuse).
• Being a right triangle we have the following relationship between its sides' lengths:
  c² = a² + b² (the Pythagorean theorem)
• The semiperimeter of this right triangle is:
• s = ^{(a + b + c)} / ₂ = ^{(3 + 4 + 5)} / ₂ = ¹²/₂ = 6