Heron’s Formula for Area of Triangle

Heron (or Hero) of Alexandria is credited with the formula, and justification can be found in his work Metrica, written around 60 AD. It has been proposed that Archimedes knew the Heron’s formula more than two centuries before, and since Metrica is a synthesis of old world mathematical knowledge, it is probable that the formula predates the reference supplied in that work. This formula can be used to prove the law of cosines or the law of cotangents, using trigonometry.

Area of a triangle can be determined when all three sides are known using Heron’s formula. This formula is not only used to find the area of a scalene triangle, but can also be applied to find the area of any triangle.

Heron’s formula states that the area of a triangle whose sides have lengths a, b, and c

Where s is a semi-perimeter of the triangle, s=a+b+c2.

herons formula

Triangle with a, b and c as it’s sides
Substituting the value of s in A, we get


Heron’s Formula Examples

Example 1: Find the area of ΔXYZ whose sides are a = 5 units, b = 6 units, and c = 10 units.

a = 5, b = 6, c = 10

To find s,


Example 2: Find the area of a triangle whose sides are 4 cm, 6 cm, and 7 cm.

Solution: a = 4 cm, b = 6 cm, c = 7 cm

Proof of Heron’s Formula

There are different ways to prove Heron’s formula. They are:

  • Using Pythagorean theorem
  • Using laws of cosines
  • Using laws of cotangents

    Proof using Pythagorean Theorem

    Pythagorean Theorem states that, “In a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.” Using this concept, the Heron’s formula can be obtained. Consider a right-angled triangle as shown in the image below.

    heron's formula
    Right-angled triangle

    Proof using Laws of Cosine

    In trigonometry, the lengths of the sides of a triangle are related to the cosine of one of its angles using the laws of cosines.

    heron's formula using laws of cosine
    Representation of length of sides and angles of a triangle

    Proof using Laws of Cotangents

    In trigonometry, the law of cotangents creates a connection between the lengths of a triangle’s sides and the cotangents of the three angles’ halves.

    In the image given below, a, b, and c are the lengths of the three sides of triangle. The angles α (alpha), β (beta), and γ (gamma) are the three angles corresponding to the vertices, A, B and C, respectively. The radius of the inscribed circle is represented by r and s is semi-perimeter, s = a + b + c2.

    Proof using laws of contangents
    Triangle with an inscribed circle

    Math Facts about Heron’s Formula

    For the area of a cyclic quadrilateral, Heron’s formula is a specific instance of Brahmagupta’s formula. Heron’s and Brahmagupta’s formulas for the area of a quadrilateral are both special instances of Bretschneider’s formula. By setting one of the quadrilateral’s sides to zero, Heron’s formula can be produced using Brahmagupta’s or Bretschneider’s formulae. The area K of a cyclic quadrilateral with sides of lengths a, b, c, and d is calculated using Brahmagupta’s formula.

    Brahmagupta's formula
    Cyclic quadrilateral

    K = (s – a)(s – b)(s – c)(s – d), where s is a semi-perimeter and defined as
    s = a + b + c + d2 .

    Heron’s formula for equilateral triangle (lengths of all sides are equal, a) is A = s(s – a)3, where s = 3a2.

    Know more about other concepts of Geometry on Geometric shapes, Perimeter of Triangle, Perimeter of Circle, What is Perimeter and Types of Triangle.

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