Heron's Formula allows us to find the area of a triangle when all three sides are known, without needing the height.
Formula: If a triangle has sides a, b and c, and the semi-perimeter s = (a + b + c)/2, then:
Area = root(s(s-a)(s-b)(s-c))
This formula was developed by the ancient Greek mathematician Heron of Alexandria.
Area using base and height (for reference): Area = (1/2) x base x height.
Area of a quadrilateral using Heron's Formula: Divide the quadrilateral into two triangles by a diagonal, find the area of each triangle using Heron's Formula, and add them.
Find the area of a triangle with sides 3, 4 and 5 cm.
s = (3+4+5)/2 = 6.
Area = root(6 x 3 x 2 x 1) = root(36) = 6 sq cm.
(This is also a right triangle: area = (1/2) x 3 x 4 = 6. Confirmed.)
A triangle has sides 7, 8 and 9 cm. Find its area.
s = (7+8+9)/2 = 12.
Area = root(12 x 5 x 4 x 3) = root(720) = 12 · root(5) approximately 26.83 sq cm.
An equilateral triangle has side 6 cm. Find area using Heron's Formula.
s = (6+6+6)/2 = 9.
Area = root(9 x 3 x 3 x 3) = root(243) = 9 · root(3) approximately 15.59 sq cm.
(Check: formula for equilateral = (root(3)/4) x 36 = 9 · root(3). Confirmed.)
A triangular park has sides 40 m, 32 m and 24 m. Find its area.
s = (40+32+24)/2 = 48.
Area = root(48 x 8 x 16 x 24) = root(147456) = 384 sq m.
(Note: 48 x 8 = 384, 16 x 24 = 384. 384 x 384 = 147456. root(147456) = 384.)
A rhombus has side 13 cm and one diagonal 10 cm. Find its area.
The diagonal divides the rhombus into two triangles. Each triangle has sides 13, 13 and 10.
s = (13+13+10)/2 = 18.
Area of one triangle = root(18 x 5 x 5 x 8) = root(3600) = 60 sq cm.
Total area = 2 x 60 = 120 sq cm.
A quadrilateral ABCD has diagonal AC = 20 m. Triangles ABC and ACD have areas found using Heron's Formula. If both triangles have the same area of 150 sq m, the quadrilateral area = 300 sq m.
Sides of a triangle are in ratio 3:5:7 and perimeter = 300 cm. Find the area.
Sides: 3k + 5k + 7k = 300, k = 20. Sides: 60, 100, 140 cm.
s = 150.
Area = root(150 x 90 x 50 x 10) = root(6750000) = 1500 · root(3) approximately 2598 sq cm.
- Key Formula Summary:
- s = (a + b + c) / 2
- Area = root(s(s-a)(s-b)(s-c))
- Equilateral triangle area = (root(3)/4) x a2
Common mistakes
Students forget to divide the perimeter by 2 when finding s. Also, all three sides must satisfy the triangle inequality before applying the formula. If the product s(s-a)(s-b)(s-c) is negative, the given sides do not form a triangle.
Summary
Heron's Formula is a universal method for triangle area using only the three side lengths. It is especially useful when the height is not given. Divide quadrilaterals into triangles to extend the method.