Heron's Formula

Heron's formula (also called Hero's formula) lets us find the area of any triangle when we know the lengths of its three sides — even if we do not know its height. This is very useful for scalene triangles, where drawing a perpendicular height is not always easy or possible with the given data.

For a triangle with sides a, b and c, we first find the semi-perimeter (half the perimeter):

s = (a + b + c) / 2

Then the area of the triangle is given by:

Area = √[s(s − a)(s − b)(s − c)]

Worked Example: Find the area of a triangle whose sides are 13 cm, 14 cm and 15 cm.

Here a = 13 cm, b = 14 cm, c = 15 cm.

s = (13 + 14 + 15) / 2 = 42 / 2 = 21 cm

s − a = 21 − 13 = 8, s − b = 21 − 14 = 7, s − c = 21 − 15 = 6

Area = √(21 × 8 × 7 × 6) = √7056 = 84 cm²

So the area of the triangle is 84 cm², found without ever drawing or measuring its height.

Heron's formula is especially powerful for scalene triangles, where no two sides are equal and finding the height directly can be difficult. It also works perfectly for isosceles and equilateral triangles, giving results that agree with their usual shortcut formulas.

Worked Example 1: A triangle has sides 5 cm, 12 cm and 13 cm. Find its area.

s = (5 + 12 + 13) / 2 = 30 / 2 = 15 cm

s − a = 15 − 5 = 10, s − b = 15 − 12 = 3, s − c = 15 − 13 = 2

Area = √(15 × 10 × 3 × 2) = √900 = 30 cm²

(Since 5² + 12² = 13², this is actually a right triangle, and 30 cm² also equals (1/2) × 5 × 12 — Heron's formula agrees with the usual right-triangle formula.)

Worked Example 2 (Equilateral triangle): Using Heron's formula on an equilateral triangle of side a, we get s = 3a/2, and each bracket (s − a) = a/2. So:

Area = √[(3a/2)(a/2)(a/2)(a/2)] = √(3a⁴/16) = (√3/4) × a²

This gives the standard shortcut formula for the area of any equilateral triangle, derived directly from Heron's formula.

Once we know a triangle's area from Heron's formula, we can use the ordinary formula Area = (1/2) × base × height to work backwards and find the height (altitude) corresponding to any chosen side — this is one of the most useful applications of Heron's formula.

Worked Example: A triangle has sides 13 cm, 14 cm and 15 cm. Find the height corresponding to the side of length 14 cm.

From the earlier calculation, the area of this triangle is 84 cm² (using s = 21 cm).

Taking the side of 14 cm as the base:

Area = (1/2) × base × height

84 = (1/2) × 14 × height

height = (84 × 2) / 14 = 168 / 14 = 12 cm

So the altitude drawn to the 14 cm side is 12 cm. Notice how Heron's formula let us find this height without ever measuring it directly — we only needed the three side lengths.

This method is used whenever a problem asks for an altitude, or when the height of a triangular plot of land, wall, or signboard needs to be calculated from its side measurements alone.

A general quadrilateral does not have a simple area formula like a rectangle or square. However, if we draw one diagonal, it splits the quadrilateral into two triangles. We can then apply Heron's formula to each triangle separately and add the two areas.

Area of quadrilateral ABCD = Area of ΔABC + Area of ΔACD (where AC is the diagonal)

Worked Example: A quadrilateral field ABCD has AB = 9 m, BC = 40 m, CD = 28 m and DA = 15 m, with diagonal AC = 41 m. Find its area.

Triangle ABC (sides 9 m, 40 m, 41 m):

s₁ = (9 + 40 + 41) / 2 = 90 / 2 = 45 m

Area = √(45 × 36 × 5 × 4) = √32400 = 180 m²

Triangle ACD (sides 41 m, 28 m, 15 m):

s₂ = (41 + 28 + 15) / 2 = 84 / 2 = 42 m

Area = √(42 × 1 × 14 × 27) = √15876 = 126 m²

Total area of quadrilateral ABCD = 180 + 126 = 306 m²

This split-by-diagonal method works for any quadrilateral — rectangles, parks, agricultural fields or irregular plots — as long as the lengths of all four sides and one diagonal are known.

A rhombus is a special quadrilateral with all four sides equal. Any one diagonal splits it into two congruent triangles (congruent by SSS, since all four sides are equal). Its two diagonals also bisect each other at right angles. Heron's formula can be applied to one of these triangles, and the result doubled to get the full area.

Worked Example: A rhombus PQRS has each side 10 cm, and one diagonal PR = 12 cm. Find its area.

The diagonal PR splits the rhombus into two congruent triangles, each with sides 10 cm, 10 cm and 12 cm.

s = (10 + 10 + 12) / 2 = 32 / 2 = 16 cm

Area of one triangle = √(16 × 6 × 6 × 4) = √2304 = 48 cm²

Area of rhombus = 2 × 48 = 96 cm²

This matches the usual diagonal formula for a rhombus: Area = (1/2) × d₁ × d₂. Here, since the diagonals bisect each other at right angles, half of the second diagonal = √(10² − 6²) = √64 = 8 cm, so the full second diagonal = 16 cm, and Area = (1/2) × 12 × 16 = 96 cm² — the same answer.

Heron's formula is widely used in real life: surveyors calculate the area of irregular triangular and quadrilateral plots of land, and it appears in problems involving triangular parks, signboards, sails and tiled floor designs, wherever only side lengths are easy to measure.