155. Heron's Secret Formula · Find any triangle's area from just its three sides

Try to break it

Drag C to make the triangle very tall, very wide, or nearly flat. Sides a, b, c shift, and the semi-perimeter s = (a+b+c)/2 updates. √(s(s−a)(s−b)(s−c)) always matches the geometric ½ × base × height. Try to find a shape where they disagree — you can't.

How you build it

Construct a triangle with sides 13, 14, 15.

• Mark point A on the canvas.
• Mark point B to the right of A.
• Draw segment AB as the base of the triangle.
• With centre A and radius 13, draw an arc above AB.
• With centre B and radius 15, draw an arc to intersect the previous arc.
• Mark point C where the two arcs intersect.
• Draw segment AC, the side of length 13.
• Draw segment BC, the side of length 15.
• From C, draw a perpendicular to line AB. This is the altitude h.

The proof, step by step

Prove that the Heron formula gives the same area as half × base × height.

• Let CD be the altitude from C to AB. Let AD = x, so DB = c - x.
• Apply Pythagoras in △ADC: h² = b² - x². In △BDC: h² = a² - (c - x)².
• Equate the two expressions for h² and solve for x: x = (b² - a² + c²) / (2c).
• Substitute x back into h² = b² - x². After extensive algebraic simplification, h² = [4b²c² - (b² - a² + c²)²] / (4c²).
• Factor the numerator as a difference of squares, then apply difference of squares again. You get 2(s-a) * 2(s-b) * 2(s-c) * 2s.
• Take the square root to find h, then Area = ½ × base × height = √[s(s-a)(s-b)(s-c)]. Q.E.D.

Worked example

A triangular plot of land has sides measuring 13 cm, 14 cm, and 15 cm. Using Heron's formula, calculate the exact area of the plot.

s = (13+14+15)/2 = 21. Area = √[21(21-13)(21-14)(21-15)] = √[21×8×7×6] = √7056 = 84 cm².

• 84 cm² — correct
• 90 cm²
• 72 cm²
• 96 cm²