Area of a Trapezium and a Polygon
189. Heron's Secret Code · Area from sides alone, every time
Heron's calculated area always matches the geometric area of the triangle.
Heron's Formula computes triangle area from side lengths only: Area = √[s(s−a)(s−b)(s−c)], where s = (a+b+c)/2 is the semi-perimeter. Powerful when the height is unknown.
What this lesson covers
Try to break it
Drag A, B, or C to reshape the triangle. Sides a, b, c shift, the semi-perimeter s = (a + b + c)/2 updates, and √(s(s − a)(s − b)(s − c)) always equals the geometric ½ × base × height. Try to find a shape where it disagrees — impossible.
How you build it
Make a triangle.
- Place point A anywhere on the canvas.
- Place point B so it is not in line with A.
- Place point C so that A, B and C are not in a straight line.
- Draw segment AB.
- Draw segment BC.
- Draw segment CA to complete triangle ABC.
- Drag any vertex to reshape the triangle, and notice that Heron's formula always matches the actual area.
The proof, step by step
Prove that the Heron formula gives the true area of the triangle.
- Start with the standard area formula: Area = ½ × base × height.
- Use the Law of Cosines to express the height in terms of the three sides.
- Substitute and simplify the algebraic expression step-by-step.
- Factor the resulting difference of squares to reveal s(s-a)(s-b)(s-c).
- Take the square root to arrive at Heron's elegant formula!
Worked example
In a triangle, the lengths of the three sides are 13 cm, 14 cm, and 15 cm. Using Heron's Formula, find the area of the triangle.
s = (13+14+15)/2 = 21 cm. Area = √[21(21-13)(21-14)(21-15)] = √[21×8×7×6] = √7056 = 84 cm².
- 84 cm² — correct
- 90 cm²
- 72 cm²
- 105 cm²