Heron's Formula
Area of a triangle from its three sides, s = (a+b+c)/2.
Formula
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Example
Triangle 3, 4, 5: s = 6, A = √(6·3·2·1) = √36 = 6.
How it works
Heron's formula gives the area of any triangle when you know all three side lengths, even if you do not know the height. It works because the side lengths determine the triangle completely, and the semi-perimeter s = (a+b+c)/2 lets the area be written in a symmetric form that comes from combining the triangle's height with the Pythagorean theorem. This is especially useful when the triangle is not drawn on a grid or when finding the height directly would be awkward.
When to use it
Use this formula when you are given the three sides of a triangle and need its area. It is common in geometry problems, surveying, and trigonometry tasks where side lengths are known but the altitude is not.
Worked example, step by step
Find the area of a triangle with sides a = 7 cm, b = 8 cm, and c = 9 cm. Step 1: Find the semi-perimeter: s = (7 + 8 + 9) / 2 = 12 cm. Step 2: Substitute into Heron's formula: A = √(s(s-a)(s-b)(s-c)) = √(12(12-7)(12-8)(12-9)). Step 3: Simplify inside the square root: A = √(12·5·4·3) = √720. Step 4: Evaluate: A = 12√5 cm² ≈ 26.8 cm².
Common mistakes to avoid
- Using the full perimeter instead of the semi-perimeter in the formula
- Forgetting to subtract each side from s before multiplying
- Leaving the answer as a plain number without square units
Did You Know?
Heron's formula finds a triangle's area from its three sides alone — no height or angle needed.
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