Heron's Formula
Triangle area from the three side lengths, where s is the semi-perimeter.
Formula
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Example
a=3,b=4,c=5 ⇒ 6
How it works
Heron's Formula lets you find the area of any triangle using only its three side lengths, even when no height is given. It works because the side lengths determine the triangle's shape, and the formula is built from the same geometric relationships that appear in the Law of Cosines and the standard area formula. The quantity s is the semiperimeter, half the perimeter, which makes the expression symmetric in a, b, and c.
When to use it
Use this when you know all three side lengths of a triangle but do not know the height or any angles. It is especially useful in geometry, surveying, and physics problems where triangle dimensions come from measurements rather than drawings.
Worked example, step by step
Given a = 7, b = 8, c = 9 Step 1: Find the semiperimeter: s = (a + b + c) / 2 = (7 + 8 + 9) / 2 = 24 / 2 = 12 Step 2: Substitute into Heron's Formula: Area = √[s(s - a)(s - b)(s - c)] Step 3: Area = √[12(12 - 7)(12 - 8)(12 - 9)] Step 4: Area = √[12 × 5 × 4 × 3] Step 5: Area = √720 Step 6: Area ≈ 26.83 Final answer: the triangle's area is about 26.83 square units
Common mistakes to avoid
- Using the full perimeter instead of the semiperimeter s
- Forgetting to subtract each side from s before multiplying
- Rounding too early and losing accuracy in the final area
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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