How it’s calculated
With all three sides known (SSS), Heron's formula gives the area directly from the semi-perimeter s, and the law of cosines gives each angle.
The sides must satisfy the triangle inequality: each side shorter than the sum of the other two.
Example: sides 5, 6 and 7 ft give s = 9, so Area = √(9 × 4 × 3 × 2) = √216 ≈ 14.70 ft². Angle A = arccos((36 + 49 − 25) ÷ 84) = arccos(0.7143) ≈ 44.42°.
Frequently asked questions
What is Heron’s formula?
Area = √(s(s−a)(s−b)(s−c)), where s is half the perimeter. It finds a triangle’s area from its three sides without needing the height.
Why do I get an error for some side lengths?
Three lengths only form a triangle if each one is shorter than the other two added together. For example 2, 3 and 6 cannot close up into a triangle.
How do I know if a triangle is a right triangle?
If the squares of the two shorter sides add up to the square of the longest side (a² + b² = c²), it has a 90° angle, like 3-4-5.
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Sources
- Precalculus 2e, 8.2 Non-right Triangles: Law of Cosines (incl. Heron’s formula) — OpenStax
- Contemporary Mathematics, 10.6 Area — OpenStax
- Heron’s Formula — Wolfram MathWorld
Formulas are taken from the free public references above. Results are provided “as is” for informational and educational purposes only. See our disclaimer.
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