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Heron's Formula

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Heron's Formula (sometimes called Hero's formula) is a formula for finding the area of a triangle given only the three side lengths.

Contents

Theorem

For any triangle with side lengths {a}, {b}, {c}, the area {A} can be found using the following formula:

A=\sqrt{s(s-a)(s-b)(s-c)}

where the semi-perimeter s=\frac{a+b+c}{2}.


Proof

[ABC]=\frac{ab}{2}\sin C

=\frac{ab}{2}\sqrt{1-\cos^2 C}

=\frac{ab}{2}\sqrt{1-\left(\frac{a^2+b^2-c^2}{2ab}\right)^2}

=\sqrt{\frac{a^2b^2}{4}\left[1-\frac{(a^2+b^2-c^2)^2}{4a^2b^2}\right]}

=\sqrt{\frac{4a^2b^2-(a^2+b^2-c^2)^2}{16}}

=\sqrt{\frac{(2ab+a^2+b^2-c^2)(2ab-a^2-b^2+c^2)}{16}}

=\sqrt{\frac{((a+b)^2-c^2)(c^2-(a-b)^2)}{16}}

=\sqrt{\frac{(a+b+c)(a+b-c)(b+c-a)(a+c-b)}{16}}

=\sqrt{s(s-a)(s-b)(s-c)}

See Also

External Links

In general, it is a good advice not to use Heron's formula in computer programs whenever we can avoid it. For example, whenever vertex coordinates are known, vector product is a much better alternative. Main reasons:

  • Computing the square root is much slower than multiplication.
  • For triangles with area close to zero Heron's formula computed using floating point variables suffers from precision problems.
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