Difference between revisions of "Heron's Formula"
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<math>=\frac{ab}{2}\sqrt{1-\cos^2 C}</math> | <math>=\frac{ab}{2}\sqrt{1-\cos^2 C}</math> | ||
− | <math>=\frac{ab}{ | + | <math>=\frac{ab}{2}\sqrt{1-\left(\frac{a^2+b^2-c^2}{2ab}\right)^2}</math> |
<math>=\sqrt{\frac{a^2b^2}{4}\left[1-\frac{(a^2+b^2-c^2)^2}{4a^2b^2}\right]}</math> | <math>=\sqrt{\frac{a^2b^2}{4}\left[1-\frac{(a^2+b^2-c^2)^2}{4a^2b^2}\right]}</math> | ||
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<math>=\sqrt{s(s-a)(s-b)(s-c)}</math> | <math>=\sqrt{s(s-a)(s-b)(s-c)}</math> | ||
+ | |||
+ | ==Isosceles Triangle Simplification== | ||
+ | |||
+ | <math>A=\sqrt{s(s-a)(s-b)(s-c)}</math> for all triangles | ||
+ | |||
+ | <math>b=c</math> for all isosceles triangles | ||
+ | |||
+ | <math>A=\sqrt{s(s-a)(s-b)(s-b)}</math> simplifies to <math>A=(s-b)\sqrt{s(s-a)}</math> <math>\blacksquare</math> | ||
+ | |||
+ | ==Example== | ||
+ | Let's say that you have a right triangle with the sides <math>3</math> ,<math>4</math> , and <math>5</math>. Your semi- perimeter would be <math>6</math> since <math>(3+4+5)</math> ÷ <math>2</math> is <math>6</math>. | ||
+ | Then you have <math>6-3=3</math>, <math>6-4=2</math>, <math>6-5=1</math>. | ||
+ | <math>1+2+3=6.</math> | ||
+ | <math> 6\cdot 6 = 36</math> | ||
+ | The square root of <math>36</math> is <math>6</math>. The area of your triangle is <math>6</math>. | ||
== See Also == | == See Also == |
Revision as of 22:49, 1 February 2021
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 , the area can be found using the following formula:
where the semi-perimeter .
Proof
Isosceles Triangle Simplification
for all triangles
for all isosceles triangles
simplifies to
Example
Let's say that you have a right triangle with the sides , , and . Your semi- perimeter would be since ÷ is . Then you have , , . The square root of is . The area of your triangle is .
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.