Difference between revisions of "Heron's Formula"
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<math>=\sqrt{s(s-a)(s-b)(s-c)}</math> | <math>=\sqrt{s(s-a)(s-b)(s-c)}</math> | ||
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+ | ==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== | ==Example== | ||
− | Let's say that you have a right triangle with the sides 3,4, and 5. Your semi- perimeter would be 6. | + | 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 6-3=3, 6-4=2, 6-5=1. | + | Then you have <math>6-3=3</math>, <math>6-4=2</math>, <math>6-5=1</math>. |
− | 1 | + | <math>1\cdot 2\cdot 3=6.</math> |
− | <math> 6 | + | <math> 6\cdot 6 = 36</math> |
− | The square root of 36 is 6. The area of your triangle is 6. | + | The square root of <math>36</math> is <math>6</math>. The area of your triangle is <math>6</math>. |
+ | |||
== See Also == | == See Also == | ||
* [[Brahmagupta's formula]] | * [[Brahmagupta's formula]] |
Revision as of 21:28, 21 April 2022
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.