Heron's Formula
Heron's Formula
\[\begin{aligned}
A &= \sqrt{s(s-a)(s-b)(s-c)} \\
s &= \frac{a+b+c}{2}
\end{aligned}\]
Variables
A = triangle area
s = semi-perimeter
a = side length a
b = side length b
c = side length c
Description
What is this formula?
Heron's formula calculates the area of a triangle using only the lengths of its three sides. It is one of the most important formulas in geometry because it does not require knowing angles or heights.
When to use it
Use this formula when the three side lengths of a triangle are known and the area needs to be determined directly.
Example
If a triangle has sides:
a = 5
b = 6
c = 7
First calculate the semi-perimeter:
s = (5+6+7)/2 = 9
Then:
A = √(9(9-5)(9-6)(9-7))
A ≈ 14.7
Applications
Used in geometry, surveying, engineering, architecture, construction, and computer graphics for triangle area calculations.
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