Heron's Formula
Description
Heron's Formula (also called Hero's Formula) is a major theorem in geometry that allows you to calculate the area of a triangle knowing *only* the lengths of its three sides ($a, b, c$). Unlike the standard formula $A = \frac{1}{2}bh$, you do not need to calculate the vertical height or measure any angles.
The key to the formula is the **Semi-perimeter ($s$)**, which is half of the triangle's total perimeter: $$s = \frac{a + b + c}{2}$$
This formula works for any type of triangle: equilateral, isosceles, or scalene. It is particularly useful in land surveying where measuring side lengths is easier than establishing a perpendicular height.
History & Origins
The formula is named after Heron of Alexandria (c. 10 – 70 AD), a Greek engineer and mathematician who proved it in his book Metrica. However, many historians believe the formula was actually known much earlier by Archimedes (c. 250 BC). Since Heron's book was a compilation of mathematical knowledge, he likely preserved Archimedes' work. Interestingly, a similar formula for the area of a cyclic quadrilateral (Brahmagupta's Formula) was discovered by Indian mathematicians in the 7th century, of which Heron's formula is a special case (where one side length is zero).
Algebraic Proof using Trigonometry
We derive this by combining the standard area formula with the Law of Cosines.
Start with Area: $Area = \frac{1}{2}ab \sin C$.
Squaring it: $Area^2 = \frac{1}{4}a^2b^2 \sin^2 C$.
Use Pythagorean Identity: $\sin^2 C = 1 - \cos^2 C = (1-\cos C)(1+\cos C)$.
Substitute Law of Cosines for $\cos C$: $\cos C = \frac{a^2+b^2-c^2}{2ab}$.
Substitute this back into the area equation. After simpler algebra (difference of squares), the terms factor neatly into the perimeter components.
The expression simplifies to $\frac{1}{16}(a+b+c)(a+b-c)(a-b+c)(-a+b+c)$.
Substituting $s = (a+b+c)/2$, this becomes $s(s-a)(s-b)(s-c)$.
Take the square root to get Heron's Formula.
Variables
| Symbol | Meaning |
|---|---|
A | Area |
a, b, c | Lengths of the three sides |
s | Semi-perimeter (half of perimeter) |
Examples
Basic Calculation
Problem: Find the area of a triangle with sides 3, 4, 5
Solution:
Land Surveying
Problem: A triangular plot of land has sides measuring 30m, 40m, and 50m. What is its area?
Solution: 600 m²
- Calculate Semi-perimeter: $s = \frac{30+40+50}{2} = \frac{120}{2} = 60$ m.
- Calculate differences: $(s-a) = 30$, $(s-b) = 20$, $(s-c) = 10$.
- Apply Formula: $A = \sqrt{60 \times 30 \times 20 \times 10}$.
- Multiply: $60 \times 30 = 1800$. $20 \times 10 = 200$. $1800 \times 200 = 360,000$.
- Square Root: $\sqrt{360,000} = 600$.
Equilateral Triangle
Problem: Find the area of an equilateral triangle with side length 6.
Solution: 15.59
- Sides: $a=6, b=6, c=6$.
- Semi-perimeter: $s = \frac{18}{2} = 9$.
- Formula: $A = \sqrt{9(9-6)(9-6)(9-6)} = \sqrt{9(3)(3)(3)}$.
- Calculate: $\sqrt{243} \approx 15.588$.
Common Mistakes
Forgetting to divide perimeter by 2
Remember that $s$ is the **Semi**-perimeter. If you use the full perimeter, the number under the square root will be huge and incorrect.
Violating Triangle Inequality
If the sum of two sides is less than the third (e.g., sides 1, 2, 10), the triangle cannot exist. In Heron's formula, this results in a negative number under the square root (imaginary area), indicating an impossible triangle.
Real-World Applications
3D Computer Graphics
In 3D modeling, objects are essentially meshes of thousands of triangles. To calculate surface properties like texture mapping or light reflection, the computer needs the area of these triangles. Since the computer knows the 3D coordinates $(x,y,z)$ of the vertices, it calculates the side lengths (distances) first, then uses Heron's formula to find the area without needing to compute a "height" vector.
GPS and Geodesy
When measuring large triangular areas on the Earth's surface (Geodesy), surveyors measure the distances between three points (sides) rather than angles. Heron's formula is the standard way to convert these distance measurements into area.
Frequently Asked Questions
Can I use this if I know the height?
You can, but the standard formula $A = \frac{1}{2}bh$ is much faster. Heron's formula is best when height is unknown.
What happens if s-a is negative?
It means your triangle is impossible! A side ($a$) cannot be longer than the sum of the other two parts of the perimeter. This is a built-in "validity check" of the formula.