Does Heron's formula work for right, isosceles, and equilateral triangles?

Yes, it applies to all planar triangles. For equilateral triangles with side length $a$, Heron's formula algebraically reduces to the standard $A = \frac{\sqrt{3}}{4}a^2$.

What if the value inside the square root is zero or negative?

A zero or negative product indicates the provided side lengths violate the triangle inequality theorem and cannot form a valid polygon with non-zero area.

Can Heron's formula be used in non-Euclidean geometry?

No. It is only valid for flat, planar triangles in Euclidean space. Spherical and hyperbolic triangles require formulas based on angular excess or defect.

How to find the area of a triangle with Heron's formula

Heron's formula calculates the area of a triangle using only the lengths of its three sides. It applies to any planar triangle when the side lengths are known, bypassing the need to determine an altitude.

The setup

Identify the three side lengths of the triangle, denoted as $a$ , $b$ , and $c$ . Ensure that the lengths satisfy the triangle inequality theorem: the sum of any two sides must be strictly greater than the third side.

The steps

Calculate the semi-perimeter $s = \frac{a + b + c}{2}$ .
Subtract each side length from the semi-perimeter to find $(s-a)$ , $(s-b)$ , and $(s-c)$ .
Multiply these three differences by the semi-perimeter $s$ .
Take the square root of the product to find the area: $A = \sqrt{s(s-a)(s-b)(s-c)}$ .

Checking the result

Verify that the argument under the square root is positive. If it is zero or negative, the side lengths do not form a valid triangle. For a quick magnitude check, the calculated area must be strictly less than $\frac{1}{2}ab$ for any two sides $a$ and $b$ .

Common errors

The most frequent error is calculating the full perimeter instead of the semi-perimeter $s$ . Another common mistake is omitting the initial $s$ term under the radical, multiplying only the three difference terms together.

Worked example

Find the exact area of a triangle with side lengths $a=7$ , $b=8$ , and $c=9$ .

Calculate the semi-perimeter: $s = \frac{7 + 8 + 9}{2} = \frac{24}{2} = 12$

Calculate the differences from the semi-perimeter: $s - a = 12 - 7 = 5$ $s - b = 12 - 8 = 4$ $s - c = 12 - 9 = 3$

Substitute into Heron's formula: $A = \sqrt{s(s-a)(s-b)(s-c)}$ $A = \sqrt{12(5)(4)(3)}$

Evaluate the product: $A = \sqrt{12(60)}$ $A = \sqrt{720}$

Simplify the radical: $A = \sqrt{144 imes 5}$ $A = 12\sqrt{5}$

FAQ

Run your own problem

Answer form

References: OpenStax Precalculus, Chapter 8 · Khan Academy, High School Geometry: Right triangles & trigonometry · Geometry by Harold R. Jacobs