Heron’s formula is the only way to find the area of a triangle when you know the three sides but not the height. It was the standard method for thousands of years because it needs nothing more than a ruler and a calculator, and it is still the fastest check on any other method. This Herons formula calculator takes three side lengths and returns the area along with the semi-perimeter, the height and the circumradius.
What Is the Herons Formula Calculator?
Heron's formula finds the area of a triangle when you know all three sides but cannot measure the height directly. That situation comes up constantly in surveying, construction and land work, where the three boundary points are known but a perpendicular drop is not.
The calculator returns the area in the unit you entered and converts it to square metres and square feet so you can compare it against a drawing or a plot plan without doing another conversion.
How Herons Formula Works
Area = √(s × (s − a) × (s − b) × (s − c)), where s is the semi-perimeter, s = (a + b + c) ÷ 2
The clever part is the substitution of the semi-perimeter, which turns the awkward general case into four simple multiplications. For a 3, 4, 5 triangle the semi-perimeter is 6, and the area is the square root of 6 × 3 × 2 × 1, which is the square root of 36, giving exactly 6 square units.
The formula only works when the three lengths genuinely form a triangle. If the longest side is longer than the other two added together, the expression under the square root goes negative and no real triangle exists.
Height and Circumradius
Once you know the area, the height onto any side follows from the ordinary triangle area formula, rearranged as height equals twice the area divided by the base. Dividing by the base you already know gives the perpendicular you could not measure directly.
The circumradius is the radius of the circle that passes through all three vertices. It comes from the relationship between the sides and the area and is useful in navigation and circle geometry. For a 3, 4, 5 triangle the circumradius is 2.5, exactly half the hypotenuse, because that triangle is right-angled.
Checks and Common Mistakes
A quick sanity check is to compare the result with the equilateral case. For three equal sides of length L the formula gives a height of L times the square root of three over two, which is the familiar 0.866 L. If your answer is wildly different, a side has probably been typed in the wrong unit.
Watch the units. If one side is in metres and another in feet the calculator cannot detect it, and the result will be plausible but wrong. Convert everything to a single unit first, then run the calculation.
Frequently Asked Questions
What is Herons formula used for?
It is used whenever three sides of a triangle are known but the height is not, which includes land area surveys, roof and gable calculations, and any irregular plot whose boundary points have been measured. It is also a handy check on an area worked out another way.
Why does the formula fail for some inputs?
If the longest side is equal to or longer than the sum of the other two, no triangle can be formed and the expression under the square root becomes zero or negative. The calculator detects this and tells you the lengths do not form a valid triangle.
How do I find the height of a triangle with three sides?
Calculate the area with Herons formula, then divide twice the area by the side you want the height onto. The calculator does this for you and reports the height onto side A.
What unit should I use for the sides?
Any unit works as long as all three sides use the same one. The area comes back in squared units of your input, and the calculator also converts it to square metres and square feet for convenience.
Does Herons formula work for obtuse and right triangles?
Yes. Unlike formulas that depend on a perpendicular height, Herons formula is valid for every non-degenerate triangle, including obtuse ones where the perpendicular falls outside the shape and right-angled ones where it lands exactly on a vertex.