What is Heron's formula?

Heron's formula computes the area of a triangle directly from its three side lengths (a, b, c) without needing any angle. First, calculate the semi-perimeter: s = (a + b + c) / 2. Then: Area = sqrt(s(s-a)(s-b)(s-c)). The formula is named after the ancient Greek mathematician Hero of Alexandria (c. 10–70 AD), who proved it in his work Metrica.

Why use Heron's formula instead of the base-times-height formula?

The base-times-height formula (Area = 0.5 x base x height) requires you to know the altitude, which you often don't if only the side lengths are given. Heron's formula works purely from the three side lengths — ideal for surveying, construction, and any situation where you measure distances rather than heights.

What is the semi-perimeter and why does the formula use it?

The semi-perimeter s = (a + b + c) / 2 is simply half the perimeter. Heron's formula can be written without it (Area = 0.25 x sqrt((a+b+c)(-a+b+c)(a-b+c)(a+b-c))), but the s notation compresses the four factors into a tidier form and makes it easier to compute by hand.

What is the triangle inequality and when does a triangle not exist?

A valid triangle requires that each side be strictly shorter than the sum of the other two: a + b > c, a + c > b, and b + c > a. If any of these conditions fails, the three lengths are collinear or degenerate and no real triangle exists. Heron's formula would yield a negative number under the square root in that case, which is why the calculator flags it as an error.

What are the inradius and circumradius?

The inradius (r) is the radius of the largest circle that fits inside the triangle, tangent to all three sides. It equals Area / s, where s is the semi-perimeter. The circumradius (R) is the radius of the circle passing through all three vertices (circumscribed circle). It equals (a x b x c) / (4 x Area). Both are useful in trigonometry, geometry proofs, and engineering design.

Is the calculator accurate for very small or very large side lengths?

Yes for most practical inputs. The calculation uses IEEE 754 double-precision floating-point arithmetic (about 15-16 significant digits). For extremely large or extremely small values (for example sides around 1e154 or 1e-154), floating-point rounding may introduce small errors. Results are rounded to 6 significant figures for display.

Heron's Formula Calculator

Heron’s formula is one of geometry’s most elegant results: given only the three side lengths of a triangle — no angles, no heights — you can calculate the area exactly. The calculator above implements the full formula, displays a labelled diagram, and shows the step-by-step working so you can follow every calculation.

The formula

Label the three sides a, b, and c. First compute the semi-perimeter:

s = (a + b + c) / 2

The area is then:

Area = sqrt( s * (s - a) * (s - b) * (s - c) )

That’s it. No trigonometry, no altitude measurement — just the three side lengths and a square root.

How this calculator works

The tool also derives every other property of the triangle from the same three inputs:

Angles — computed from the law of cosines: angle A = arccos((b^2 + c^2 - a^2) / (2bc)), and similarly for B and C.
Altitudes — each altitude equals 2 * Area / (base side), so h_a = 2A / a.
Inradius — the radius of the inscribed circle: r = Area / s.
Circumradius — the radius of the circumscribed circle: R = (a * b * c) / (4 * Area).
Triangle type — classified as equilateral, isosceles, or scalene by comparing sides, and as acute, right, or obtuse by comparing the largest angle to 90 degrees.

The SVG diagram scales to your inputs so the relative proportions are always correct.

Worked example

Suppose a triangle has sides a = 5, b = 6, c = 7:

Semi-perimeter: s = (5 + 6 + 7) / 2 = 9
Factors: (9 - 5) = 4, (9 - 6) = 3, (9 - 7) = 2
Product under the root: 9 * 4 * 3 * 2 = 216
Area: sqrt(216) = 14.6969... (approximately 14.70 units squared)

Cross-checks:

Perimeter = 18 units
Largest angle (opposite c = 7): arccos((25 + 36 - 49) / 60) = arccos(0.2) = approx 78.46 degrees — acute triangle.
Inradius: 14.697 / 9 = approx 1.633
Circumradius: (5 * 6 * 7) / (4 * 14.697) = 210 / 58.79 = approx 3.572

Formula note

Heron’s formula is algebraically equivalent to the determinant form and to the cross-product form: Area = 0.5 * |b x c| * sin(A). It can also be written without the semi-perimeter variable as: Area = 0.25 * sqrt((a + b + c) * (-a + b + c) * (a - b + c) * (a + b - c)). All three forms give identical results; the semi-perimeter version is used here because it matches the classical statement of the theorem and makes the working easiest to read.

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