Heron's Formula Calculator

About the Heron's Formula Calculator

**Heron's formula** computes the area of any triangle when you know all three side lengths — no height or angle measurements required. Given sides a, b, c, compute the semi-perimeter s = (a + b + c) / 2, then the area is √[s(s − a)(s − b)(s − c)]. The formula is attributed to Heron of Alexandria (c. 60 AD) and remains one of the most elegant results in elementary geometry.

This calculator goes far beyond the area. It verifies the triangle inequality, classifies the triangle by side and angle type, computes all three interior angles via the law of cosines, derives altitudes (heights), medians, the inradius of the inscribed circle, and the circumradius of the circumscribed circle. A proportional SVG sketch and colour-coded inequality bars make the geometry tangible.

Whether you are a student learning triangle properties, an engineer double-checking structural measurements, or a competitive-math contestant needing quick triangle data, the page keeps the derived quantities attached to the same three sides, fully explained with adjustable precision and selectable units.

When This Page Helps

Heron's formula is the go-to method whenever you have three measured side lengths and need the area — a situation that arises in surveying, construction, navigation, and classroom geometry alike. Because it avoids trigonometric functions in the core computation, it is both numerically stable and easy to explain.

This calculator bundles every derived triangle property into a single page: area, perimeter, angles, altitudes, medians, inradius, circumradius, triangle classification, inequality verification, and a proportional sketch. Students see formulas in action with step-by-step intermediate values; professionals get a quick cross-check for design measurements without opening a CAD tool.

How to Use the Inputs

1. Enter the three side lengths a, b, c, or click a preset triangle.
2. Select a measurement unit (cm, in, m, ft, mm, yd) for labelling.
3. Set the number of decimal places for all displayed values.
4. Read the area, perimeter, and semi-perimeter output cards.
5. Review the triangle type classification (scalene/isosceles/equilateral, acute/right/obtuse).
6. Check the angles table, altitudes & medians table, and triangle inequality bars.
7. View the SVG sketch for a proportional visualization of your triangle.

Example Calculation

Tips & Best Practices

• Use the 5-12-13 preset to verify right-triangle properties — the area should be exactly 30.
• An equilateral triangle (all sides equal) always has all angles at 60° and the highest possible area-to-perimeter ratio.
• If the triangle inequality bars show a ratio near 100 %, the triangle is close to degenerate (flat).
• The inradius × semi-perimeter always equals the area — a quick sanity check.
• Increase decimal places to reveal rounding details for irrational values like √3.
• Try swapping two side labels — the area stays the same because Heron's formula is symmetric in a, b, c.

Historical Background

Hero (Heron) of Alexandria published the formula around 60 AD in his work *Metrica*, though some historians believe Archimedes knew it two centuries earlier. The formula was rediscovered independently by Chinese mathematicians in the 7th century. Its proof can be given algebraically by expanding the expression, geometrically using the inscribed circle, or via the Cayley–Menger determinant. The beauty of the formula lies in its symmetry — it treats all three sides equally, reflecting the fact that a triangle's area depends on its shape rather than its orientation.

Numerical Stability

For very flat triangles (one side nearly equal to the sum of the other two), the naive square-root computation can lose precision because the factors (s − a), (s − b), (s − c) include near-zero terms. A numerically stable rearrangement, due to W. Kahan, sorts the sides so that a ≥ b ≥ c, then computes A = ¼√[(a+(b+c))(c−(a−b))(c+(a−b))(a+(b−c))]. This calculator uses standard floating-point evaluation, which is accurate for all non-degenerate triangles encountered in practice.

Generalizations

Heron's formula generalizes to cyclic quadrilaterals via Brahmagupta's formula, and further to general quadrilaterals with Bretschneider's formula. In higher dimensions, the Cayley–Menger determinant extends the idea to compute the volume of simplices from edge lengths alone.

Frequently Asked Questions

• What is Heron's formula?

  Heron's formula calculates a triangle's area from its three side lengths: A = √[s(s−a)(s−b)(s−c)], where s is the semi-perimeter (a + b + c)/2. No height measurement is needed.

• When does the triangle inequality fail?

  The triangle inequality fails when any one side is greater than or equal to the sum of the other two. Such side lengths cannot form a triangle, and the expression under the square root in Heron's formula becomes zero or negative.

• How do I find the inradius from the area?

  The inradius r = Area / s, where s is the semi-perimeter. This gives the radius of the largest circle that fits inside the triangle (inscribed circle, or incircle).

• What is the circumradius?

  The circumradius R = abc / (4 × Area). It is the radius of the circle passing through all three vertices (circumscribed circle). For a right triangle, R equals half the hypotenuse.

• Can Heron's formula handle degenerate triangles?

  A degenerate triangle has zero area — the three points are collinear. Heron's formula returns 0 when one side exactly equals the sum of the other two, correctly reflecting the degenerate case.

• How is the altitude calculated?

  The altitude from vertex A to side a is hₐ = 2 × Area / a. Each altitude creates a right angle with its base and represents the perpendicular distance from the opposite vertex to that side.