Calculate triangle area from three side lengths using Heron's formula. See semi-perimeter, inradius, circumradius, altitudes, medians, angles, and a visual sketch.
**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, this tool gives you every derived quantity in one place, fully explained with adjustable precision and selectable units.
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.
Heron's Formula: A = √[s(s − a)(s − b)(s − c)] where s = (a + b + c) / 2. Inradius r = A / s. Circumradius R = abc / (4A). Altitude hₐ = 2A / a. Median mₐ = √[(2b² + 2c² − a²) / 4].
Result: Area = 30 cm²
s = (5 + 12 + 13) / 2 = 15. Area = √[15 × 10 × 3 × 2] = √900 = 30. This is a right triangle (5² + 12² = 13²). Inradius r = 30/15 = 2, circumradius R = 5×12×13/(4×30) = 6.5.
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.
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.
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.
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.
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.
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).
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.
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.
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.