Calculate triangle area from three sides using Heron's formula. Also computes perimeter, all angles, altitudes, medians, circumradius, inradius, and classifies by angle and side type.
Heron's formula is one of the oldest and most elegant results in geometry: it computes the area of a triangle purely from the lengths of its three sides, with no need to find a height first. Named after Hero of Alexandria (1st century AD), the formula uses the semi-perimeter s = (a + b + c)/2 and gives Area = √[s·(s−a)·(s−b)·(s−c)].
The SSS (Side-Side-Side) case is the most common triangle specification in practical problems. Surveyors measure three distances, carpenters cut three boards, engineers specify three beam lengths. From those three numbers, every other property of the triangle can be derived — angles via the law of cosines, altitudes as 2·Area/side, medians via the median formula, and the radii of the inscribed and circumscribed circles.
This calculator walks through all of those computations step by step. It validates that the three sides satisfy the triangle inequality (the sum of any two sides must exceed the third), classifies the triangle by its angles (acute, right, or obtuse) and its sides (scalene, isosceles, or equilateral), and presents a detailed Heron's formula breakdown table so you can follow each intermediate value.
Preset buttons load famous Pythagorean triples and other well-known triangles, and a reference table lets you compare your result against common configurations. Whether you're a student learning geometry or an engineer sizing structure members, Heron's formula remains the go-to tool for three-side triangle problems.
This calculator is useful when the only measurements you have are the three side lengths. Heron's formula turns that SSS data directly into area, and once the area is known the rest of the triangle properties follow naturally, including altitudes, medians, radii, and angle classification. That makes it practical for surveying checks, fabrication work, and geometry problems where no angle or height was measured in advance.
s = (a + b + c) / 2 Area = √[s·(s−a)·(s−b)·(s−c)] Angle A = arccos((b²+c²−a²) / (2bc)) Altitude hₐ = 2·Area / a Median mₐ = ½·√(2b²+2c²−a²) Circumradius R = abc / (4·Area) Inradius r = Area / s
Result: Area ≈ 26.83, Perimeter = 24, Angles ≈ 48.19°, 58.41°, 73.40°
With sA = 7, sB = 8, and sC = 9, the semi-perimeter is 12. Heron's formula gives area = √(12 × 5 × 4 × 3) = √720 ≈ 26.83. From there, the law of cosines gives angle A ≈ 48.19°, and the other two angles confirm that the triangle is acute and scalene.
Heron's formula starts with the semi-perimeter s = (a + b + c) / 2 and then multiplies the four factors s, s−a, s−b, and s−c under one square root. That structure is what makes the method so useful: three side lengths are enough to recover area without dropping an altitude or knowing any angle first. A worked breakdown is especially helpful for spotting invalid input, because one negative factor means the triangle inequality has failed.
Once the area is known, side-only data becomes surprisingly rich. The law of cosines turns the same three lengths into all three angles, letting you decide whether the triangle is acute, right, or obtuse. Altitudes, medians, the inradius, and the circumradius all follow from those core values, so an SSS solve is often enough to characterize the entire triangle from a single measurement set.
SSS calculations show up whenever distances are easier to measure than heights or angles. Surveyors can record three edge lengths around a parcel corner, fabricators can verify a triangular brace from cut lengths alone, and students can solve competition problems where only side relationships are given. In each case, Heron's formula is the bridge from raw lengths to a complete geometric picture.
Heron's formula calculates the area of a triangle from its three side lengths: Area = √[s(s−a)(s−b)(s−c)], where s = (a+b+c)/2 is the semi-perimeter. Understanding this concept helps you apply the calculator correctly and interpret the results with confidence.
The semi-perimeter is half the perimeter: s = (a+b+c)/2. It simplifies many triangle formulas, including Heron's, the inradius, and Euler's line relations.
Yes — if the three sides don't satisfy the triangle inequality (a+b>c, etc.), the product under the square root is negative, meaning no valid triangle exists. Understanding this concept helps you apply the calculator correctly and interpret the results with confidence.
Use the law of cosines. For angle A: cos(A) = (b²+c²−a²)/(2bc). Then A = arccos of that value. Repeat for each angle, or compute the third as 180°−A−B.
R = abc/(4·Area). It's the radius of the unique circle passing through all three vertices.
r = Area/s. It's the radius of the largest circle that fits inside the triangle, tangent to all three sides.