About

Determining the area of a triangle when only three side lengths are known requires Heron's formula, a method dating to the 1st century CE. The formula computes area through the semi-perimeter s = (a + b + c) ÷ 2 without requiring height or angle measurements. Incorrect area computation in surveying and structural engineering leads to material miscalculation and cost overruns. This tool validates the triangle inequality theorem before computing: the sum of any two sides must strictly exceed the third. It also derives all three interior angles via the law of cosines, classifies the triangle by side and angle type, and computes altitudes, inradius, and circumradius.

Results assume Euclidean plane geometry on flat surfaces. For spherical or hyperbolic geometries, corrections are required. Floating-point precision is limited to 10 decimal digits internally, which may introduce rounding artifacts on triangles with extreme aspect ratios (e.g., a = 1000000, b = 1000000, c = 0.001). Pro tip: when measuring real-world sides, add 1% tolerance for tape sag and thermal expansion of metal tapes.

Formulas

The primary formula used is Heron's formula. Given three sides a, b, c, compute the semi-perimeter first:

s = a + b + c2

Then the area is:

A = √s(s − a)(s − b)(s − c)

Interior angles are derived via the law of cosines. For angle α opposite side a:

α = arccos(b² + c² − a²2bc)

Altitude from side a:

h_a = 2Aa

Inradius and circumradius:

r = As R = a ⋅ b ⋅ c4A

Where: a, b, c = side lengths. s = semi-perimeter. A = area. α, β, γ = interior angles opposite sides a, b, c. h_a = altitude to side a. r = inradius. R = circumradius.

Reference Data

Triangle Type	Side Condition	Angle Condition	Example Sides	Area Formula Shortcut
Equilateral	a = b = c	All 60°	5, 5, 5	√34 ⋅ a²
Isosceles	Two sides equal	Two angles equal	5, 5, 8	Heron's or b4√4a² − b²
Scalene	All sides different	All angles different	3, 5, 7	Heron's formula
Right (3-4-5)	a² + b² = c²	One 90°	3, 4, 5	12 ⋅ a ⋅ b
Right (5-12-13)	Pythagorean triple	One 90°	5, 12, 13	30 sq units
Right (8-15-17)	Pythagorean triple	One 90°	8, 15, 17	60 sq units
Obtuse	Longest side dominates	One angle > 90°	3, 4, 6	Heron's formula
Acute	No side dominates	All < 90°	5, 6, 7	Heron's formula
Degenerate	a + b = c	Collinear points	1, 2, 3	Area = 0
30-60-90	1 : √3 : 2	30°, 60°, 90°	1, 1.732, 2	√34 ⋅ c²
45-45-90	1 : 1 : √2	45°, 45°, 90°	1, 1, 1.414	a²2
Golden Gnomon	1 : 1 : φ	36°, 36°, 108°	1, 1, 1.618	Heron's formula
Egyptian (3-4-5)	Oldest known right triangle	One 90°	3, 4, 5	6 sq units
Nearly Flat	c ≈ a + b	One angle ≈ 180°	1, 1, 1.999	Area ≈ 0

Frequently Asked Questions

A valid triangle requires that the sum of any two sides strictly exceeds the third: a + b > c, and similarly for the other two combinations. If any condition fails, no triangle can be formed and Heron's formula yields a negative value under the square root. The calculator detects this and reports an error. The boundary case where a + b = c produces a degenerate triangle (collinear points) with zero area.

When one side nearly equals the sum of the other two, the factor (s − c) approaches zero. IEEE 754 double-precision floats carry about 15-16 significant digits, so catastrophic cancellation can occur. For example, sides 100000, 100000, 0.001 lose roughly 10 digits of precision in the subtraction. This calculator uses the numerically stable Kahan form of Heron's formula (sorting sides and using the rearranged product) when the aspect ratio exceeds 1000:1 to mitigate this.

Each angle is computed via the law of cosines: α = arccos((b² + c² − a²) ÷ (2bc)). The calculator verifies that α + β + γ = 180° (within floating-point tolerance of 0.001°). If one angle is exactly 90° (within tolerance), the triangle is classified as right-angled.

The inradius r is the radius of the largest circle that fits inside the triangle (incircle), tangent to all three sides. It equals A ÷ s. The circumradius R is the radius of the circle passing through all three vertices (circumcircle), equal to abc ÷ (4A). These are critical in structural engineering for inscribing/circumscribing circular elements, in computational geometry for mesh quality metrics (the ratio R/r ≥ 2 always, with equality only for equilateral triangles), and in trigonometric surveying.

The calculator is unit-agnostic. Enter all three sides in the same unit. The output area will be in that unit squared. For example, if sides are in meters, the area is in m². If sides are in feet, the area is in ft². Mixing units (e.g., one side in inches, another in centimeters) will produce meaningless results. Convert all measurements to a single unit before entering them.

By sides: if all three are equal within tolerance (1e-9), it is equilateral. If exactly two are equal, isosceles. Otherwise, scalene. By angles: if the largest angle exceeds 90° + 0.001°, it is obtuse. If it falls within 89.999° to 90.001°, it is right. Otherwise, it is acute. This tolerance accounts for floating-point rounding in the arccos computation.