Triangle Calculator
Calculate the area, perimeter, angles, and height of a triangle from three sides using Heron's formula.
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How to use this calculator
Heron's formula computes triangle area from all three side lengths without needing the height.
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Enter the lengths of all three sides of the triangle.
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The calculator verifies the triangle inequality and computes area using Heron's formula.
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All three angles are calculated using the law of cosines.
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Height to side a is also shown (useful for area verification).
Frequently asked questions
What is Heron's formula?
Heron's formula calculates triangle area knowing only the three side lengths. First compute the semi-perimeter s = (a+b+c)/2, then Area = √(s(s−a)(s−b)(s−c)). Named after Hero of Alexandria (~60 AD), though possibly known to Archimedes earlier.
What is the triangle inequality?
For a valid triangle, the sum of any two sides must be strictly greater than the third side. If a=3, b=4, c=10: 3+4=7 < 10, so no triangle can be formed. This is why the calculator checks this before computing.
How do I find a triangle's area from base and height?
Area = (base × height) / 2. Heron's formula is used when the height is unknown. Both methods give the same result — Heron's formula is essentially computing the height implicitly from the side lengths.
What type of triangle do my sides form?
Equilateral: all sides equal. Isosceles: two sides equal. Scalene: all sides different. Right triangle: if a² + b² = c². Obtuse: if the largest angle > 90°. Acute: all angles < 90°. Check the computed angles to classify.
Triangle area and angle calculations
The law of cosines for finding angles
When all three sides are known, the law of cosines finds any angle: cos(A) = (b² + c² − a²) / (2bc). This generalizes the Pythagorean theorem — when angle A = 90°, cos(A) = 0 and the formula reduces to a² = b² + c².
Types of triangles and their properties
Right triangles obey the Pythagorean theorem and appear in navigation, construction, and physics. Equilateral triangles (all sides equal, all angles 60°) are inherently rigid — used in trusses and bridges. Isosceles triangles appear in roof designs and isometric projection.
Triangle area in surveying and land measurement
Land parcels are often subdivided into triangles for area calculation. GPS coordinates can be used as vertices; the shoelace formula (a generalization of Heron's approach) computes area from coordinate pairs. Heron's formula is the go-to for field surveyors working with physical measurements.
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