Triangle Calculator
This free triangle calculator computes the edges, angles, area, height, perimeter, and median of the resulting triangle.
How to Use the Triangle Calculator
Provide any 3 values from the following 6: side lengths (a, b, c) and angles (A, B, C). At least one side must be provided. The calculator solves the triangle using the Law of Sines, Law of Cosines, and Pythagorean theorem (for right triangles). Results include all missing sides, angles, area, perimeter, height, median, inradius, and circumradius.
Triangle Input Combinations
Different known-value combinations (SSS, SAS, ASA, AAS, SSA) require different solving methods and produce different certainty levels.
Area Methods Overview
Area can be derived from side-only data via Heron formula or from two sides with included angle via trigonometric area formula.
Angle-Side Consistency
Larger angles are opposite larger sides, which provides a quick reasonableness check for solved triangles.
Geometric Radius Metrics
Inradius and circumradius summarize triangle geometry and connect side/area values to circle-based constructions.
Practical Validation Steps
Always validate sum of angles equals 180 degrees and side lengths satisfy triangle inequality before using results in downstream work.
Frequently Asked Questions
What is the Law of Sines and when is it used?+
The Law of Sines states that the ratio of a side to the sine of its opposite angle is constant for all three sides. Use it when you know two angles and one side (AAS/ASA) or two sides and a non-opposite angle (SSA, ambiguous case).
What is the Law of Cosines?+
The Law of Cosines (c² = a² + b² − 2ab×cos(C)) relates sides and angles. Use it for SAS (two sides + included angle) or SSS (all three sides) cases.
What is Heron's Formula?+
Heron's Formula calculates the area of any triangle given only its three side lengths: Area = √[s(s−a)(s−b)(s−c)], where s is the semi-perimeter (a+b+c)/2.
What are inradius and circumradius?+
Inradius is the radius of the largest circle that fits inside the triangle. Circumradius is the radius of the circle that passes through all three vertices. Both are useful in geometry and trigonometry.
What is the difference between acute, right, and obtuse triangles?+
Acute: all angles < 90°. Right: one angle = 90° (uses Pythagorean theorem). Obtuse: one angle > 90°. Sum of angles in any triangle always = 180°.
What is the SSA ambiguous case?+
With two sides and a non-included angle, there may be zero, one, or two valid triangles depending on side-angle relationships.
Can I solve a triangle with only angles?+
No. At least one side length is required to establish scale and compute actual side values.
How do I verify triangle validity?+
For side inputs, check triangle inequality: each side must be less than the sum of the other two sides.
What is semi-perimeter used for?+
Semi-perimeter s = (a+b+c)/2 is used directly in Heron formula for area computation.
When should I use right-triangle shortcuts?+
If one angle is 90 degrees, right-triangle identities and Pythagorean relationships simplify computations.
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