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Triangle & Trigonometry Calculator

開発者

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入力

自動処理

出力

クライアント側

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ガイド

Triangle & Trigonometry Calculator

Solve any triangle from three known values — enter sides and angles in any valid combination (SSS, SAS, ASA, AAS, SSA) and instantly get all missing measurements, area, perimeter, heights, inradius, circumradius, and triangle classification. Includes step-by-step solutions, an SVG diagram, and a quick reference for special right triangles.

使い方

Enter at least three known values (sides and/or angles), with at least one side. The calculator auto-detects the configuration (SSS, SAS, ASA, AAS, or SSA) and solves for all unknowns using the Law of Cosines, Law of Sines, and Pythagorean theorem. Toggle between degrees and radians for angle units. Scroll down to see the step-by-step solution showing every formula and intermediate calculation, plus an SVG diagram of your triangle.

特徴

Triangle Solver — Enter any valid combination of 3+ sides and angles. Solves for all 3 sides, all 3 angles, area, perimeter, heights, inradius, and circumradius.
Auto Configuration Detection — Automatically identifies SSS, SAS, ASA, AAS, or SSA and applies the correct solving method. Handles the ambiguous SSA case with two possible solutions.
Step-by-Step Solution — See exactly which formulas were used and how each value was calculated. Great for learning or verifying homework.
SVG Triangle Diagram — Visual representation of the solved triangle with labeled sides and angles, approximately proportional to actual measurements.
Triangle Classification — By sides (equilateral, isosceles, scalene) and by angles (acute, right, obtuse).
Special Right Triangles — Quick reference for common triangles (30-60-90, 45-45-90, 3-4-5, 5-12-13, 8-15-17) with clickable presets to populate the solver.
Degrees/Radians Toggle — Switch between degree and radian angle units.
入力検証 — Triangle inequality check, angle sum validation, and clear error messages for impossible triangles.

Formulas Used

Law of Cosines: c² = a² + b² − 2ab·cos(C)
Law of Sines: a/sin(A) = b/sin(B) = c/sin(C)
Pythagorean Theorem: a² + b² = c² (right triangles)
Area (SAS): ½·a·b·sin(C)
Area (Heron’s): √(s(s−a)(s−b)(s−c)) where s = (a+b+c)/2
Height: hₐ = 2·Area/a
Inradius: r = Area/s
Circumradius: R = a/(2·sin(A))

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What triangle configurations can this calculator solve?

This calculator solves any valid triangle from three or more known values, as long as at least one side is provided. Supported configurations: SSS (three sides), SAS (two sides and the included angle), ASA (two angles and the included side), AAS (two angles and a non-included side), and SSA (two sides and a non-included angle). The SSA case is special because it can produce zero, one, or two valid triangles — the calculator handles this ambiguous case and shows all possible solutions.

What is the ambiguous case (SSA)?

When you know two sides and an angle opposite one of them (SSA), there may be zero, one, or two valid triangles. This happens because the given information doesn’t uniquely determine the triangle. For example, if side a = 8, side b = 6, and angle A = 30°, there are two triangles that satisfy these conditions. The calculator detects this ambiguity and shows both solutions when they exist, or explains why no valid triangle is possible.

What are special right triangles?

Special right triangles have angle or side ratios that produce clean, memorable numbers. The 30-60-90 triangle has sides in ratio 1:√3:2 and the 45-45-90 triangle has sides in ratio 1:1:√2. Pythagorean triples like 3-4-5, 5-12-13, and 8-15-17 are right triangles where all three sides are whole numbers. These are commonly used in geometry, construction, and standardized tests because they simplify calculations.

How is triangle area calculated?

Triangle area can be calculated several ways depending on known values. With two sides and the included angle (SAS): Area = ½·a·b·sin(C). With all three sides (SSS), Heron’s formula: Area = √(s(s−a)(s−b)(s−c)) where s is the semi-perimeter (a+b+c)/2. With base and height: Area = ½·base·height. This calculator uses whichever formula fits the given input and shows the calculation in the step-by-step solution.

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