Euclidean Interior Angle Sum TheoremGeometric Invariant Constant [A + B + C = 180° / π rad]
In flat Euclidean geometry, the sum of all three interior angles in any non-collapsed triangle strictly equals exactly 180 degrees (π radians or 200 gradians). If two angles are known, the third angle is uniquely established as 180° - (A + B). In practical carpentry, roof framing, and athletic field layout, this invariant serves as the primary verification check against construction tilt or surveying drift.
Default: 180° (π radians)
Law of Cosines (Generalized Pythagorean Theorem)Trigonometric Invariant [c² = a² + b² - 2ab cos(C)]
Extends the Pythagorean theorem to all triangles by relating any side length to the cosine of its opposite angle: c² = a² + b² - 2ab × cos(C). It solves Side-Angle-Side (SAS) layouts to discover missing boundary lengths, and Side-Side-Side (SSS) layouts via cos(C) = (a² + b² - c²) / (2ab). Because arccosine uniquely maps between 0° and 180°, it resolves obtuse and acute angles without quadrant confusion.
Default: c² = a² + b² - 2ab cos(C)
Law of Sines (Sine Ratio Invariant)Proportional Ratio Invariant [a/sin(A) = b/sin(B) = c/sin(C) = 2R]
Establishes that the ratio between any side length and the sine of its opposite interior angle remains constant across all three vertices, directly equaling twice the circumradius (2R). This proportional rule is the primary mathematical engine for solving Angle-Side-Angle (ASA) and Angle-Angle-Side (AAS) configurations in outdoor navigation, land parceling, and astronomy.
Default: a / sin(A) = 2R
SSA Ambiguous Case Discriminant CriteriaGeometric Bifurcation Parameter
Evaluates the geometric feasibility and number of possible triangles when given two sides and a non-included acute angle (a, b, and Angle A). Comparing side a to the vertical altitude h = b × sin(A) defines the topology: if a < h, 0 triangles exist; if a = h, exactly 1 right-angled triangle forms; if h < a < b, exactly 2 valid triangles exist (one acute, one obtuse); and if a ≥ b, exactly 1 triangle forms.
Default: h = b × sin(A) altitude baseline
Heron's Formula & Semi-Perimeter Area ModelArea Invariant [Area = √(s(s - a)(s - b)(s - c))]
Calculates the exact enclosed area of any triangle directly from its three boundary edges without requiring height or angle measurements, where s = (a + b + c) / 2 is the semi-perimeter. In precision calculation, sorting side lengths so a ≥ b ≥ c prevents severe rounding cancellation when measuring extremely elongated or needle-thin plots such as yard wedges and roof gables.
Default: Area = √(s(s - a)(s - b)(s - c))
Inradius and Circumradius Fundamental RadiiConcentric Geometric Radii [r = Area / s, R = abc / (4 × Area)]
Quantifies the inner and outer circular boundaries of a triangle. The inradius (r = Area / s) represents the largest circle that can fit inside the triangle touching all three walls, essential for structural packing and mechanical clearance. The circumradius (R = abc / (4 × Area)) defines the circle passing through all three vertices, satisfying Euler's distance relation d² = R(R - 2r).
Default: r = Area / s, R = abc / (4 × Area)
