Pythagorean Theorem — 3D

A right-angle tetrahedron: three mutually perpendicular edges p, q, r from one corner, exactly like the legs of the 2D original — extended one dimension further.

Edge p (OA)

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Edge q (OB)

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Edge r (OC)

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Box diagonal d

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Edge p3.0

Edge q4.0

Edge r5.0

Auto-rotate: On

Box + diagonal: On

De Gua's theorem — the 3D Pythagorean theorem: the square of the slanted face's area equals the sum of the squares of the three right-angle faces.

S₁²+S₂²+S₃² = —

S(ABC)² = —

Face OAB — S₁ = ½pq

Face OBC — S₂ = ½qr

Face OCA — S₃ = ½rp

Face ABC — the "hypotenuse" face

De Gua's Theorem — The Real 3D Pythagorean Theorem

The 2D original computes a right triangle's hypotenuse from c = √(a²+b²). This companion builds the actual solid that generalizes it: a right-angle tetrahedron with three mutually perpendicular edges p, q, r meeting at corner O, exactly like the two catheti of the flat triangle plus one more perpendicular dimension. The three faces touching O (OAB, OBC, OCA) are ordinary right triangles with areas S₁=½pq, S₂=½qr, S₃=½rp — direct 3D analogues of the a² and b² squares drawn in the 2D visualization.

The fourth, slanted face ABC plays the role of the hypotenuse — not as a length this time, but as an area. De Gua's theorem states S(ABC)² = S₁² + S₂² + S₃², the true 3D generalization of a²+b²=c². The readout panel computes both sides live from the cross-product area of triangle ABC and confirms they match to floating-point precision for any p, q, r. The panel also tracks the box's main space diagonal d = √(p²+q²+r²) — the 3D extension of the distance formula referenced in the 2D original's FAQ, applying the same theorem one more time along the third axis. Drag to orbit the camera; scroll to zoom.