The Pythagorean theorem is the one equation almost everyone remembers from school: in a right triangle, a² + b² = c², where c is the hypotenuse (the side opposite the right angle). Once you know any two sides, the third drops out by algebra. The Toolenza calculator does the algebra in both directions — given two sides it returns the third, or given all three it confirms whether they actually form a right triangle.

When you actually use this

• Construction. Squaring up a floor or wall? The 3-4-5 trick — measure 3 ft along one edge, 4 ft along the perpendicular, and the diagonal should be exactly 5 ft if your corner is square. Builders use 6-8-10 or 12-16-20 for larger checks. All of these are Pythagorean triples — integer side lengths where a² + b² = c² works out exactly.
• Screen and TV diagonals. TV manufacturers quote the diagonal, but you usually need to know height × width to fit a slot. A 55-inch 16:9 TV is 47.9 in × 27.0 in — that's the theorem solved for the legs given the hypotenuse and the aspect ratio.
• Navigation and surveying. Straight-line distance between two GPS points (on a small enough scale that the Earth's curvature doesn't matter) is just a Pythagorean calculation on the lat/lng deltas.
• Drone and camera framing. The hypotenuse of the field-of-view triangle from drone altitude tells you how wide the captured image is on the ground.

Pitfalls

The theorem only works on right triangles. For any other triangle (acute, obtuse, or scalene without a 90° corner) you need the Law of Cosines: c² = a² + b² − 2ab·cos(C). A frequent mistake is plugging in three random side lengths and expecting a right-triangle result; if a² + b² ≠ c², the triangle isn't right-angled.