Pythagorean Theorem Calculator — hypotenuse, missing leg & point-to-point distance

Real-world use cases

• Square a foundation with a 3-4-5 layout

  You are setting out the corner of a deck and need it dead square without a framing square big enough to trust. Measure 3 ft along one edge and 4 ft along the other; the diagonal between those marks must be exactly 5 ft for a true 90° corner. Punch 3 and 4 into the leg mode and it confirms the hypotenuse is 5 and flags it as an exact Pythagorean triple. Want a longer baseline for more accuracy? Use 6-8-10 or 9-12-15 — the same triple scaled up, and the tool will confirm each one.

• Find a TV or monitor's true width from its diagonal

  A screen advertised as 27 inches is measured corner to corner — that is the hypotenuse. A common 16:9 panel means width and height are in a 16:9 ratio, so you can treat the diagonal as c and solve the legs. Use the leg mode with the 16- and 9-unit sides to see the diagonal-to-width relationship, or the find-a-leg mode if you already know one dimension and want the other. It is the fastest way to check whether a screen actually fits the slot in your desk before you buy.

• Measure straight-line distance on a map grid

  You have two points on a coordinate map — a trailhead at (2, 3) and a summit at (11, 15) in kilometers. Switch to distance mode, enter both points, and read d = 15 km as the crow flies. The tool also shows the legs Δx = 9 and Δy = 12, so you can see the 9-12-15 right triangle hiding in the calculation. For a drone flight that also climbs, flip on the Z axis and add the altitude change for the true 3D path length.

• Check a geometry homework answer

  The worksheet says "a right triangle has legs 6 and 8 — find the hypotenuse and both acute angles." Enter 6 and 8 in the leg mode: the hypotenuse is 10, the area is 24, and the acute angles are about 36.87° and 53.13°, which sum to 90°. Because the diagram is drawn to scale, a student can also see that the longer leg sits opposite the larger angle — the kind of intuition a flat list of numbers never builds.

• Size a diagonal brace for a gate or frame

  You are building a rectangular gate 1.2 m wide and 0.9 m tall and want a diagonal brace to stop it sagging. The brace spans corner to corner, so its length is the hypotenuse of a 1.2 by 0.9 right triangle. Enter those in the leg mode and read 1.5 m — cut the timber to that and the brace fits exactly. The tool's area and angle outputs also tell you the brace meets each corner at about 36.87° and 53.13°, useful when you mitre the ends.

Common pitfalls

• Plugging the hypotenuse in as a leg. The hypotenuse is always the longest side and sits opposite the right angle. If you put it where a leg goes, a²+b²=c² gives the wrong answer — use the find-a-leg mode and enter it in the hypotenuse field instead.

• Using a²+b²=c² on a triangle that is not right-angled. The theorem only holds when two sides meet at exactly 90°. For other triangles use the law of cosines; this tool's leg/hypotenuse modes assume a right angle.

• Forgetting to square the differences in the distance formula. The distance is √((x₂−x₁)²+(y₂−y₁)²), not (x₂−x₁)+(y₂−y₁). Each gap is a leg that must be squared first; the distance mode does this for you and shows the legs so the step is visible.

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