How to Find the Missing Side of a Right Triangle with the Pythagorean Theorem

A right triangle has one 90° corner, two legs that meet at that corner, and a hypotenuse — the long side facing the right angle. The reason the Pythagorean theorem matters is that knowing any two of those three sides lets you compute the third, with no protractor and no guessing. That single fact shows up far more often than the geometry textbook lets on: it is how a carpenter checks that a deck corner is square, how you work out a TV's real width from the number on the box, and how a map app measures the straight-line gap between two points.

This guide walks through the two questions people actually ask — find the hypotenuse and find a missing leg — then shows where the same equation hides inside distances and real measurements. You can follow along in the Pythagorean theorem calculator, which solves all three cases and draws the triangle to scale.

The one equation: a² + b² = c²

Everything below is one relationship:

a² + b² = c²

Here a and b are the two legs, and c is the hypotenuse. Square each leg, add them, and the sum equals the square of the hypotenuse. It only works when a and b meet at exactly 90° — if the corner is anything else, you need the law of cosines instead, which is a separate tool.

The equation is symmetric in the legs, so it does not matter which leg you call a and which you call b. What does matter is that c is always the longest side. Mix that up and every answer comes out wrong.

Finding the hypotenuse: the 3-4-5 example

When you know both legs, solving for the hypotenuse is direct:

c = √(a² + b²)

Take the classic example. A right triangle has legs of 3 and 4. Square them: 3² = 9 and 4² = 16. Add: 9 + 16 = 25. The hypotenuse is the square root of 25, which is exactly 5. Those three whole numbers — 3, 4, 5 — form a Pythagorean triple, meaning all three sides are integers. Other clean triples include 5-12-13, 8-15-17, and 7-24-25, plus every scaled-up multiple like 6-8-10.

Here is one I lean on constantly off the page. When I am setting out the corner of a deck and have no framing square big enough to trust, I use the 3-4-5 method to square the corner by hand. I measure 3 feet along one edge, 4 feet along the other, then adjust the boards until the diagonal between those two marks reads exactly 5 feet. The moment the diagonal hits 5, the corner is a true 90° — because the only triangle with legs 3 and 4 and a hypotenuse of 5 is a right one. For a bigger, more accurate baseline I scale the whole thing up to 6-8-10 or 9-12-15. It is the cheapest carpenter's square ever invented, and it is just a² + b² = c² standing up in the real world.

Finding a missing leg: rearrange the equation

The reverse question is just as common: you know the hypotenuse and one leg, and you want the other leg. Rearranging a² + b² = c² gives:

b = √(c² − a²)

Worked example: suppose the hypotenuse is 13 and one leg is 5. Square them: 13² = 169 and 5² = 25. Subtract: 169 − 25 = 144. The missing leg is √144 = 12 — and 5-12-13 is another exact triple.

There is one rule the math enforces, and it trips people up. The hypotenuse is always the longest side, so c must be greater than the leg you enter. If you try to subtract a leg that is equal to or longer than the hypotenuse, you get a negative number under the square root — an impossible triangle. A good calculator refuses that case and tells you why, rather than printing an imaginary number you would have to puzzle over.

The fastest way to keep the two cases straight: are you adding the squares or subtracting them? Two legs known → add → take the root for the hypotenuse. Hypotenuse plus one leg known → subtract → take the root for the missing leg.

Real uses: decks, screens, and braces

These two cases cover an enormous amount of everyday measuring.

Squaring a corner. Covered above — the 3-4-5 layout, scaled to whatever baseline gives you the accuracy you need. The longer the legs, the smaller the error in the angle.

A screen's true width. A "27-inch" monitor is measured corner to corner; that number is the hypotenuse, not the width. A standard 16:9 panel has its width and height in a 16:9 ratio, so you can treat the diagonal as c and back out the legs. This is the quickest way to check whether a screen actually fits the slot in your desk before you buy it.

A diagonal brace. Building a rectangular gate 1.2 m wide and 0.9 m tall? The diagonal brace that stops it sagging runs corner to corner, so its length is the hypotenuse of a 1.2 × 0.9 right triangle: √(1.2² + 0.9²) = √(1.44 + 0.81) = √2.25 = 1.5 m. Cut the timber to 1.5 m and the brace drops in exactly.

In every one of these, the triangle is hiding in plain sight; the theorem just turns two measurements you have into the one you need.

Distance is the Pythagorean theorem in disguise

The distance between two points on a coordinate grid looks like a different formula, but it is the same equation wearing coordinates:

d = √((x₂ − x₁)² + (y₂ − y₁)²)

Draw the horizontal gap Δx = x₂ − x₁ and the vertical gap Δy = y₂ − y₁ between your two points. They meet at a right angle and form a right triangle whose hypotenuse is the straight-line distance. So d = √(Δx² + Δy²) is literally a² + b² = c² with a = Δx and b = Δy.

Concrete case: a trailhead at (2, 3) and a summit at (11, 15), in kilometers. The legs are Δx = 11 − 2 = 9 and Δy = 15 − 3 = 12, and the distance is √(9² + 12²) = √(81 + 144) = √225 = 15 km as the crow flies — the 9-12-15 triangle again, the same 3-4-5 scaled by three. Add a Z axis for altitude and it extends straight to 3D: d = √(Δx² + Δy² + Δz²), the space diagonal.

If your geometry problem also involves slopes or angles rather than pure lengths, two related tools pick up where this one leaves off: the slope calculator handles the rise-over-run side of coordinate geometry, and the scientific calculator covers the square roots and trig functions when the numbers stop being clean.

A quick checklist before you trust an answer

• Confirm the corner is actually 90°. No right angle, no a² + b² = c².
• Put the longest side in the hypotenuse slot, never in a leg slot.
• Adding the squares finds the hypotenuse; subtracting them finds a leg.
• In the distance formula, square each gap before adding — Δx + Δy is not the distance.

Get those four right and you will solve right triangles faster than you can find the protractor.

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Made by Toolora · Updated 2026-06-13