Pythagorean theorem:
finding the hypotenuse and the missing leg

What the theorem says and when it applies

In a triangle with one right angle, the side opposite that angle is called the hypotenuse. It is always the longest side. The theorem says the square built on it has the same area as the two squares built on the other sides, the legs, put together.

The statement is only true for right triangles. If you are unsure whether the corner is a right angle, the converse helps: when the three sides satisfy a² + b² = c², the triangle must be right-angled. Builders use this as the 3-4-5 check for square corners.

Finding the hypotenuse

Square both legs, add, then take the square root. Taking the root is the step people forget; the sum itself is c², not c.

That one was a whole number. Most are not. With legs of 5 and 7 the sum is 25 + 49 = 74, and √74 = 8.602 to three decimals, so you round to the precision the job needs.

Keep the unrounded square root in your calculator until the last step. If you round √74 to 8.6 and then use it in a later calculation, the error compounds. Round only the final figure, and to the precision your tape measure can really support; quoting 8.602325 m on a building site implies a precision nobody can cut to.

Finding a missing leg

When the hypotenuse is known, the legs are what is left. Subtract the known leg's square from the hypotenuse square, then take the root.

A sanity check always works: the answer must be shorter than the hypotenuse. If you get something longer, you added when you should have subtracted.

Everyday uses: ladders, screens and diagonals

Ladders are the classic case. A 5 m ladder with its foot 1.5 m from the wall forms a right triangle with the wall and the ground, so it reaches √(25 − 2.25) = √22.75 = 4.77 m up.

• A rectangular room measuring 4 m by 3 m has a diagonal of 5 m, useful for checking that a frame is square.
• A TV advertised as 55 inches is a diagonal; with a 16:9 shape the width and height follow from the same theorem.
• A roof rise and run give the length of the rafter before any cutting allowance.

Integer triples worth remembering

Every multiple of a triple is also a triple, so 9-12-15 is just the 3-4-5 triangle tripled. Spotting one lets you skip the arithmetic entirely.

The converse can also sort triangles by their corner. For sides a ≤ b ≤ c, if a² + b² equals c² the triangle is right-angled, if it is greater the triangle is acute, and if it is less the triangle is obtuse. Sides of 4, 5 and 7 give 16 + 25 = 41, which is below 49, so the largest angle is obtuse.

Diagonals, screen sizes and checking a square corner

A rectangle is two right triangles sharing the diagonal, so the theorem gives its diagonal directly: diagonal = √(length² + width²). A door frame 0.9 m wide and 2.1 m tall has a diagonal of √(0.81 + 4.41) = √5.22 = 2.28 m. If you measure both diagonals of a door frame and they differ, the frame is out of square.

Screens are sold by diagonal. A display with a 16:9 shape and a 55-inch diagonal has a width of 55 × 16 ÷ √(16² + 9²) = 55 × 16 ÷ 18.36 = 47.9 inches and a height of 55 × 9 ÷ 18.36 = 27.0 inches. The sum of the squares of those, 2,294 + 729, is 3,023, close to 55² = 3,025 and the difference is just rounding.

Builders scale the check up for accuracy: 6 m and 8 m on the walls with a 10 m diagonal. The longer the sides, the more a small misalignment shows.

Going to three dimensions

The same idea applies twice in a box. First find the diagonal of the floor, then use that as one leg of a vertical triangle with the height as the other. The result is the space diagonal: √(l² + w² + h²).

A box that is 40 cm by 30 cm by 120 cm has a floor diagonal of √(1,600 + 900) = 50 cm and a space diagonal of √(2,500 + 14,400) = √16,900 = 130 cm. A 125 cm rod would therefore fit diagonally, while a 135 cm one would not. Shipping and packing sometimes depend on exactly this.

Slips to avoid

• Using it on a triangle that has no right angle.
• Adding the legs and forgetting the square root.
• Subtracting in the wrong order when finding a leg.
• Treating a leg as the hypotenuse; the hypotenuse is always the longest side.
• Mixing units between the two legs.

A calculator that returns the hypotenuse or the missing leg is a quick way to confirm a hand calculation, and it handles the decimals when the result is not a whole number.