Overview
The Pythagorean theorem is one of the oldest and most widely used results in mathematics: in any right triangle, the square of the hypotenuse equals the sum of the squares of the other two sides. Written as a² + b² = c², it lets you find a missing side of a right triangle whenever the other two are known, with no protractor or trigonometry required.
This guide walks through identifying a right triangle correctly, applying the formula in both directions (solving for the hypotenuse and solving for a leg), and using the theorem for practical problems like construction layout, screen diagonals, and distance measurement. Once you understand the steps, the Pythagorean Theorem Calculator can check your work in seconds.
What You Need
Before applying the theorem, gather:
- Confirmation that the triangle has a right angle: exactly one angle measuring 90 degrees
- The lengths of any two sides: either both legs, or one leg and the hypotenuse
- A consistent unit of measurement: convert all sides to the same unit (metres, feet, inches) before calculating
- A calculator or square root table for any answer that isn't a perfect square
Step 1: Identify a Right Triangle
The Pythagorean theorem only applies to right triangles, meaning triangles with exactly one 90-degree angle. Before doing anything else, confirm the triangle qualifies. In diagrams, a right angle is usually marked with a small square symbol at the corner.
Once you've confirmed the right angle, identify the three sides:
- The hypotenuse sits directly opposite the right angle. It's always the longest side of the triangle.
- The two legs are the sides that meet at the 90-degree corner and form the right angle itself.
Getting this identification wrong is the single most common mistake when using the theorem, because the formula treats the hypotenuse differently from the two legs. Mislabel a leg as the hypotenuse, and every calculation after that point will be wrong.
Step 2: Understand the Formula
The Pythagorean theorem is written as:
a² + b² = c²
Where a and b are the lengths of the two legs, and c is the length of the hypotenuse. In words: square each leg, add the results together, and you get the square of the hypotenuse.
The formula can be rearranged depending on which side is unknown:
c = √(a² + b²) → solving for the hypotenuse
a = √(c² − b²) → solving for a missing leg
b = √(c² − a²) → solving for the other missing leg
Only one of these three forms applies to any given problem, depending on which two sides you already know.
Step 3: Solve for the Hypotenuse (When Both Legs Are Known)
This is the most common use case. If you know both legs, square each one, add the squares together, then take the square root of the total.
Worked example: A right triangle has legs of 3 and 4 units.
c² = a² + b² = 3² + 4² = 9 + 16 = 25
c = √25 = 5
The hypotenuse is 5 units. This particular combination, 3-4-5, is famous as the simplest Pythagorean triple, a set of whole numbers that satisfies the theorem exactly with no rounding. It shows up constantly in textbooks and real-world layout work because the numbers are clean.
If your legs don't form a perfect-square combination, your final answer will be an irrational decimal. Legs of 5 and 7, for example, give c = √(25+49) = √74 ≈ 8.60. That's still a correct and usable answer, it just won't be a whole number.
Step 4: Solve for a Missing Leg (When Hypotenuse and One Leg Are Known)
If you already know the hypotenuse and one leg, rearrange the formula to isolate the missing leg. Subtract the known leg's square from the hypotenuse's square, then take the square root.
a = √(c² − b²)
Worked example: A right triangle has a hypotenuse of 13 units and one leg of 5 units.
a² = c² − b² = 13² − 5² = 169 − 25 = 144
a = √144 = 12
The missing leg is 12 units, giving the triple (5, 12, 13), another common whole-number combination. Notice that this step subtracts rather than adds, the opposite of Step 3. Mixing up addition and subtraction here is a frequent source of errors, since both steps use squares and square roots but combine them differently depending on which side is missing.
Step 5: Apply It to Real-World Problems
The theorem shows up well beyond the classroom, wherever two perpendicular measurements meet a third diagonal one.
Screen and TV sizes. A television's advertised size is always its diagonal measurement, which is the hypotenuse of the right triangle formed by its height and width. A screen that's 24 inches wide and 13.5 inches tall has a diagonal of √(24² + 13.5²) = √(576 + 182.25) ≈ √758.25 ≈ 27.5 inches, close to a standard "27-inch" monitor once you account for rounding and bezel.
Diagonal of a room or rectangle. A room that measures 12 feet by 16 feet has a corner-to-corner diagonal of √(12² + 16²) = √(144+256) = √400 = 20 feet. That's useful for fitting furniture, running cables, or checking whether a rectangular space is actually square.
Construction, the 3-4-5 method. Builders verify that a corner is exactly 90 degrees without a protractor by measuring 3 units along one wall and 4 units along the adjoining wall, then checking that the diagonal between those two marked points is exactly 5 units (or any consistent multiple, like 6-8-10 for larger layouts). If the diagonal is off, the corner isn't square, and the framing needs adjustment before work continues.
Navigation and shortest-distance problems. Travel 8 km east and then 6 km north, and your straight-line distance from the starting point is √(8² + 6²) = √(64+36) = √100 = 10 km, shorter than the 14 km you actually walked, because the direct path cuts across the right angle formed by the two legs of the journey.
Common Mistakes to Avoid
Confusing which side is the hypotenuse. The hypotenuse is always opposite the right angle and always the longest side, never one of the two sides forming the 90-degree corner. If your "answer" for the hypotenuse comes out shorter than one of the legs you started with, you've set up the equation incorrectly. Go back and recheck which side is actually opposite the right angle.
Forgetting to take the square root at the end. The formula a² + b² = c² produces c², not c. A calculation that stops at "c² = 25" isn't finished; the actual side length is √25 = 5. Leaving an answer in squared form is a frequent error on tests and in practical work, and it will be off by a wide margin from the true measurement since squared units grow much faster than linear ones.
Applying the theorem to a triangle that isn't a right triangle. This theorem only works for right triangles. For any other triangle, acute or obtuse, with no 90-degree angle, you need the Law of Cosines (c² = a² + b² − 2ab·cos(C)) instead. Using a² + b² = c² on a non-right triangle silently produces a wrong number with no error or warning, so confirm the right angle first.
Mixing units before calculating. Entering one leg in centimetres and the other in metres without converting will corrupt the result, since the squaring operation amplifies the inconsistency. Always convert every measurement to the same unit before squaring and adding.
Formula & Methodology
The core relationship, a² + b² = c², can be proven several ways, and the proof makes the formula much easier to remember and trust.
Rearrangement (geometric) proof. Take four identical copies of the same right triangle and arrange them inside a square whose side length is (a + b). Arranged one way, the four triangles surround a smaller square in the middle with side length c, so the total area is 4 × (½ab) + c². Arranged another way, the same four triangles can be repositioned to leave two smaller squares, one with side a, one with side b, uncovered, giving total area 4 × (½ab) + a² + b². Both arrangements fill the exact same (a+b)² square, so the leftover areas must be equal: c² = a² + b².
Algebraic check. Expanding (a+b)² = a² + 2ab + b² and comparing it against the area arrangement above is the standard way textbooks formalize the rearrangement proof without relying purely on a diagram.
The converse, used for verification. The converse of the Pythagorean theorem states that if a triangle's three sides satisfy a² + b² = c² (where c is the longest side), the triangle contains a 90-degree angle, even if no one ever measured an angle directly. This is the basis of the 3-4-5 construction method. Rather than assuming a corner is square and calculating the diagonal, builders measure all three sides and use the converse to confirm the angle is exactly 90 degrees. If the measured diagonal doesn't match the calculated value, the angle isn't square, and the structure needs correcting before work continues.
Together, the direct theorem and its converse make the Pythagorean relationship a two-way tool. Use a² + b² = c² to predict an unknown side from a known right angle, and run the same equation in reverse to prove a right angle exists from three measured sides.