Find the missing side of a right triangle using the Pythagorean theorem.
The Pythagorean theorem states that in a right triangle, a squared plus b squared equals c squared, where c is the hypotenuse (the longest side) and a, b are the two legs.
Given any two sides of a right triangle, this calculator instantly finds the missing side - whether that means solving for the hypotenuse or for one of the legs.The theorem (a² + b² = c², relating the two legs of a right triangle to its hypotenuse) is one of the most practically useful pieces of geometry taught in school, showing up constantly in construction and carpentry - checking that a wall or foundation corner is truly square using the 3-4-5 rule, calculating the length of a diagonal brace needed for a rectangular frame, or figuring out the exact length of a staircase stringer given its rise and run. TV and monitor sizes are also measured diagonally, so the theorem is what actually connects a screen's stated size to its width and height dimensions.
Navigation and surveying historically relied on this same relationship to calculate distances that couldn't be measured directly, by breaking a path into right-triangle components using known reference points - a technique still used today in GPS systems and land surveying.
It's worth noting this relationship specifically requires a 90-degree angle between the two legs - for triangles without a right angle, the more general Law of Cosines is needed instead, since the simple a² + b² = c² relationship doesn't hold for non-right triangles.
The Pythagorean theorem — a² + b² = c², where c is the hypotenuse of a right triangle — is one of the most widely applied formulas outside of pure mathematics classrooms, showing up constantly in construction, navigation, and design wherever a diagonal distance needs to be calculated from two known perpendicular distances. Builders use it to verify that a corner is truly square (a 3-4-5 triangle, where sides of 3 and 4 units produce a hypotenuse of exactly 5 units, is a classic on-site check for a right angle), and it underlies distance calculations in everything from TV screen sizing to GPS navigation.
The theorem only applies to right triangles specifically — triangles containing one 90-degree angle — and using it on a non-right triangle produces an incorrect result, which is a common error for anyone applying the formula without first confirming the triangle actually has a right angle.
Television and monitor screen sizes are advertised by diagonal measurement, calculated using the Pythagorean theorem from the screen's width and height. Construction workers use it to calculate rafter lengths for roofs, ensuring the pitch and dimensions work together correctly. Navigation systems use a three-dimensional extension of the same principle to calculate straight-line distances that account for elevation changes, not just horizontal position.
Certain sets of three whole numbers satisfy the Pythagorean theorem exactly, without any rounding — these are called Pythagorean triples, with 3-4-5 being the most well-known example. Other common triples include 5-12-13 and 8-15-17, and recognizing these patterns can speed up manual calculations or serve as a quick sanity check when a calculated hypotenuse happens to land on a familiar whole number.
Yes, the Pythagorean theorem applies only to right triangles (triangles with a 90 degree angle).
No, the Pythagorean theorem only applies to right triangles - for other triangle types, the Law of Cosines is needed instead.
Checking that a rectangular frame or foundation corner is truly square using the 3-4-5 rule is one of the most common real-world applications, especially in construction.
Yes, given any two of the three sides (two legs, or one leg and the hypotenuse), the third can always be calculated using the theorem.
