Hypotenuse Calculator
Free Hypotenuse Calculator for triangle. Enter values to get step-by-step solutions with formulas and graphs. Free to use with no signup required.
Calculator
Adjust values & calculate9.0000 + 16.0000 = 25.0000
Formula
Where c is the hypotenuse (the side opposite the right angle), and a and b are the two legs of the right triangle. The theorem states that the square of the hypotenuse equals the sum of the squares of the other two sides.
Last reviewed: December 2025
Worked Examples
Example 1: Classic 3-4-5 Right Triangle
Example 2: Building Foundation Diagonal
Background & Theory
The Hypotenuse Calculator applies the following established principles and formulas. Mathematics rests on a hierarchy of number systems, each extending the previous. The natural numbers (1, 2, 3, ...) support counting and ordering. The integers add negative values and zero, enabling subtraction without restriction. The rational numbers, expressible as p/q where p and q are integers and q is nonzero, close the system under division. The real numbers fill the gaps left by irrationals such as the square root of 2 or pi, forming a complete ordered field. The complex numbers, written as a + bi where i is the square root of negative one, complete the algebraic closure of the reals and allow every polynomial to have a root. Prime factorization states that every integer greater than one is uniquely expressible as a product of primes, a result known as the Fundamental Theorem of Arithmetic. Computing the greatest common divisor (GCD) of two integers relies most efficiently on the Euclidean algorithm: repeatedly replace the larger number with the remainder when it is divided by the smaller, until the remainder is zero. The last nonzero remainder is the GCD. The least common multiple (LCM) follows from the identity LCM(a, b) = |a * b| / GCD(a, b). Modular arithmetic defines equivalence classes of integers that share the same remainder under division by a modulus n. Fermat's Little Theorem and Euler's Theorem arise from this structure and underpin modern cryptography. Logarithms are the inverses of exponential functions. If b raised to the power x equals y, then the logarithm base b of y equals x. The natural logarithm uses base e, approximately 2.71828. Combinatorics counts arrangements and selections. The number of ordered arrangements (permutations) of r objects from n distinct objects is nPr = n! / (n - r)!. The number of unordered selections (combinations) is nCr = n! / (r! * (n - r)!). Pascal's triangle arranges these binomial coefficients so that each entry equals the sum of the two entries directly above it. The Fibonacci sequence, defined by F(1) = 1, F(2) = 1, and F(n) = F(n-1) + F(n-2), appears throughout nature and connects deeply to the golden ratio via Binet's formula.
History
The history behind the Hypotenuse Calculator traces back through the following developments. Mathematics as a systematic discipline traces to ancient Mesopotamia. Babylonian clay tablets dating to around 1800 BCE demonstrate knowledge of quadratic equations, Pythagorean triples, and base-60 arithmetic, suggesting a practical mathematical tradition far preceding Greek formalism. Euclid of Alexandria compiled the Elements around 300 BCE, establishing the axiomatic method that would define rigorous mathematics for over two thousand years. His work organized plane geometry, number theory, and proportion into logically chained propositions derived from a small set of postulates. The algorithm bearing his name for computing GCDs appears in Book VII and remains in use today. In the 9th century, the Persian scholar Muhammad ibn Musa Al-Khwarizmi wrote Al-Kitab al-mukhtasar fi hisab al-jabr wal-muqabala, the treatise whose title gave algebra its name. He systematized the solution of linear and quadratic equations and described procedures that operated on unknowns as objects, a conceptual leap away from purely numerical calculation. Rene Descartes introduced coordinate geometry in 1637 by uniting algebra and Euclidean geometry, allowing curves to be studied through equations. This synthesis set the stage for calculus. Isaac Newton and Gottfried Wilhelm Leibniz independently developed calculus during the 1660s and 1670s, triggering a priority dispute that lasted decades and divided British and Continental mathematicians. Carl Friedrich Gauss proved the Fundamental Theorem of Algebra in 1799, showing that every nonconstant polynomial has at least one complex root. His Disquisitiones Arithmeticae of 1801 established modern number theory. David Hilbert's formalist program at the turn of the 20th century sought to place all of mathematics on an explicit axiomatic foundation, a project that Kurt Godel's incompleteness theorems of 1931 showed to be fundamentally limited. Alan Turing's work in the 1930s on computability introduced the theoretical model of the stored-program computer and linked mathematical logic directly to the limits of algorithmic calculation. His proof that no algorithm can decide in general whether an arbitrary program will halt or run forever placed fundamental boundaries on what mathematics can mechanically determine, and it opened the discipline now known as theoretical computer science.
Frequently Asked Questions
Formula
c = sqrt(a^2 + b^2)
Where c is the hypotenuse (the side opposite the right angle), and a and b are the two legs of the right triangle. The theorem states that the square of the hypotenuse equals the sum of the squares of the other two sides.
Worked Examples
Example 1: Classic 3-4-5 Right Triangle
Problem: Find the hypotenuse and all properties of a right triangle with legs of 3 and 4 units.
Solution: Hypotenuse = sqrt(3^2 + 4^2) = sqrt(9 + 16) = sqrt(25) = 5\nAngle A = arctan(3/4) = 36.87 degrees\nAngle B = arctan(4/3) = 53.13 degrees\nArea = 0.5 x 3 x 4 = 6 square units\nPerimeter = 3 + 4 + 5 = 12 units\nAltitude to hypotenuse = (3 x 4) / 5 = 2.4
Result: Hypotenuse = 5, Area = 6, Perimeter = 12, Pythagorean Triple = Yes
Example 2: Building Foundation Diagonal
Problem: A rectangular foundation measures 12 feet by 16 feet. Calculate the diagonal to verify squareness.
Solution: Diagonal = sqrt(12^2 + 16^2) = sqrt(144 + 256) = sqrt(400) = 20 feet\nAngle at 12-foot side = arctan(12/16) = 36.87 degrees\nAngle at 16-foot side = arctan(16/12) = 53.13 degrees\nThis is a 3-4-5 triple scaled by 4 (12-16-20)
Result: Diagonal = 20 feet. If both diagonals measure 20 feet, the foundation is square.
Frequently Asked Questions
What is the hypotenuse of a right triangle?
The hypotenuse is the longest side of a right triangle, located directly opposite the 90-degree right angle. It is always longer than either of the other two sides, which are called legs or catheti. The hypotenuse plays a central role in trigonometry and geometry because many fundamental relationships depend on it. The word hypotenuse comes from the Greek word hypoteinousa, meaning stretching under, referring to the side that stretches under the right angle. Every right triangle has exactly one hypotenuse, and knowing its length along with one leg allows you to calculate all other properties of the triangle including angles, area, and perimeter.
How does the Pythagorean theorem calculate the hypotenuse?
The Pythagorean theorem states that in a right triangle, the square of the hypotenuse equals the sum of the squares of the other two sides, expressed as c squared equals a squared plus b squared. To find the hypotenuse, you square each leg, add the results together, and then take the square root of that sum. For example, with legs of 3 and 4, you compute 9 plus 16 equals 25, and the square root of 25 is 5. This theorem was known to ancient Babylonians over 1,000 years before Pythagoras, and it has been proven in hundreds of different ways throughout history. It only applies to right triangles, making the 90-degree angle a prerequisite for using this formula.
Can the hypotenuse be used to find a missing leg?
Yes, the Pythagorean theorem can be rearranged to find a missing leg when you know the hypotenuse and one leg. If you know c (hypotenuse) and a (one leg), the missing leg b equals the square root of c squared minus a squared. For example, if the hypotenuse is 13 and one leg is 5, the other leg is the square root of 169 minus 25, which equals the square root of 144, giving you 12. This rearrangement is extremely useful in real-world applications such as construction, navigation, and engineering where you might know the diagonal measurement and one side but need to calculate the other. Always verify that c is larger than either leg to ensure valid inputs.
What is the altitude to the hypotenuse?
The altitude to the hypotenuse is a line segment drawn from the right angle vertex perpendicular to the hypotenuse. Its length equals the product of the two legs divided by the hypotenuse (h = ab/c). This altitude creates two smaller right triangles that are both similar to the original triangle and to each other, which is a powerful geometric relationship. For a 3-4-5 triangle, the altitude to the hypotenuse is 3 times 4 divided by 5, equaling 2.4. This altitude also represents the shortest distance from the right angle vertex to the hypotenuse. The geometric mean relationships in these similar triangles provide elegant proofs of the Pythagorean theorem itself.
What is the relationship between the hypotenuse and the circumscribed circle?
In any right triangle, the hypotenuse is always the diameter of the circumscribed circle, also known as the circumcircle. This means the circumradius (radius of the circumscribed circle) equals exactly half the hypotenuse. This remarkable property was known to the ancient Greek mathematician Thales and is sometimes called Thales theorem. It also works in reverse: if a triangle is inscribed in a circle with one side being a diameter, then the angle opposite that side must be exactly 90 degrees. For a 3-4-5 triangle, the circumradius is 5/2 = 2.5. This relationship is fundamental in circle geometry and has practical applications in engineering, optics, and computer graphics where circular arcs and right angles intersect.
What are some real-world applications of hypotenuse calculations?
Hypotenuse calculations appear constantly in construction, engineering, navigation, and everyday life. Builders use the Pythagorean theorem to ensure walls are square by measuring diagonals, and the classic 3-4-5 method is used on virtually every construction site worldwide. Surveyors calculate distances across obstacles like rivers by measuring angles and using right triangle relationships. Pilots and sailors use right triangle calculations to determine ground speed, wind correction angles, and shortest routes. In computer graphics, the distance between two pixels is calculated as the hypotenuse of a right triangle formed by horizontal and vertical pixel differences. Even smartphone screens are measured diagonally, which is the hypotenuse of the width and height dimensions.
References
Reviewed by Manoj Kumar, Mathematics Educator · Editorial policy