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.
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.