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.
Reviewed for accuracy by Manoj Kumar, Mathematics Educator
Hypotenuse Calculator
Calculator
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Formula: c = sqrt(a^2 + b^2)
Worked example — Hypotenuse = 5, Area = 6, Perimeter = 12, Pythagorean Triple = Yes
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 Angle A = arctan(3/4) = 36.87 degrees Angle B = arctan(4/3) = 53.13 degrees Area = 0.5 x 3 x 4 = 6 square units Perimeter = 3 + 4 + 5 = 12 units Altitude 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 Angle at 12-foot side = arctan(12/16) = 36.87 degrees Angle at 16-foot side = arctan(16/12) = 53.13 degrees This 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 are the angles in a right triangle given two legs?
In a right triangle, one angle is always 90 degrees, and the other two acute angles can be calculated using inverse trigonometric functions. Given legs a and b, the angle opposite side a is arctan(a/b) and the angle opposite side b is arctan(b/a). These two acute angles always sum to exactly 90 degrees. For example, a 3-4-5 triangle has acute angles of approximately 36.87 degrees and 53.13 degrees. The relationship between sides and angles is governed by trigonometric ratios: sine equals opposite over hypotenuse, cosine equals adjacent over hypotenuse, and tangent equals opposite over adjacent. These ratios are foundational for surveying, navigation, and physics calculations.
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.
How do you calculate the inradius of a right triangle?
The inradius of a right triangle, which is the radius of the inscribed circle that touches all three sides, can be calculated using the simple formula r = (a + b - c) / 2, where a and b are the legs and c is the hypotenuse. This formula is derived from the general triangle inradius formula r = Area / s, where s is the semi-perimeter. For a right triangle with area equal to ab/2 and semi-perimeter equal to (a + b + c) / 2, the algebra simplifies beautifully to the formula above. For a 3-4-5 triangle, the inradius is (3 + 4 - 5) / 2 = 1. The inscribed circle center, called the incenter, is equidistant from all three sides and always lies inside the triangle.
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.
How accurate is this hypotenuse calculator for decimal inputs?
Hypotenuse Calculator uses JavaScript floating-point arithmetic with double precision, providing approximately 15-16 significant digits of accuracy, which is more than sufficient for virtually all practical applications. The Pythagorean theorem calculation itself is mathematically exact, so the only source of error is the finite precision of computer floating-point representation. For integer inputs that form Pythagorean triples like 3-4-5 or 5-12-13, the result will be exactly correct. For irrational results like the hypotenuse of a 1-1 right triangle (which equals the square root of 2), the calculator provides a decimal approximation accurate to many decimal places. For most engineering and construction applications, 4-6 decimal places far exceed measurement precision requirements.
What is the difference between hypotenuse, median, and altitude in a right triangle?
The hypotenuse is the side opposite the right angle and is always the longest side of the triangle. The median to the hypotenuse is a line segment from the right angle vertex to the midpoint of the hypotenuse, and in a right triangle it always equals exactly half the hypotenuse length, making it equal to the circumradius. The altitude to the hypotenuse is a line segment from the right angle vertex perpendicular to the hypotenuse, and it equals ab/c where a and b are the legs. These three measurements serve different geometric purposes: the hypotenuse defines the triangle shape, the median connects to the center of the circumscribed circle, and the altitude determines the height for area calculations relative to the hypotenuse as a base.
References
Reviewed for accuracy by Manoj Kumar, Mathematics Educator · Editorial policy
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