Hypotenuse Calculator
Find the hypotenuse of any right triangle from two legs, one leg and an angle, or area and one leg. Uses the Pythagorean theorem and trigonometric ratios with full step-by-step solutions.
Calculation Examples
How to Find the Hypotenuse of a Right Triangle
Pick the input method that matches what you know. The calculator has three modes:
1. Two legs (a and b): The most common case. Enter both perpendicular sides and the calculator applies the Pythagorean theorem directly.
2. One leg and one acute angle: Enter one side and either acute angle (\(\alpha\) or \(\beta\)). The calculator picks sine or cosine depending on whether the known leg is opposite or adjacent to that angle.
3. Area and one leg: The calculator finds the second leg from \(b = 2A/a\), then applies the Pythagorean theorem.
Keep all units the same (do not mix inches and centimeters) and click Calculate. The result shows the hypotenuse, any missing leg, both acute angles, and the area. A quick check: legs of 3 and 4 always give a hypotenuse of exactly 5. These formulas work only for right triangles (one 90° angle). For other triangles, use the Law of Cosines.
Geometric Formulas and Principles
All three modes rest on two ideas from Euclidean geometry:
1. Pythagorean theorem (both legs known): \[c = \sqrt{a^2 + b^2}\] where \(a\) and \(b\) are the legs (the two sides that form the right angle) and \(c\) is the hypotenuse. Euclid gives a proof in Elements (Book I, Proposition 47, c. 300 BCE), and the theorem remains the standard tool for right-triangle work in engineering and construction.
2. Trigonometric ratios (one leg and one acute angle known): \[c = \frac{a}{\sin\alpha}\] when \(a\) is the leg opposite \(\alpha\), and \[c = \frac{b}{\cos\alpha}\] when \(b\) is the leg adjacent to \(\alpha\).
3. Area method (area and one leg known): recover the second leg with \(b = 2A/a\), then apply the Pythagorean theorem.
Two special triangles give exact answers without a calculator. In a 30-60-90 triangle, the sides follow the ratio \(1 : \sqrt{3} : 2\), so the hypotenuse is twice the shortest leg. In a 45-45-90 triangle (isosceles right triangle), both legs are equal and the hypotenuse is \(a\sqrt{2}\).
Useful Tips 💡
- The hypotenuse is always the longest side of a right triangle. If you measure a "right" corner and the diagonal comes out too long or too short for the 3-4-5 check, the corner is not square or a measurement is off.
- For a 30-60-90 triangle, the hypotenuse is exactly twice the shortest leg. A 7 cm short leg gives a 14 cm hypotenuse with no calculator needed.
- To find the diagonal of a physical object, measure the two perpendicular sides (width and height) and calculate. Perpendicular measurements are easier to take accurately than a diagonal across the object.
- Memorize a few integer triples for quick checks: 3-4-5, 5-12-13, 8-15-17, and 7-24-25.
📋Steps to Calculate
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Select the input mode: two legs, one leg and one angle, or area and one leg.
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Enter the values and confirm all inputs use the same unit (cm, m, inches, or feet).
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Click Calculate to see the hypotenuse, missing leg, both acute angles, and the triangle area.
Mistakes to Avoid ⚠️
- Applying the Pythagorean theorem to a triangle without a 90° angle: the formula c = square root of (a squared plus b squared) holds only for right triangles. For other triangles, use the Law of Cosines.
- Forgetting the square root: adding the squares of the legs gives c squared, not c. For legs of 3 and 4, the sum is 25, but the hypotenuse is 5.
- Confusing opposite and adjacent sides in trigonometry: the opposite side faces the reference angle and the adjacent side touches it. Swapping them swaps sine and cosine and gives the wrong hypotenuse.
- Adding when you should subtract: to find a missing leg from the hypotenuse and the other leg, subtract the square of the known leg from the square of the hypotenuse, then take the square root.
Practical Applications in Construction and Design📊
Construction and framing: Get rafter, stair stringer, and brace lengths from rise and run. A roof with 7 ft of rise over a 14 ft run needs a rafter of about 15.65 ft (15 ft 7.8 in), before overhang and cuts.
Architecture and surveying: Check that a foundation or room corner is square with the 3-4-5 rule. Larger multiples such as 6-8-10 or 9-12-15 are more accurate: a 108 in by 144 in layout must have a diagonal of exactly 180 in.
Display technology: Find the diagonal screen size from width and height. A 16:9 panel 48 inches wide and 27 inches tall has a diagonal of about 55.1 inches, sold as a 55 inch TV.
Ladder safety: Find how high a ladder reaches. OSHA (29 CFR 1926.1053) sets the base one quarter of the working length from the wall, so a 20 ft ladder with a 5 ft base reaches \(\sqrt{20^2 - 5^2} \approx 19.4\) ft, at about 75.5° from the ground.