A carpenter checking whether a corner is truly square reaches for the 3-4-5 trick: mark 3 feet along one wall, 4 feet along the other, and if the diagonal between the marks measures exactly 5 feet, the corner is a perfect right angle. That trick works because of the Pythagorean theorem, the most useful fact in all of practical geometry. In a right triangle, the square of the hypotenuse equals the sum of the squares of the two legs, written a squared plus b squared equals c squared. The right angle forces the whole relationship, which is why roofers, stair builders, surveyors, and deck builders trust it with real lumber and real money, and why it still shows up on every geometry test decades after students first meet it.

Naming the sides before you touch a number

A right triangle has exactly one angle of 90 degrees. The two sides that meet at that right angle are called legs, and the side opposite the right angle is called the hypotenuse. The hypotenuse is always the longest side, and that single fact is the cheapest error check in the subject: any answer in which a leg comes out longer than the hypotenuse is wrong, full stop. It also decides which version of the formula you need. If the two numbers you know are both legs, you add their squares. If one number is the hypotenuse and the other is a leg, you subtract the leg square from the hypotenuse square. Everything else is arithmetic, and every worked example below follows that fork in the road, so name the hypotenuse first and the rest of the problem organizes itself.

The three formulas that solve almost every right triangle

The hypotenuse formula is c equals the square root of a squared plus b squared. The missing-leg formulas are a equals the square root of c squared minus b squared, and b equals the square root of c squared minus a squared, with the whole subtraction inside the square root. The area formula is equally direct: multiply the two legs and halve the product, because two copies of the triangle make a rectangle with sides a and b. The perimeter is simply a plus b plus c. Once you have the three sides, the acute angles follow from arctangents: the angle opposite leg a is the arctangent of a divided by b, and the other acute angle is 90 degrees minus that angle. The two acute angles always add to 90 degrees, which is a free check on every angle answer you ever produce.

Worked example: a stair stringer and its triangular face

A stair builder cuts a stringer with a 7-inch rise and an 11-inch run. The two legs are 7 and 11, so their product is 77, and halving gives a triangular face of 38.5 square inches. On this calculator, type 7 in the Variable A box and 11 in the Variable B box, and the Result box shows 77, the leg product at the heart of the area formula. Divide that Result by 2 yourself and you have the 38.5 square inches of face per step, before the saw kerf and the tread overhang complicate the carpentry. The hypotenuse of that same step triangle is the square root of 49 plus 121, the square root of 170, about 13.04 inches, which is the sloped length the stringer actually spans. One pair of leg measurements thus prices both the face area and the cut length, which is the practical reason stair builders measure rise and run before they ever pick up a saw.

Worked example: the full 3-4-5 solve, side by side

Take legs of 3 and 4. Square them: 9 and 16. Add them: 25. Take the square root: a hypotenuse of 5. The area is 3 times 4 divided by 2, which is 6 square units, and the perimeter is 3 plus 4 plus 5, which is 12 units. The angle opposite the 3 leg is the arctangent of 3 divided by 4, about 36.87 degrees, and the other acute angle is 53.13 degrees, and 36.87 plus 53.13 is exactly 90, as it must be. The altitude from the right-angle corner to the hypotenuse is the leg product divided by the hypotenuse, 12 divided by 5, which is 2.4 units. That altitude splits the hypotenuse into two projections: 9 divided by 5, which is 1.8, and 16 divided by 5, which is 3.2, and 1.8 plus 3.2 is 5, the whole hypotenuse, which confirms the split. A complete triangle solution is never just one number; it is this bundle of sides, angles, area, perimeter, and altitude, all checking against each other.

Worked example: finding a missing leg, three ways

Take a hypotenuse of 13 and a known leg of 5. Square them: 169 and 25. Subtract: 144. Take the square root: a missing leg of 12. That is the 5-12-13 triangle, with an area of 5 times 12 divided by 2, which is 30 square units. A second case works the same way: a hypotenuse of 25 with a leg of 7 gives 625 minus 49, which is 576, and the square root is 24. A third: a hypotenuse of 10 with a leg of 6 gives 100 minus 36, which is 64, and the square root is 8, the familiar 6-8-10 triangle in disguise. The subtraction order matters and it only works one way. If the number you subtract from is not the largest, the result under the square root goes negative, which is the arithmetic telling you that no such right triangle exists: a hypotenuse shorter than one of its own legs is a contradiction, not a rounding problem.

Pythagorean triples worth memorizing

A Pythagorean triple is a set of three whole numbers that satisfies the theorem exactly. The 3-4-5 triple scales to 6-8-10, 9-12-15, and 12-16-20, so recognizing a multiple of 3-4-5 lets you answer without a calculator: 6-8-10 has an area of 24 and a perimeter of 24 with an altitude of 4.8, 9-12-15 has an area of 54, a perimeter of 36, and an altitude of 7.2, and 12-16-20 has an area of 96, a perimeter of 48, and an altitude of 9.6. The 5-12-13 triple gives an area of 30 and a perimeter of 30, a coincidence students love, with an altitude of about 4.62. The 8-15-17 triple gives an area of 60 and a perimeter of 40 with an altitude of about 7.06. The 7-24-25 triple covers a 24-foot height with a 7-foot setback and a 25-foot ladder, with an area of 84 and an altitude of 6.72. The 20-21-29 triple is the largest common classroom triple, with an area of 210, a perimeter of 70, and an altitude of about 14.48. Carpenters use the 3-4-5 triple to square a foundation: mark 3 feet along one string and 4 feet along the other, and adjust until the diagonal between the marks is exactly 5 feet, which guarantees a right angle at the corner.

Special right triangles: the two that recur everywhere

The 45-45-90 triangle is half a square: its legs are equal, and its hypotenuse is a leg times the square root of 2, about 1.4142. Legs of 7 give a hypotenuse of about 9.90 and an area of 24.5, while legs of 10 give a hypotenuse of about 14.14 and an area of 50. The 30-60-90 triangle is half an equilateral triangle: its short leg, long leg, and hypotenuse sit in the ratio 1 to the square root of 3 to 2. A short leg of 5 gives a long leg of about 8.66, a hypotenuse of 10, and an area of about 21.65. These two triangles are worth recognizing on sight, because test questions and roof pitches use them constantly, and spotting one turns a square-root problem into a multiplication you can do in your head.

Testing whether a triangle is right at all

The theorem also works in reverse, and this reverse test settles arguments about prefabricated corners, picture frames, and deck footings without an angle gauge. Sort the three sides, call the largest the candidate hypotenuse, and test whether the squares of the two smaller sides add to the square of the largest. Sides of 5, 6, and 8 fail: 25 plus 36 is 61, not 64, so no right angle exists, however square the corner looks. Sides of 6, 8, and 10 pass: 36 plus 64 is exactly 100, so the triangle is right with hypotenuse 10. Sides of 9, 12, and 15 pass: 81 plus 144 is exactly 225. If the squares do not balance, the corner is not square, and no amount of wishful measuring changes that verdict. Note the boundary: the test needs an exact right angle, so a triangle that is nearly right but off by a degree will fail by a small margin, which is why framers re-measure rather than argue.

Angles from sides, and the altitude shortcut

Once the three sides are known, the acute angles follow from arctangents. In the 5-12-13 triangle, the angle opposite the 5 leg is the arctangent of 5 divided by 12, about 22.62 degrees, and the other acute angle is 67.38 degrees. In the 9-12-15 triangle, the angles are 36.87 and 53.13 degrees, the same angles as the 3-4-5 triangle, because scaling a triangle changes its size but never its angles. The altitude to the hypotenuse deserves its own habit: altitude equals the leg product divided by the hypotenuse. For 9-12-15 that is 108 divided by 15, which is 7.2. For a 15-20-25 triangle it is 300 divided by 25, which is 12. The altitude also splits the hypotenuse into projections of a squared divided by c and b squared divided by c: for 9-12-15 those are 81 divided by 15, which is 5.4, and 144 divided by 15, which is 9.6, and 5.4 plus 9.6 is 15. When a roof or ramp problem asks for the height above a sloped surface rather than a side length, the altitude is the number being asked for, and it comes free from the same two legs.

Real jobs: ladders, ramps, roofs, and grid distances

A 41-foot extension ladder leaning against a house forms a right triangle with the ground and the wall. With the foot 9 feet out and the top resting 40 feet up, the ladder length is the square root of 81 plus 1,600, the square root of 1,681, which is exactly 41 feet. The angle at the ground is the arctangent of 40 divided by 9, about 77.32 degrees. Ladder safety guidance in the United States uses a 4-to-1 rule, one foot out for every four feet up, which is an angle of about 75.96 degrees; the 9-and-40 ladder sits slightly steeper than the rule, so a careful crew moves the foot out to 10 feet and checks a ladder of the square root of 100 plus 1,600, about 41.23 feet, instead. A wheelchair ramp that rises 3 feet over a 36-foot run follows the 1-to-12 slope used by the Americans with Disabilities Act: its sloped length is the square root of 9 plus 1,296, about 36.12 feet, and its angle is the arctangent of 3 divided by 36, about 4.76 degrees. A roof with a 12-foot horizontal run and a 9-foot rise has a rafter length of 15 feet by the 9-12-15 triple. The same theorem measures distance on a grid: the straight-line distance between points (1, 2) and (10, 14) has horizontal change 9 and vertical change 12, so the distance is the square root of 81 plus 144, which is 15 units, a step surveyors, game developers, and delivery-route planners use thousands of times a day.

How to use this calculator, honestly

This calculator performs the multiplication at the heart of the right-triangle formulas. To get a leg product for an area, type one leg in Variable A and the other in Variable B, read the Result, and divide by 2: 7 and 11 give 77, and halving gives the 38.5-square-inch stair face. To square a leg for the Pythagorean theorem, type the same leg in both boxes: 9 twice gives 81, and 12 twice gives 144, and adding those two Results gives 225, whose square root is the 15-foot hypotenuse. The tool does not take square roots for you, and it does not decide add versus subtract, so naming the hypotenuse correctly before you start is the entire skill. Keep every measurement in the same unit: a leg in feet and a leg in inches produces a hypotenuse in neither, because the squares corrupt by a factor of 144 before the square root even enters the picture.

Five mistakes that break the answer

The first mistake is adding the legs and then taking the square root, skipping the squaring, which understates every hypotenuse. The second is treating a leg as the hypotenuse, which makes the subtraction step go negative or the addition step overshoot. The third is forgetting the final square root, which reports 225 when the answer is 15. The fourth is applying the theorem to a triangle that is not right: the reverse test above exists precisely because not every triangle qualifies, and a general triangle needs the law of cosines instead. The fifth is mixing units, feet with inches or meters with centimeters. A result that fails the longest-side check, a hypotenuse shorter than a leg, is the fingerprint of one of these five, and it should never be trusted to a cut list.

Related calculators on this site

If the right triangle is part of a bigger job, companion tools carry the next step. The right triangle calculator page works the same half-diagonal and ladder problems from the squaring angle. The Pythagorean triples calculator covers the whole-number families listed above when you want to recognize a triple on sight. The area of triangle calculator handles triangles that are not right, using base and height or three sides. The triangle angles calculator works the angle side of the same problem when an angle, rather than a second side, is the known quantity. Each one starts from the same discipline: name the hypotenuse first, then compute.

A classroom note on why the squares add

The theorem is easiest to believe from the picture of squares built on each side. The square on the hypotenuse has exactly the area of the two leg squares combined, and the classic rearrangement proof slides four copies of the triangle inside a larger square to show it with no algebra at all. That picture also explains why the theorem fails for other triangles: only the right angle makes the four-triangle rearrangement line up. Once students see the squares, the formula stops being a chant and becomes a fact about area, and the ladder, ramp, and foundation examples stop looking like separate tricks.

With the pre-filled defaults — Variable A 10, Variable B 5 — the Right Angle Triangle Calculator returns Result 50.0000. Change any input and those results update instantly, so you can see exactly how each number you enter moves the answer.