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Free Online Pythagorean Theorem Calculator With Steps

Give any two sides of a right angled triangle and get the third, together with both acute angles, the area, the perimeter and the height to the hypotenuse. The subtraction is arranged so that two nearly equal sides keep their accuracy, and a hypotenuse shorter than a leg is explained rather than returning not a number.

Free Forever Nothing Uploaded Any Two Sides Every Step Shown

Free Online Pythagorean Theorem Calculator With Steps

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📏 Pythagorean Theorem Any two sides · the third, the angles and the area · every step shown

The sum runs in your browser. Nothing you type is sent anywhere.

What you know

📏 The sides you have

In a right angled triangle the two short sides are the legs and the long one opposite the right angle is the hypotenuse. Any two of the three are enough.

Leg a

Leg b

Hypotenuse c

Try one of these

The fourth loses its accuracy in the obvious arrangement of the formula, and the last one cannot be a triangle at all.

The missing side

—

The three sides—

Angle opposite a—

Angle opposite b—

Area—

Perimeter—

Height to the hypotenuse—

Each step with your own numbers in it, in the order you would write it out.

Step	With your numbers

Recent triangles, kept in this browser only.

Nothing here yet Press Log to history on the Calculate tab to keep a triangle here.

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How to Use the Pythagorean Theorem

Three steps, and the third is the one that teaches.

Say which two sides you know Either the two legs, or one leg and the hypotenuse. The box for the side being worked out turns into the answer, so it is always clear which number came from you and which came from the page.

Read the missing side Along with both acute angles, the area, the perimeter and the height to the hypotenuse, since a right angled triangle is usually wanted for one of those rather than for the side alone.

Open the working Every step with your own numbers in it. When a leg is being found the subtraction is shown in the factored form, which is the arrangement that keeps its accuracy.

What to Know About Right Angled Triangles

Which side is which, and where the arithmetic goes wrong.

Two nearly equal sides need the subtraction rearranged. Finding a leg means taking the square root of c squared minus a squared. If c and a are close together those two squares are almost identical and subtracting them throws away the digits that carry the answer. Writing it as c minus a, times c plus a, gives the same value with none of that loss. With c of a million and a of 999999 the difference is visible, and it is why the page uses the factored form.

The hypotenuse is always the longest side. If you enter a hypotenuse shorter than one of the legs, no such triangle exists, and the formula asks for the square root of a negative number. Printing not a number tells you nothing. The page says which side is too long and by how much, which is usually enough to spot the typing mistake.

Whole number answers are rare, and that is normal. Sets of three whole numbers that fit, such as 3 4 5 and 5 12 13, are called Pythagorean triples and there are not many small ones. Most right angled triangles have an irrational side, so a long decimal is the correct answer rather than a sign that something went wrong.

The angles come free with the sides. Once all three sides are known the two acute angles follow from the inverse sine or cosine, and they add to ninety degrees. Both are given, because knowing the triangle usually means knowing its shape rather than just its measurements.

The height to the hypotenuse is the one people forget. It is the product of the two legs divided by the hypotenuse, and it is what you need for the distance from the right angle to the longest side. It appears constantly in construction and in geometry problems, so it is given alongside the area.

It runs in your browser with nothing uploaded. The arithmetic is in the page, so it works offline once loaded and nothing you type is sent anywhere.

Key Features & Capabilities

What it does, and what it deliberately does not.

Any two sides Two legs or one leg and the hypotenuse, with the box being solved for marked as the answer.

Accurate on near equal sides The difference of two squares is factored, which keeps the digits the answer depends on.

Angles, area and height Both acute angles, the area, the perimeter and the height to the hypotenuse, all at once.

Impossible triangles explained A hypotenuse shorter than a leg is named as the problem rather than returning not a number.

The working shown Each step with your own numbers, ready to copy as a method.

Nothing is uploaded It runs in this tab and keeps working offline.

About the Pythagorean Theorem

The Pythagorean theorem says that in a right angled triangle the square on the hypotenuse equals the sum of the squares on the other two sides. It is the oldest piece of mathematics most people can still state from memory, and it is the basis of every distance calculation there is: the straight line distance between two points on a grid is a Pythagorean calculation, and so is the length of a diagonal, the reach of a ladder and the shortest path across a rectangle.

Using it forwards is easy. Square the two legs, add them, take the square root, and you have the hypotenuse. Using it backwards, to find a leg from the hypotenuse and the other leg, is where the arithmetic gets delicate. The formula asks for c squared minus a squared, and when c and a are close to each other those two numbers agree in most of their digits. Subtracting them cancels all the agreeing digits and leaves only the few that differed, so the answer is computed from far less information than it looks.

The repair is old and simple: c squared minus a squared is the same as c minus a, times c plus a. Both factors are computed exactly, and nothing cancels. With a hypotenuse of a million and a leg of 999999 the two arrangements give visibly different answers, and the factored one is right. It costs nothing, so it is what the page uses.

The other case worth handling properly is the impossible triangle. If someone types a hypotenuse that is shorter than a leg there is no such shape, and the formula asks for the square root of a negative number. Returning not a number is technically honest and practically useless. Naming the side that is too long, and by how much, points straight at the typing mistake, which is almost always what has happened.

Frequently Asked Questions

Legs, hypotenuse, triples and what counts as impossible.

The one opposite the right angle, and it is always the longest of the three. The other two, which meet at the right angle, are the legs.

Yes. Choose the option that says you know a leg and the hypotenuse, and the other leg is worked out. That direction is the one that needs care with the arithmetic, and the page uses the arrangement that keeps its accuracy.

Three whole numbers that satisfy the theorem, such as 3 4 5, 5 12 13 and 8 15 17. They are useful because they give exact answers, but most right angled triangles are not of this kind and have an irrational side.

Because the hypotenuse you entered is shorter than one of the legs. The longest side of a right angled triangle is always the hypotenuse, so no such triangle exists and the formula would ask for the square root of a negative number.

No. It holds only when one angle is exactly ninety degrees. For any other triangle the law of cosines is the general version, and the triangle calculator on this hub uses it.

The perpendicular distance from the right angle to the hypotenuse. It equals the two legs multiplied together and divided by the hypotenuse, and it is what you need when a length has to be measured across the triangle rather than along a side.

No. The arithmetic runs in your browser, so nothing you type leaves the page and it works offline.