How do I know which side is the hypotenuse?

It is the side directly opposite the right angle, and it never forms one of the right angle’s arms. It is also always the longest side, so if your answer for the hypotenuse comes out shorter than a leg, you have mixed them up.

What if the answer is not a whole number?

That is normal — most right triangles are not Pythagorean triples. Leave the answer as an exact surd like \sqrt{74} unless the question asks for a decimal, and simplify where possible: \sqrt{50} = 5\sqrt{2}.

Does the Pythagorean theorem work on any triangle?

No — it requires a right angle. For triangles without one you need the law of cosines, c^2 = a^2 + b^2 - 2ab\cos C, which reduces to the Pythagorean theorem exactly when C = 90° and the cosine term vanishes.

What is the converse of the Pythagorean theorem?

It runs the logic backwards: if the squares of the two shorter sides add up to the square of the longest, the triangle must be right-angled. It is how you classify a triangle from three side lengths alone.

Why is the distance formula the same thing?

Because the horizontal and vertical gaps between two points form the legs of a right triangle whose hypotenuse is the distance you want. d = \sqrt{(\Delta x)^2 + (\Delta y)^2} is a^2 + b^2 = c^2 rearranged.

Can Photo Math Solver read the triangle diagram itself?

Yes — screenshot the figure with its labels and the right-angle mark included, and the solver uses the values written on the picture, not just the sentence underneath it.

You can start solving Pythagorean theorem problems for free by adding the Chrome extension — try it on your own worksheet before deciding if you need anything more.

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Solve Pythagorean Theorem Problems Step by Step

Screenshot any right triangle and get the substitution, the square root, and the missing side — hypotenuse or leg.

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Photo Math Solver Solved

a = 3,\; b = 4,\; c = ?

1Both legs are known, so square and add: $3^2 + 4^2 = 9 + 16 = 25$ .
2Take the square root: $c = \sqrt{25} = 5$ .
3This is the 3-4-5 triple — worth memorising, it appears constantly.

On this page01Definition 02Formulas 03Methods 04Examples 05Practice 06Mistakes 07FAQ

01Definition

What is the Pythagorean theorem?

The Pythagorean theorem says that in any right triangle, the squares of the two shorter sides add up to the square of the longest side: $a^2 + b^2 = c^2$ . The two shorter sides are the legs, the one opposite the right angle is the hypotenuse, and it is always the longest.

The word "squares" is literal. Build a square on each side of a right triangle and the two smaller squares have exactly the same combined area as the big one — that is the picture the algebra is describing, and it is why the theorem only works when there is a right angle.

It runs in both directions. Given two sides you can find the third; given all three sides you can test whether the triangle is right-angled at all — that reverse use is called the converse, and it is where the 3-4-5 and 5-12-13 triples come from.

Where you'll actually use this

Ladder safety: a 10 m ladder with its base 6 m out reaches exactly 8 m up the wall.
Checking a room or foundation is square by measuring 3 m, 4 m, and confirming the diagonal is 5 m.
TV and monitor sizing: a screen advertised as 55" is the diagonal, recovered from width and height.
Navigation and GPS: straight-line distance from east–west and north–south displacements.
Roof pitch and staircase stringers: rise and run give the rafter or stringer length directly.

Hypotenuse: The side opposite the right angle — always the longest side of a right triangle.
Leg: Either of the two shorter sides that meet at the right angle.
Pythagorean triple: Three whole numbers satisfying $a^2 + b^2 = c^2$ , such as 3-4-5 or 5-12-13.
Converse: The reversed statement: if $a^2 + b^2 = c^2$ holds, the triangle is right-angled.

02Formulas

The formulas you need

Keep these on hand — every method below builds on them.

The theorem

a^2 + b^2 = c^2

a and b are the legs; c is the hypotenuse, always opposite the right angle.

Finding the hypotenuse

c = \sqrt{a^2 + b^2}

Both legs known — square, add, then take the square root.

Finding a leg

a = \sqrt{c^2 - b^2}

Hypotenuse known — subtract, do not add. The hypotenuse term is always the one being reduced.

The converse

a^2 + b^2 = c^2 \;\Rightarrow\; \text{right triangle}

If the sum falls short the triangle is obtuse; if it overshoots, acute.

Common triples

(3,4,5),\; (5,12,13),\; (8,15,17),\; (7,24,25)

Any multiple works too — 6-8-10 and 9-12-15 are both 3-4-5 scaled up.

Distance between points

d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}

The distance formula is the theorem applied to the horizontal and vertical gaps.

03Methods

How to solve pythagorean theorem

Pick the method that fits the problem in front of you.

When to use it: Both legs are given and the unknown is the side opposite the right angle — the longest side.

1Identify the hypotenuse: it is opposite the right angle, never touching it.
2Square both legs and add them: for legs $6$ and $8$ , $36 + 64 = 100$ .
3Take the positive square root: $c = \sqrt{100} = 10$ .
4Sanity-check that the answer is larger than either leg — if it is not, something went wrong.

04Worked examples

Pythagorean Theorem, solved step by step

From easy to hard — pick a problem to see its full solution.

a = 3,\; b = 4,\; c = ?

1Both legs are known, so square and add: $3^2 + 4^2 = 9 + 16 = 25$ .
2Take the square root: $c = \sqrt{25} = 5$ .
3This is the 3-4-5 triple — worth memorising, it appears constantly.

Try it on your own pythagorean theorem homework

Photo Math Solver reads a right-triangle problem straight from a screenshot — a labelled diagram, a word problem about a ladder, a distance question on a grid — and shows the full Pythagorean working: which side is the hypotenuse, the substitution into $a^2 + b^2 = c^2$, and the square root at the end. No mistyping squares into a calculator and no guessing whether to add or subtract.

05Practice

Now you try

Work each one on paper first, then check your answer.

Problem 1

a = 9,\; b = 12,\; c = ?

Problem 2

a = ?,\; b = 24,\; c = 25

Problem 3

a = 2,\; b = 3,\; c = ?

Problem 4

\text{Distance from } (2,1) \text{ to } (7,13)

Problem 5

\text{Are } 6, 7, 10 \text{ the sides of a right triangle?}

Stuck on your pythagorean theorem homework? Screenshot it

Right-triangle problems put half the information on the diagram — side labels, the little square marking the right angle, which side is opposite it. Screenshot the whole figure with Photo Math Solver rather than retyping the numbers, and the solution identifies the hypotenuse for you before substituting.

06Common mistakes

Where points get lost

Each of these shows up on real graded work — and each has a simple fix.

The mistake	Why it happens	The fix
Adding when the hypotenuse is the known side: getting $c = \sqrt{13^2 + 5^2}$ for a missing leg.	The formula is memorised as "square, add, square root" as a single procedure, without checking which side is unknown.	Ask first: is the unknown the longest side? If yes, add. If the hypotenuse is already given, subtract — the answer has to come out smaller than the hypotenuse.
Forgetting the square root and reporting $c^2$ as the answer.	The addition step produces a satisfying round number like $100$ , which looks like a finished answer.	Write the unknown as $c^2 = \ldots$ on its own line so the missing final step is visible. A hypotenuse of $100$ for legs of $6$ and $8$ is obviously far too long.
Applying the theorem to a triangle that has no right angle.	The problem gives three convenient numbers and the theorem is the most familiar tool available.	Look for the right-angle square in the diagram or the words "right triangle" in the text. Without a right angle you need the law of cosines instead.
Assuming $(a + b)^2 = a^2 + b^2$ and computing $\sqrt{6^2 + 8^2}$ as $6 + 8 = 14$ .	Squaring feels distributive over addition, and $14$ is quicker to produce than the real answer.	Square each side separately before adding: $36 + 64 = 100$ , giving $10$ , not $14$ . The hypotenuse is always shorter than the two legs combined.
Picking the wrong side as the hypotenuse in a rotated diagram.	The hypotenuse is remembered as "the bottom one" or "the slanted one", which stops being true once the figure is turned.	Find the right angle first, then take the side that does not touch it. That side is the hypotenuse no matter how the triangle is drawn.

Photo Math Solver shows every intermediate step, so slips like these are easy to catch before they cost you marks.

07FAQ