Pythagorean Theorem Calculator

Solve a missing side of a right triangle or check whether three side lengths satisfy the Pythagorean theorem. Follow the addition or subtraction steps with unit conversion.

Choose Your Calculation

Find an unknown leg or hypotenuse, or test three known side lengths. All lengths must be positive.

In missing-side mode, c is the hypotenuse and must be longer than the known leg.

a b c 90°
Right-triangle reference diagram, not a drawing of the entered measurements. Check mode does not assume the sides form a right triangle.

Select the matching unit for each side. Changing an input unit converts its value. Switching modes or the missing side clears the measurements.

Only calculating the longest side from two legs? The hypotenuse calculator provides a focused two-input tool.

QUICK START

How to Use the Pythagorean Theorem Calculator

Choose whether you need to solve an unknown side or test three known lengths. The calculator shows the working for your selected task.

  1. Select a calculation mode

    Choose Find a missing side when your triangle is known to be right-angled. Choose Check three sides to test whether three lengths form a triangle and satisfy the theorem.

  2. Identify the unknown side or enter all three

    In missing-side mode, select a, b or c and enter the other two lengths. Here, c is the hypotenuse. In check mode, enter all three sides in any order; the tool identifies the longest side automatically.

  3. Match the units and choose display precision

    Select the unit beside each positive measurement. Mixed units are converted before calculating. Choose result units and decimal places. Display rounding does not change the right-triangle check.

  4. Review the result and its explanation

    Follow the addition or subtraction used to find a side. In check mode, read the triangle validity and right-angle comparison. Select Copy Result to copy the result summary.

Keep solving and checking separate: Missing-side mode assumes a right triangle. Check mode tests the entered side lengths, using a small numerical tolerance. Rounded physical measurements may not match the theorem exactly.

WORKED EXAMPLES

Solve a Missing Leg or Check Three Sides

Subtract squares to find an unknown leg, or compare squared lengths to test a complete set of sides. These tasks use the same theorem in different ways.

EXAMPLE 01 · MISSING LEG

Find an unknown leg from 8 cm and 17 cm

A right triangle has a hypotenuse of 17 centimeters and one leg of 8 centimeters. Find the other leg by rearranging the theorem.

GIVEN MEASUREMENTS
Known leg b
8 cm
Hypotenuse c
17 cm
  1. Isolate the missing leg’s square

    a² + b² = c²

    a² = c² − b²

  2. Substitute and subtract

    a² = 17² − 8²

    a² = 289 − 64 = 225 cm²

  3. Take the positive square root

    a = √225 = 15 cm

RESULT Missing leg a = 15 cm

Verification: 15² + 8² = 225 + 64 = 289 = 17².

EXAMPLE 02 · THREE-SIDE CHECK

Check whether 7, 24 and 25 form a right triangle

All three side lengths are known in centimeters. First confirm they can form a triangle, then compare the longest side’s square with the other two.

GIVEN MEASUREMENTS
Shorter sides
7 cm, 24 cm
Longest side
25 cm
  1. Check the triangle inequality

    7 + 24 = 31 > 25

    The lengths can form a triangle.

  2. Add the squares of the shorter sides

    7² + 24² = 49 + 576 = 625 cm²

  3. Compare with the longest side squared

    25² = 625 cm²

    7² + 24² = 25²

RESULT These lengths form a right triangle

The 25 cm side is the hypotenuse, opposite the 90° angle.

A valid triangle is not necessarily right-angled

Sides of 4, 5 and 6 cm satisfy 4 + 5 > 6, so they can form a triangle. However, 4² + 5² = 41 while 6² = 36. The values are unequal; this triangle is acute, rather than right-angled. Select Check three sides in the calculator above to test another set.

UNDERSTAND THE THEOREM

Pythagorean Theorem Formulas and Checks

The theorem relates the areas of squares built on a right triangle’s sides. Rearrange that relationship to solve an unknown side, or use its converse to check three known lengths.

a² b² c² a² + b² = c²
Two leg-square areas equal the hypotenuse-square area a² and b² represent areas, not the leg lengths themselves. Their sum equals c² when the legs meet at 90°.
FIND THE HYPOTENUSE c = √(a² + b²) FIND A LEG a = √(c² − b²)
Add for the hypotenuse; subtract for a leg A valid missing-leg calculation requires the hypotenuse to be longer than the known leg. Take the positive square root for a side length.
In solve mode, a and b are perpendicular legs and c is the hypotenuse. For three-side comparisons, c denotes the longest side after sorting.
Task Formula or condition Interpretation
Find hypotenuse c c = √(a² + b²) Both legs are known
Find leg a a = √(c² − b²) Requires c > b
Find leg b b = √(c² − a²) Requires c > a
Confirm a valid triangle a + b > c c is the longest side; all lengths are positive
Right triangle a² + b² = c² The largest angle is 90°
Acute triangle a² + b² > c² All angles are less than 90°
Obtuse triangle a² + b² < c² The largest angle is greater than 90°

Check triangle validity before classifying angles

Positive lengths alone are not enough. After identifying the longest side, the two shorter lengths must add to more than it. For example, 2, 3 and 5 do not form a nondegenerate triangle because 2 + 3 = 5.

The converse tests a triangle’s right angle

The theorem starts with a right triangle and relates its sides. The converse works in the other direction: if a valid triangle’s shorter-side squares add to the longest-side square, the angle opposite the longest side is 90°.

Exact mathematics and numerical checks: The table uses exact equality. The calculator checks equality within its stated numerical tolerance. For geometric explanations and proofs, explore the Pythagorean theorem reference on Wolfram MathWorld.

COMMON QUESTIONS

Frequently Asked Questions

Answers about missing legs, triangle validity, three-side checks and numerical precision.

When should I add squares and when should I subtract them?

Add the leg squares when finding the hypotenuse: c = √(a² + b²). To find a leg, subtract the known leg’s square from the hypotenuse’s square: a = √(c² − b²) or b = √(c² − a²). Do not subtract the lengths themselves before squaring.

Why must the hypotenuse be longer than the known leg?

In a right triangle, the hypotenuse is the longest side. If it equals the known leg, subtraction gives zero for the other leg, so the triangle collapses. If it is shorter, the expression under the square root is negative and does not give a real side length.

Do I have to enter the longest side in field c when checking?

No. In Check three sides mode, enter the lengths in any order. The calculator identifies the longest side before testing the theorem. In missing-side mode, however, c specifically means the hypotenuse, while a and b are the perpendicular legs.

Why does the calculator check the triangle inequality first?

Three positive lengths do not always form a triangle. The two shorter sides must add to more than the longest side. For example, 3, 4 and 8 fail because 3 + 4 < 8. Equality also fails to form a nondegenerate triangle. Angle classification is only meaningful after this validity check passes.

What does “not right-angled” mean in check mode?

The lengths form a valid triangle but do not satisfy the right-angle equality within the tool’s numerical tolerance. Let c be the longest side. If a² + b² > c², the triangle is acute. If a² + b² < c², it is obtuse. Check mode explains which case applies.

How does the numerical tolerance affect the right-triangle check?

The calculator divides the shorter sides by the longest side and compares (a/c)² + (b/c)² with 1. A difference of at most 10⁻¹⁰ is treated as numerical equality. This handles small floating-point differences; it is not a measurement-error allowance. Rounded physical measurements may change the result.

Can I use different units or enter fractions?

Each side can use a different unit; the calculator converts the lengths before solving or checking. The input fields accept numbers and decimals, not fraction or radical notation. Enter 0.5 for 1/2, for example. A rounded decimal approximation produces results based on that entered value.

Can one side alone determine a right triangle?

Not in general. Many right triangles can share one side length. You need another side or additional information, such as an acute angle or a known special-triangle ratio. This calculator requires two known sides for solving and three for checking.

PUT THE THEOREM TO WORK

Solve a Side or Check Your Triangle

Find a missing leg or hypotenuse from two known sides, or test three lengths for a right angle. Follow the calculation steps to understand your result.

Use Pythagorean Calculator