Pythagorean theorem calculator
Missing hypotenuse or leg, exact radical form, angles, area and diagram.
- Instant
- Free
- Private (processed locally)
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Find any side of a right triangle with the Pythagorean theorem
The Pythagorean theorem links the three sides of a right triangle: a² + b² = c², where c is the hypotenuse. Know two sides and you can find the third. This calculator does it instantly, whether you need the hypotenuse or a missing leg, and gives the exact value (such as 5√2) alongside the decimal approximation.
It is one of the most widely used results in mathematics, from grade 8 in the US Common Core and Key Stage 3 or GCSE in the UK up to university: students checking homework, builders squaring a corner with the 3-4-5 rule, physics students adding perpendicular vectors. The tool also returns the angles, area, perimeter, altitude, projections, inradius and circumradius, a scale drawing and a written step-by-step solution.
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Choose the unknown
Click “Find hypotenuse c” (the default), “Find leg a” or “Find leg b”. Legs a and b form the right angle.
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Enter the two known lengths
Fill in the two visible fields (Leg a, Leg b or Hypotenuse c); decimals are fine. The Unit menu adds the chosen unit (cm, m, ft…) to every result, and the area is shown in square units (cm²).
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Read the result
The missing side appears in large type, with its exact value (such as √50 = 5√2) when the squared result is a whole number, and a badge when the three sides form a Pythagorean triple. The “Copy” button puts the answer on your clipboard.
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Check the drawing and the steps
The triangle is drawn to scale with the computed side dashed. Open “Step-by-step solution” to copy the working: formula, substitution, square root and, when needed, simplifying the radical.
The Pythagorean formula and what the tool computes
In a right triangle, the square of the hypotenuse equals the sum of the squares of the other two sides: c² = a² + b². For a leg, isolate its square: a = √(c² − b²). Every other measurement then follows from the three sides:
- a and b: the legs that form the right angle; c: the hypotenuse, the longest side, opposite the right angle.
- Acute angles:
tan α = a / bandβ = 90° − α, shown in degrees (2 decimals) and radians. α is the angle opposite leg a. - Area and perimeter:
A = a × b / 2(the legs act as base and height) andP = a + b + c. - Altitude to the hypotenuse:
h = a × b / c, found by writing the area two ways. - Projections onto the hypotenuse:
a² / candb² / c; their product equalsh²(for 3, 4, 5: 1.8 × 3.2 = 5.76 = 2.4²), the geometric mean relation. - Circles: inradius
r = (a + b − c) / 2; circumradiusR = c / 2, because the hypotenuse is a diameter of the circumscribed circle (Thales’s theorem).
When the squared result is a whole number that is not a perfect square, the tool simplifies the radical by pulling out the largest square factor: √50 = √(25 × 2) = 5√2. The square of the unknown side is computed with exact decimal arithmetic, so 0.3 and 0.4 give exactly c = 0.5 with no floating-point noise.
Pythagorean theorem worked examples
| Unknown | Given | Tool output | Takeaway |
|---|---|---|---|
| c | a = 3, b = 4 | c = 5 | primitive triple (3, 4, 5), area 6, α ≈ 36.87° |
| c | a = 5, b = 12 | c = 13 | primitive triple, altitude h ≈ 4.615385 |
| c | a = 1, b = 1 | c = √2 ≈ 1.414214 | isosceles right triangle (half a square): α = β = 45° |
| c | a = 5, b = 5 | c = √50 = 5√2 ≈ 7.071068 | area 12.5 |
| b | c = 10, a = 6 | b = 8 | (6, 8, 10) = 2 × (3, 4, 5), not primitive |
| b | c = 2, a = 1 | b = √3 ≈ 1.732051 | 30-60-90 triangle: α = 30°, β = 60° |
Real-life uses of the Pythagorean theorem
- Construction and DIY: the 3-4-5 rule checks a right angle; measure 60 cm and 80 cm along two walls and the diagonal must be exactly 100 cm.
- Screens: a TV panel measuring 120 × 67.5 cm has a diagonal of 137.681698 cm, about 54.21 inches (1 in = 2.54 cm).
- Ladders: a 5 m ladder with its foot 1.5 m from the wall reaches √22.75 ≈ 4.769696 m up the wall.
- Ramps and roofs: a ramp running 6 m horizontally and rising 0.5 m is 6.020797 m long, at an angle of 4.76°. For a staircase, the stair calculator also gives the number of risers and the opening.
- Coordinates and physics: two points 3 units apart in x and 4 units apart in y are 5 units apart (the distance formula); perpendicular forces of 30 N and 40 N have a 50 N resultant.
Common mistakes
- Adding the lengths instead of the squares: c is not a + b (3 + 4 = 7, yet c = 5).
- Adding when you should subtract: for a leg, compute
c² − b². If that is zero or negative, the hypotenuse you entered is too short. - Mixing units: legs of 1 m and 50 cm must first become 100 cm and 50 cm, which gives c = √12500 = 50√5 ≈ 111.803399 cm.
- Rounding too early: 7.07² = 49.9849, not 50. Keep the exact value 5√2 until the end, then round.
Every calculation runs locally in your browser: no data is sent anywhere. Limits: lengths between 10⁻¹⁰⁰ and 10¹⁰⁰, results rounded to 6 decimals (angles to 2), exact radical shown when the squared result is a whole number below 2⁵³. The drawing is no longer to scale when one side is much shorter than the other.
Frequently asked questions
What is the Pythagorean theorem formula?
In a right triangle with legs a and b and hypotenuse c, a² + b² = c². For a = 3 and b = 4: c² = 9 + 16 = 25, so c = √25 = 5. The hypotenuse is always the side opposite the right angle and the longest of the three.
How do I find a missing leg?
Isolate its square: b² = c² − a², then b = √(c² − a²). With c = 10 and a = 6: b² = 100 − 36 = 64, so b = 8. The hypotenuse must be strictly longer than the known leg: with c = 5 and a = 5 no triangle exists, and the tool shows an error.
Why does the tool show √50 = 5√2?
That is the exact answer teachers expect. For a = b = 5, c² = 50 = 25 × 2, so c = √25 × √2 = 5√2 ≈ 7.071068. The exact radical appears whenever the squared result is a whole number that is not a perfect square (√2, √3, 5√2…); the decimal is only a rounded value.
What is a Pythagorean triple?
Three positive integers with a² + b² = c², such as (3, 4, 5), (5, 12, 13), (8, 15, 17) or (7, 24, 25). A triple is primitive when the three numbers have no common factor other than 1 (GCD = 1): (6, 8, 10) = 2 × (3, 4, 5) is not. Euclid’s formula (m² − n², 2mn, m² + n²), with whole numbers m > n > 0, produces as many as you like: m = 3 and n = 2 give (5, 12, 13).
How can I tell whether a triangle is a right triangle?
Use the converse of the theorem: compare the square of the longest side with the sum of the squares of the other two. For 5, 12, 13: 25 + 144 = 169 = 13², so it is a right triangle. For 4, 5, 6: 16 + 25 = 41 ≠ 36, so it is not (and since 36 < 41, all its angles are acute).
How are the angles and the altitude calculated?
With trigonometry: tan α = a / b, then β = 90° − α. For 3, 4, 5: α = arctan(0.75) ≈ 36.87° and β ≈ 53.13°. The altitude to the hypotenuse comes from writing the area two ways (a × b = c × h), so h = 12 / 5 = 2.4.