The Pythagorean theorem is the cornerstone of right-triangle geometry: for any right triangle, the square of the hypotenuse equals the sum of the squares of the two legs (a² + b² = c²). This calculator solves for any missing side — hypotenuse or either leg — and also doubles as a 2D distance calculator using the same identity. It shows full step-by-step algebra, computes triangle area (½ab), perimeter, the acute angles via arctangent, and flags exact Pythagorean triples like 3-4-5, 5-12-13, 8-15-17, 7-24-25, and 9-40-41. Useful for homework, carpentry and framing, squaring a foundation, navigation, physics vector magnitudes, screen-diagonal sizing, and any engineering task that reduces to a right triangle.
Rearrange to solve for the unknown: c = √(a² + b²), a = √(c² − b²), or b = √(c² − a²). When solving for a leg, the hypotenuse must be larger than the known leg.
a² + b² = c²
The hypotenuse is always the longest side — it's opposite the 90° angle.
Common Pythagorean triples: 3-4-5, 5-12-13, 8-15-17, 7-24-25, 9-40-41, 20-21-29.
Any multiple of a triple is also a triple (e.g., 6-8-10, 9-12-15).
To verify a right angle on-site, measure 3 ft, 4 ft, and 5 ft — if the diagonal is exactly 5, the corner is square.
The theorem extends to 3D: d = √(x² + y² + z²) for diagonals of a box.
Distance between two points (x₁,y₁) and (x₂,y₂): d = √((x₂−x₁)² + (y₂−y₁)²).
Solve any right triangle in seconds. Enter two sides, pick the unknown, and get the missing side along with area, perimeter, angles, and a full step-by-step solution. Also handles 2D distance between two points.
Anytime you have a right triangle and know two of the three sides.
Common real-world uses include:
Integer solutions to a² + b² = c².
The primitive triples include:
Any scalar multiple is also a valid triple, which is why 6-8-10 and 9-12-15 work too.
The hypotenuse is the longest side of a right triangle, opposite the 90° angle, and you find it with c = √(a² + b²). Square each leg, add the two squares, then take the square root.
For legs of 6 and 8: c = √(6² + 8²) = √(36 + 64) = √100 = 10. For legs of 5 and 12: c = √(25 + 144) = √169 = 13. The result is always larger than either leg but smaller than their sum.
Set the calculator mode to 'Hypotenuse c (given legs a and b)', enter both legs in the same unit, and it returns c plus the triangle's area, perimeter, and acute angles.
This is the most common use of the theorem — squaring a corner, sizing a diagonal brace, or finding a ramp's slope length from its rise and run.
When you know the hypotenuse and one leg, rearrange the theorem to solve for the missing leg: a = √(c² − b²). Subtract the known leg's square from the hypotenuse's square, then take the square root.
For a hypotenuse of 13 and a leg of 5: a = √(13² − 5²) = √(169 − 25) = √144 = 12. The hypotenuse must always be larger than the known leg — if c ≤ b, no real right triangle exists and the calculator will flag it.
Choose the 'Leg a' or 'Leg b' mode, enter the hypotenuse and the known leg, and the tool returns the missing side with the full c² − b² working shown step by step.
This is how you find a ladder's safe base distance from its length and the wall height it reaches.
The 2D distance formula d = √((x₂ − x₁)² + (y₂ − y₁)²) is the Pythagorean theorem in disguise: the horizontal gap (x₂ − x₁) and vertical gap (y₂ − y₁) are the two legs of a right triangle, and the straight-line distance is the hypotenuse.
From (1, 2) to (4, 6): d = √((4−1)² + (6−2)²) = √(9 + 16) = √25 = 5. Switch the calculator to 'Distance between two 2D points', enter the four coordinates, and it returns the exact distance.
This underpins:
Once all three sides are known, a right triangle's area and perimeter follow immediately. Because the two legs meet at 90°, they act as base and height, so the area is simply ½ × a × b — no trigonometry needed.
A 3-4-5 triangle has area ½ × 3 × 4 = 6 square units and perimeter 3 + 4 + 5 = 12. A 6-8-10 triangle has area ½ × 6 × 8 = 24 and perimeter 24.
This calculator reports area, perimeter, and both acute angles (found with arctangent, e.g. angle A = arctan(a/b)) alongside the missing side, giving a complete solution of the triangle from just two inputs. For a triangle that has no right angle, our triangle calculator solves the same area, perimeter, and angles using the law of sines and cosines.
The theorem turns two everyday measurements into a diagonal.
A TV or monitor's screen size is its diagonal: a screen 1.21 m wide and 0.68 m tall measures √(1.21² + 0.68²) ≈ 1.39 m, about a 55-inch TV.
A rectangle's diagonal is the hypotenuse of the triangle formed by its length and width, so a 16:9 frame's corner-to-corner length is √(16² + 9²) ≈ 18.36 units.
Carpenters square a foundation or deck with the 3-4-5 method: measure 3 ft along one edge, 4 ft along the perpendicular edge, and if the diagonal between those marks is exactly 5 ft, the corner is a true right angle. Scaling up to 6-8-10 or 9-12-15 gives more accuracy over longer walls.
The converse of the Pythagorean theorem lets you test whether a triangle has a right angle:
This is how you confirm a corner is square from three measured lengths alone.
The theorem also extends into three dimensions: the space diagonal of a rectangular box with edges x, y, and z is d = √(x² + y² + z²), derived by applying the theorem twice — first across the base, then up to the opposite corner.
A 3 × 4 × 12 box has a space diagonal of √(9 + 16 + 144) = √169 = 13 units, useful for checking whether a long item fits diagonally inside a carton.
Worked example for a hypotenuse: legs 9 and 12 → 9² + 12² = 81 + 144 = 225 → c = √225 = 15.
Worked example for a leg: hypotenuse 25, leg 7 → 25² − 7² = 625 − 49 = 576 → a = √576 = 24.
The calculator displays each of these steps so you can follow or check the algebra by hand.
Data sourced from trusted institutions
All formulas verified against official standards.