पाइथागोरस कैलकुलेटर

Q: What is the Pythagorean theorem?

The Pythagorean theorem states that in a right triangle, the square of the hypotenuse equals the sum of the squares of the other two sides: a² + b² = c². It lets you find any missing side when you know the other two.

Q: How do I find the hypotenuse?

Use c = √(a² + b²). Square both legs, add the results, then take the square root. For example, if a = 6 and b = 8: c = √(36 + 64) = √100 = 10.

Q: Can I use this for non-right triangles?

No, the Pythagorean theorem only applies to right triangles (triangles with a 90° angle). For non-right triangles, use the Law of Cosines: c² = a² + b² − 2ab·cos(C).

Q: What are Pythagorean triples?

Pythagorean triples are sets of three positive integers that satisfy a² + b² = c². Common examples include (3, 4, 5), (5, 12, 13), (8, 15, 17), and (7, 24, 25).

Q: How do I know which side is the hypotenuse?

The hypotenuse is always the longest side and is always opposite the right angle (90°). In the formula a² + b² = c², c is always the hypotenuse.

किसी भी समकोण त्रिभुज को हल करें। दो ज्ञात भुजाएँ दर्ज करें।

पाइथागोरस कैलकुलेटर क्या है?

The Pythagorean theorem is one of the most fundamental and widely used results in all of mathematics. It states that in any right-angled triangle, the square of the hypotenuse (the side opposite the right angle) equals the sum of the squares of the other two sides: a² + b² = c².

This calculator lets you solve a right triangle in three ways: find the hypotenuse when you know both legs, find one leg when you know the other leg and the hypotenuse, or verify whether three given sides form a valid right triangle. Every result includes the area, perimeter, and a complete step-by-step derivation so you can follow the algebra.

सूत्र

Pythagorean Theorem: a² + b² = c²

Finding the hypotenuse: c = √(a² + b²)

Finding leg a: a = √(c² - b²)

Finding leg b: b = √(c² - a²)

Area: A = ½ab

Perimeter: P = a + b + c

कैसे गणना करें

Choose what you want to find: the hypotenuse or one of the legs.
Enter the two known side lengths.
The calculator applies the Pythagorean theorem and shows the full algebraic derivation.
Review the area and perimeter of the right triangle.

उदाहरण

For a right triangle with legs a = 3 and b = 4: c = √(3² + 4²) = √(9 + 16) = √25 = 5. Area = ½ × 3 × 4 = 6 square units. Perimeter = 3 + 4 + 5 = 12 units.

मुख्य लाभ

Three solving modes: find hypotenuse, find leg a, or find leg b
Animated SVG right triangle visualization with labeled sides
Full step-by-step algebraic derivation using KaTeX-rendered formulas
Instant area and perimeter calculation
Copy, share, and export results with one click

सामान्य गलतियाँ जिनसे बचना चाहिए

Applying the theorem to non-right triangles — it only works for 90° angles
Confusing which side is the hypotenuse — it is always the longest side, opposite the right angle
Subtracting when you should add (or vice versa) — remember: c² = a² + b² to find c, but a² = c² − b² to find a leg
Forgetting to take the square root after adding or subtracting the squares

प्रो टिप्स

Use Pythagorean triples (3-4-5, 5-12-13, 8-15-17) to quickly check your work
The hypotenuse is always the longest side of a right triangle
If a² + b² > c², the triangle is acute; if a² + b² < c², it is obtuse
For coordinate geometry, the distance between two points is just the hypotenuse of a right triangle formed by the horizontal and vertical differences

मुख्य शर्तें समझाई गईं

कर्ण: The longest side of a right triangle, opposite the right angle
भुजा: Either of the two shorter sides of a right triangle that form the right angle
पाइथागोरस त्रिपद: Three positive integers a, b, c that satisfy a² + b² = c²
समकोण त्रिभुज: A triangle with exactly one 90° angle
वर्गमूल: The inverse operation of squaring a number

सामान्य उपयोग के मामले

Solving geometry homework and exam problems
Construction — calculating diagonal distances
Navigation — finding straight-line distances between two points
Architecture — verifying right angles in building layouts
Physics — resolving vector components

अक्सर पूछे जाने वाले प्रश्न

What is the Pythagorean theorem?

The Pythagorean theorem states that in a right triangle, the square of the hypotenuse equals the sum of the squares of the other two sides: a² + b² = c². It lets you find any missing side when you know the other two.

How do I find the hypotenuse?

Use c = √(a² + b²). Square both legs, add the results, then take the square root. For example, if a = 6 and b = 8: c = √(36 + 64) = √100 = 10.

Can I use this for non-right triangles?

No, the Pythagorean theorem only applies to right triangles (triangles with a 90° angle). For non-right triangles, use the Law of Cosines: c² = a² + b² − 2ab·cos(C).

What are Pythagorean triples?

Pythagorean triples are sets of three positive integers that satisfy a² + b² = c². Common examples include (3, 4, 5), (5, 12, 13), (8, 15, 17), and (7, 24, 25).

How do I know which side is the hypotenuse?

The hypotenuse is always the longest side and is always opposite the right angle (90°). In the formula a² + b² = c², c is always the hypotenuse.

पाइथागोरस कैलकुलेटर

पाइथागोरस कैलकुलेटर क्या है?

सूत्र

कैसे गणना करें

उदाहरण

मुख्य लाभ

सामान्य गलतियाँ जिनसे बचना चाहिए

प्रो टिप्स

मुख्य शर्तें समझाई गईं

सामान्य उपयोग के मामले

अक्सर पूछे जाने वाले प्रश्न

संबंधित उपकरण

पाइथागोरस कैलकुलेटर

पाइथागोरस कैलकुलेटर क्या है?

सूत्र

कैसे गणना करें

उदाहरण

मुख्य लाभ

सामान्य गलतियाँ जिनसे बचना चाहिए

प्रो टिप्स

मुख्य शर्तें समझाई गईं

सामान्य उपयोग के मामले

अक्सर पूछे जाने वाले प्रश्न

संबंधित उपकरण