Pythagorean Theorem Calculator Calculate the missing side of a right triangle using the Pythagorean theorem — with step-b...

Pythagorean Theorem Calculator

Calculate the missing side of a right triangle using the Pythagorean theorem — with step-by-step solutions and visual representations.

Find Hypotenuse (c)

Find Leg (a)

Find Leg (b)

Verify Triple

Leg a

Leg b

Decimal Places

Hypotenuse c

Leg b

Decimal Places

Hypotenuse c

Leg a

Decimal Places

Side a

Side b

Side c

a² + b² = c²
3.00² + 4.00² = 5.00²

3.00

Leg a

4.00

Leg b

5.00

Hypotenuse c

6.00

Area

12.00

Perimeter

36.87°

Angle A

53.13°

Angle B

2.40

Height to c

Pythagorean Theorem Formulas

Main Theorem

a² + b² = c²

Find c

c = √(a² + b²)

Find a

a = √(c² - b²)

Find b

b = √(c² - a²)

Interpretation

—

Right Triangle Visualization

Pythagorean Theorem Fundamentals

The Pythagorean theorem states that in a right triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.

Mathematical Formula**:

a² + b² = c²

Where:

• a, b = lengths of the legs (sides forming the right angle)
• c = length of the hypotenuse (side opposite the right angle)

Historical Context**:

Although named after the ancient Greek mathematician Pythagoras (c. 570–495 BCE), evidence suggests the theorem was known to Babylonian, Indian, and Chinese mathematicians centuries earlier.

✅ Key Insight**: This theorem only applies to right triangles (triangles with one 90° angle).

Common Pythagorean Theorem Mistakes

⚠️ Avoid these frequent errors:

• Applying to non-right triangles**: The theorem only works for right triangles
• Confusing hypotenuse**: The hypotenuse is always the longest side and opposite the right angle
• Forgetting to square**: Both sides must be squared before adding
• Not taking square root**: Remember to take the square root of the sum to find the hypotenuse
• Negative results**: Lengths cannot be negative; always use the positive square root

✅ Best Practices**:

• Always identify the right angle and hypotenuse first
• Verify your answer by plugging back into the original equation
• Use the theorem to verify if a triangle is right-angled
• Remember that the hypotenuse is always the longest side

Real-World Applications

The Pythagorean theorem is essential in:

• Construction**: Ensuring corners are square, calculating roof pitches, stair dimensions
• Navigation**: GPS systems, distance calculations, route planning
• Computer Graphics**: Collision detection, distance calculations in 2D/3D space
• Surveying**: Land measurement, elevation calculations, mapping
• Physics**: Vector calculations, force resolution, kinematics

📊 Example Use Cases**:

• Construction**: Verify a corner is square using the 3-4-5 rule
• Navigation**: Calculate straight-line distance between two GPS coordinates
• Gaming**: Determine if two objects are within a certain distance of each other
• Architecture**: Calculate diagonal measurements for room layouts

How to Use This Calculator

➡️ Find Hypotenuse (c)

"Legs 3 and 4" → Hypotenuse = 5.00

➡️ Find Leg (a)

"Hypotenuse 5, leg 4" → Missing leg = 3.00

➡️ Find Leg (b)

"Hypotenuse 5, leg 3" → Missing leg = 4.00

➡️ Verify Triple

"Sides 3, 4, 5" → Valid Pythagorean triple (3² + 4² = 5²)

Note: All calculations use the Pythagorean theorem formula. Results are rounded to the specified decimal places. The calculator validates triangle constraints automatically.