What is the Pythagorean Theorem?

The Pythagorean Theorem is a fundamental geometric relationship among the three sides of any right triangle. It states that the square of the hypotenuse (the side opposite the 90° right angle, c) is equal to the sum of the squares of the other two legs (a and b): a² + b² = c².

How do you calculate the hypotenuse (c)?

To find the hypotenuse c given the lengths of legs a and b, take the square root of the sum of their squares: c = √(a² + b²). For example, if leg a = 3 and leg b = 4, then c = √(3² + 4²) = √(9 + 16) = √25 = 5.

How do you calculate a missing leg (a or b)?

To find a missing leg given hypotenuse c and known leg b, subtract the square of the known leg from the square of the hypotenuse and take the square root: a = √(c² − b²). The hypotenuse c must always be strictly greater than either leg.

What is the 3-4-5 rule in construction and carpentry?

The 3-4-5 rule is a practical application of the Pythagorean Theorem used by builders, carpenters, and landscapers to verify that a corner is exactly square (90°). By measuring 3 units along one wall, 4 units along the perpendicular wall, and checking if the diagonal distance between the two marks measures exactly 5 units (or multiples like 6-8-10 or 9-12-15), builders ensure a true right angle.

What is the Converse of the Pythagorean Theorem?

The Converse states that if the side lengths of a triangle satisfy a² + b² = c² (where c is the longest side), the triangle must be a right triangle. If a² + b² > c², the triangle is acute (all angles 90°).

What is a Pythagorean triple?

A Pythagorean triple consists of three positive integers (a, b, c) that satisfy a² + b² = c². When the greatest common divisor gcd(a, b, c) = 1, it is called a 'primitive' Pythagorean triple. Famous examples include (3, 4, 5), (5, 12, 13), (8, 15, 17), and (7, 24, 25).

How does the Pythagorean Theorem extend to 3D space?

In three dimensions, the theorem calculates the space diagonal (or Euclidean distance) across a rectangular prism or box: d = √(x² + y² + z²). This is derived by applying Pythagoras twice: first along the 2D floor diagonal (d_base = √(x² + y²)), and then vertically to the opposite top corner.

Is this Pythagorean Theorem Calculator free to use?

Yes. This Pythagorean theorem calculator is 100% free, requires no login or download, and provides step-by-step mathematical solutions, exact simplified radicals, 3D diagonals, and triangle classification.

Geometry & Trigonometry

Pythagorean Theorem Calculator

Calculate hypotenuse (c), legs (a or b), angles, area, perimeter, and 3D space diagonals using the Pythagorean theorem. Includes simplified radical expressions, step-by-step mathematical derivations, dynamic interactive triangle diagrams, and Pythagorean triples.

Solve Right Triangle

What do you want to find?

Common Presets

Leg (a)Vertical side

Leg (b)Horizontal base

Unit (Optional)

Decimals

The Pythagorean Equation:

a² + b² = c²

c = √(a² + b²)

Calculated Hypotenuse (c)

5.0000

Exact: 5

Step-by-Step Mathematical Derivation:

1. Apply Theorem: c = √(a² + b²)

2. Substitute: c = √(3² + 4²)

3. Square terms: c = √(9.00 + 16.00)

4. Sum: c = √(25.00)

5. Result: c = 5.0000 (5)

Dynamic Right Triangle Diagram

α + β = 90.0°•γ = 90° (Right Angle)

Complete Right Triangle Geometry Properties

Area

6.0000

½ × a × b

Perimeter

12.0000

a + b + c

Altitude to Hypotenuse

2.4000

hc = (a × b) / c

Incircle Radius (r)

1.0000

(a + b − c) / 2

Circumcircle Radius (R)

2.5000

c / 2

Acute Angles

36.87° & 53.13°

0.644 & 0.927 rad

The Fundamental Principles of the Pythagorean Theorem

Named after the ancient Greek philosopher and mathematician Pythagoras of Samos (c. 570–495 BC), the Pythagorean Theorem is one of the most foundational principles in all of mathematics. It establishes an exact relationship between the three sides of any right-angled Euclidean triangle:

The Standard Equation

a² + b² = c²

Where a and b represent the lengths of the two perpendicular legs forming the 90° right angle, and c represents the hypotenuse (the longest side opposite the right angle).

Remarkably, evidence of this geometric insight predates Pythagoras by centuries. Babylonian cuneiform tablets such as Plimpton 322 (c. 1800 BC), Indian Vedic Sulba Sutras (c. 800 BC), and the ancient Chinese mathematical treatise Zhoubi Suanjing all documented practical methods for generating Pythagorean triples long before ancient Greek geometry was formalized.

The Visual Geometric Proof of Pythagoras

The most intuitive proof of the theorem relies upon the conservation of geometric area. Imagine a large square with side length (a + b):

Arrangement 1: Two Inner Squares

Place four identical right triangles (each with area ½ab) into the corners of the (a + b) square. The remaining unfilled central area consists of two smaller squares: one of area a² and one of area b². The total empty area is a² + b².

Arrangement 2: One Tilted Square

Now rearrange the four identical right triangles along the perimeter so their hypotenuses face inward. The central empty space now forms a single tilted square with side length c and area c². Because the total bounding area and the four triangles remain unchanged, it follows algebraically that a² + b² = c².

The Converse of the Pythagorean Theorem: Triangle Classification

While the theorem proves that every right triangle satisfies a² + b² = c², the Converse works in reverse: if three positive lengths satisfy the equation, the triangle must be a right triangle. Furthermore, by comparing a² + b² against c² (where c is the longest side), you can instantly classify any valid triangle:

Condition	Triangle Type	Largest Angle	Example Sides
a² + b² = c²	Right Triangle	Exactly 90°	3, 4, 5 (9 + 16 = 25)
a² + b² > c²	Acute Triangle	< 90° (All angles acute)	5, 6, 7 (25 + 36 = 61 > 49)
a² + b² < c²	Obtuse Triangle	> 90° (One obtuse angle)	4, 5, 8 (16 + 25 = 41 < 64)

Practical Real-World Applications

The Pythagorean Theorem is an everyday practical tool utilized across engineering, architecture, sports, and technology:

Construction & Framing (3-4-5)

Carpenters square building foundations, deck framing, and tile layouts by measuring 3 feet along one side, 4 feet along the other, and ensuring the diagonal connects at exactly 5 feet.

Screen Dimensions (TVs & Monitors)

Display manufacturers advertise diagonal screen measurements. A 55-inch TV with a 16:9 widescreen ratio has a width of approximately 47.9 inches and a height of 27.0 inches.