Pythagorean Theorem Calculator & Equation Solver - Free Online Tool with Examples

What is the Pythagorean Theorem?

The Pythagorean Theorem (also known as Pythagoras theorem) is one of the most fundamental equations in mathematics. This Pythagorean theorem equation solver helps you understand how the formula works: the sum of the squares of the two shorter sides (legs) equals the square of the longest side (hypotenuse). Learning how to do Pythagorean theorem is essential for geometry, trigonometry, and real-world applications.

a^2 + b^2 = c^2

Where a and b are the legs, and c is the hypotenuse

Understanding the Formula

The Legs (a and b)

The two sides that form the right angle (90°) are called legs. They are adjacent to the right angle and can be of any length, as long as they create a valid triangle.

The Hypotenuse (c)

The longest side of a right triangle, located opposite to the right angle. It is always longer than either leg and connects the endpoints of both legs.

Mathematical Proof of Pythagoras Theorem

Understanding the mathematical proof of Pythagoras theorem helps you truly grasp why a² + b² = c² always works. There are over 400 ways of proving Pythagoras theorem, but here is one of the most elegant geometric proofs:

Large square (a+b)² with inner square c²

Sum of squares on legs = a² + b²

The inner tilted square (c²) equals the total area minus the four triangles, proving a² + b² = c²

Create a square with side length (a + b)

Start with a large square where each side measures (a + b), the sum of the two legs.

Place four identical right triangles inside

Arrange four copies of the right triangle (with legs a and b) inside the square. Each triangle has area = ½ab.

The remaining space forms a square of side c

After placing the triangles, the empty space in the middle is a square with side length c (the hypotenuse).

Calculate the areas

Large square area =

(a + b)^2 = a^2 + 2ab + b^2

Four triangles area =

4 \times \frac{1}{2}ab = 2ab

Inner square area =

c^2

Set up and solve the equation

Large square = Four triangles + Inner square

a^2 + 2ab + b^2 = 2ab + c^2

a^2 + b^2 = c^2 \quad \checkmark

Pythagorean Theorem Examples with Solutions

Here are detailed Pythagorean theorem examples showing exactly how to solve problems step by step. These theorem of Pythagoras examples cover the most common problem types.

Pythagorean Theorem Formula Example 1: Finding the Hypotenuse

Given: $a = 3$ , $b = 4$ . Find $c$ .

Step 1:

a^2 + b^2 = c^2

Step 2:

3^2 + 4^2 = c^2

Step 3:

9 + 16 = c^2

Step 4:

25 = c^2

Step 5:

c = \sqrt{25} = 5

Example of a Pythagorean Theorem 2: Finding a Leg

Given: $a = 5$ , $c = 13$ . Find $b$ .

Step 1:

a^2 + b^2 = c^2

Step 2:

b^2 = c^2 - a^2

Step 3:

b^2 = 13^2 - 5^2

Step 4:

b^2 = 169 - 25 = 144

Step 5:

b = \sqrt{144} = 12

Pythagorean Theorem Word Problems

Pythagorean theorem word problems apply the formula to real-life situations. Here are common Pythagorean theorem problems you might encounter:

Problem 1: The Ladder Problem

A ladder leans against a wall. The base is 6 feet from the wall, and it reaches 8 feet up the wall. How long is the ladder?

a = 6

ft (distance from wall)

b = 8

ft (height on wall)

c^2 = 6^2 + 8^2 = 36 + 64 = 100

c = \sqrt{100} = 10

feet

Problem 2: The TV Size Problem

A TV screen is 48 inches wide and 36 inches tall. What is the diagonal measurement (advertised size)?

a = 48

in (width)

b = 36

in (height)

c^2 = 48^2 + 36^2 = 2304 + 1296 = 3600

c = \sqrt{3600} = 60

inches

Problem 3: The Walking Distance Problem

You walk 9 blocks east and 12 blocks north. How far are you from your starting point (straight line)?

a = 9

blocks (east)

b = 12

blocks (north)

c^2 = 9^2 + 12^2 = 81 + 144 = 225

c = \sqrt{225} = 15

blocks

Triples Pythagorean Theorem - Complete Reference

Triples Pythagorean theorem (or Pythagorean triples) are sets of three positive integers (a, b, c) that perfectly satisfy $a^2 + b^2 = c^2$ . These special number sets are valuable because all values are whole numbers, making calculations exact without decimals.

a	b	c	Verification
3	4	5	9 + 16 = 25 ✓
5	12	13	25 + 144 = 169 ✓
8	15	17	64 + 225 = 289 ✓
7	24	25	49 + 576 = 625 ✓
9	40	41	81 + 1600 = 1681 ✓
11	60	61	121 + 3600 = 3721 ✓
20	21	29	400 + 441 = 841 ✓

Tip: If (a, b, c) is a Pythagorean triple, then any multiple (ka, kb, kc) is also a triple. For example, (6, 8, 10) = 2 × (3, 4, 5).

Converse of the Pythagorean Theorem

The converse of the Pythagorean theorem is equally important and works in reverse: If the sum of the squares of two sides of a triangle equals the square of the third side, then the triangle is a right triangle.

The Converse Statement:

If $a^2 + b^2 = c^2$ for the three sides of a triangle, then the angle opposite to side c is a right angle (90°).

Practical Use: Construction workers use the converse to verify right angles. By measuring 3 feet along one edge, 4 feet along another, and checking if the diagonal is exactly 5 feet, they confirm a perfect 90° corner.

Extended Triangle Classification:

$a^2 + b^2 = c^2$ → Right triangle (90° angle)
$a^2 + b^2 > c^2$ → Acute triangle (all angles < 90°)
$a^2 + b^2 < c^2$ → Obtuse triangle (one angle > 90°)

Pythagorean Theorem Identities

Pythagorean theorem identities extend the basic formula into related mathematical concepts, especially in trigonometry.

Trigonometric Pythagorean Identity

\sin^2\theta + \cos^2\theta = 1

This fundamental trig identity comes directly from the Pythagorean theorem applied to a unit circle.

Related Trigonometric Identities

1 + \tan^2\theta = \sec^2\theta

1 + \cot^2\theta = \csc^2\theta

These identities are derived from dividing the main identity by $\cos^2\theta$ and $\sin^2\theta$ respectively.

Distance Formula (2D)

d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}

Finding the distance between two points is a direct application of the Pythagorean theorem.

3D Distance Formula

d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2 + (z_2-z_1)^2}

Extended to three dimensions by applying the theorem twice.

Real-World Applications

Construction & Architecture

Ensuring walls are perpendicular, calculating roof pitches, and determining diagonal bracing lengths.

Navigation & GPS

Calculating distances between two points, determining shortest paths, and triangulating positions.

Computer Graphics

Calculating distances between pixels, determining collision detection, and rendering 3D objects.

Surveying & Mapping

Measuring land areas, determining property boundaries, and calculating elevations.

Historical Background

Although named after the Greek mathematician Pythagoras of Samos (c. 570 - c. 495 BCE), evidence suggests that the relationship was known to the Babylonians and Indians centuries earlier. Clay tablets from ancient Babylon (c. 1800 BCE) show that they used the theorem for practical calculations.

Pythagoras and his followers (the Pythagoreans) are credited with providing the first known proof of the theorem, elevating it from an empirical observation to a mathematical truth. Since then, over 400 different proofs have been discovered, including ones by Leonardo da Vinci, President James Garfield, and even a visual proof using just folding paper.

Calculator.

Pythagorean Theorem Calculator.

What is the Pythagorean Theorem?

Understanding the Formula

The Legs (a and b)

The Hypotenuse (c)

Mathematical Proof of Pythagoras Theorem

Pythagorean Theorem Examples with Solutions

Pythagorean Theorem Formula Example 1: Finding the Hypotenuse

Example of a Pythagorean Theorem 2: Finding a Leg

Pythagorean Theorem Word Problems

Problem 1: The Ladder Problem

Problem 2: The TV Size Problem

Problem 3: The Walking Distance Problem

Triples Pythagorean Theorem - Complete Reference

Converse of the Pythagorean Theorem

Extended Triangle Classification:

Pythagorean Theorem Identities

Trigonometric Pythagorean Identity

Related Trigonometric Identities

Distance Formula (2D)

3D Distance Formula

Real-World Applications

Construction & Architecture

Navigation & GPS

Computer Graphics

Surveying & Mapping

Historical Background

Frequently Asked Questions

Calculator.

Calculators

QUICK LINKS

Contact Us

🇨🇦 Kitchener, Ontario

Follow Us

Calculator.

What is the Pythagorean Theorem?

Understanding the Formula

The Legs (a and b)

The Hypotenuse (c)

Mathematical Proof of Pythagoras Theorem

Pythagorean Theorem Examples with Solutions

Pythagorean Theorem Formula Example 1: Finding the Hypotenuse

Example of a Pythagorean Theorem 2: Finding a Leg

Pythagorean Theorem Word Problems

Problem 1: The Ladder Problem

Problem 2: The TV Size Problem

Problem 3: The Walking Distance Problem

Triples Pythagorean Theorem - Complete Reference

Converse of the Pythagorean Theorem

Extended Triangle Classification:

Pythagorean Theorem Identities

Trigonometric Pythagorean Identity

Related Trigonometric Identities

Distance Formula (2D)

3D Distance Formula

Real-World Applications

Construction & Architecture

Navigation & GPS

Computer Graphics

Surveying & Mapping

Historical Background

Frequently Asked Questions

1. How to do Pythagorean theorem step by step?

2. What is a Pythagorean theorem example with solution?

3. What are Pythagorean triples (triples Pythagorean theorem)?

4. What is the mathematical proof of Pythagoras theorem?

5. What is the converse of the Pythagorean theorem?

6. How to solve Pythagorean theorem word problems?

7. What are Pythagorean theorem identities?

8. Can Pythagorean theorem equation solver find any missing side?

9. What are common Pythagorean theorem problems for practice?

10. Is there a Pythagorean theorem worksheet for practice?

11. How is proving Pythagoras theorem done geometrically?

12. What's the difference between Pythagoras theorem calculator and manual calculation?

Related Calculators

Calculator.

Calculators

QUICK LINKS

Contact Us

🇨🇦 Kitchener, Ontario

Follow Us