Pythagorean Theorem Calculator

About Pythagorean Theorem Calculator

Find the missing side of a right triangle using the Pythagorean theorem (a² + b² = c²). Enter any two sides to calculate the third, or enter all three sides to check if they form a right triangle. Every calculation shows a full step-by-step solution with an interactive diagram.

The Pythagorean Theorem

In any right 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 gives three formulas depending on which side you are solving for:

Find	Formula	Example (a=3, b=4)
Hypotenuse (c)	c = √(a² + b²)	c = √(9 + 16) = √25 = 5
Leg a	a = √(c² - b²)	a = √(25 - 16) = √9 = 3
Leg b	b = √(c² - a²)	b = √(25 - 9) = √16 = 4

Worked Examples

Example 1: A ladder leans against a wall. The base is 6 feet from the wall and the ladder is 10 feet long. How high does it reach?

• a = 6 (base), c = 10 (ladder = hypotenuse)
• b = √(10² - 6²) = √(100 - 36) = √64 = 8 feet

Example 2: A TV is advertised as 55 inches (diagonal). It is 48 inches wide. What is the height?

• c = 55 (diagonal), a = 48 (width)
• b = √(55² - 48²) = √(3025 - 2304) = √721 = 26.9 inches

Example 3: Walking diagonally across a rectangular field that is 300m by 400m:

• Diagonal = √(300² + 400²) = √(90,000 + 160,000) = √250,000 = 500m
• The diagonal saves 200m compared to walking two sides (300 + 400 = 700m)

Common Pythagorean Triples

Pythagorean triples are sets of three whole numbers that satisfy a² + b² = c². Memorizing a few helps with quick mental calculations:

Triple	a²	b²	c²	Common Multiples
3 - 4 - 5	9	16	25	6-8-10, 9-12-15, 12-16-20, 15-20-25
5 - 12 - 13	25	144	169	10-24-26, 15-36-39
8 - 15 - 17	64	225	289	16-30-34
7 - 24 - 25	49	576	625	14-48-50
20 - 21 - 29	400	441	841	40-42-58
9 - 40 - 41	81	1600	1681	18-80-82

Any multiple of a Pythagorean triple is also a triple. The 3-4-5 triple (and its multiples) is by far the most commonly used in construction and everyday calculations.

The 3-4-5 Rule in Construction

Builders use the 3-4-5 triple to check right angles. To verify a corner is square: measure 3 feet along one wall, 4 feet along the other, and the diagonal should be exactly 5 feet. For larger areas, use multiples like 6-8-10 or 9-12-15 for better accuracy. This technique dates back thousands of years - ancient Egyptian rope-stretchers used knotted ropes in 3-4-5 ratios to survey land along the Nile.

Brief History

The theorem is named after the Greek mathematician Pythagoras (~570-495 BC), but the relationship was known much earlier. A Babylonian clay tablet (Plimpton 322, ~1800 BC) lists 15 Pythagorean triples. Ancient Indian and Chinese mathematicians also knew the theorem. Over 400 distinct proofs have been published, including one by US President James Garfield (1876) using a trapezoid construction.

Beyond Right Triangles

For non-right triangles, the Pythagorean theorem does not apply directly. Instead, use:

• Acute triangle (all angles less than 90°): a² + b² > c² for all sides
• Obtuse triangle (one angle greater than 90°): a² + b² < c² where c is opposite the obtuse angle
• The law of cosines generalizes the theorem: c² = a² + b² - 2ab cos(C). When C = 90°, cos(C) = 0 and you get the Pythagorean theorem

For general triangle solving, the triangle calculator handles SSS, SAS, and ASA modes. For function plotting including geometric shapes, the graphing calculator provides a full coordinate system.

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