🔺 Advanced Pythagorean Theorem Calculator
a² + b² = c² | Calculate any side of a right triangle
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These are integer combinations that satisfy a² + b² = c²
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Pythagorean Theorem Explained
Master a² + b² = c² and learn to calculate missing sides in right‑angled triangles with proofs, real‑world examples & interactive practice.
What is the Pythagorean Theorem?
The Pythagorean Theorem states that in a right‑angled triangle, the square of the length of the hypotenuse (c) is equal to the sum of the squares of the other two sides (a and b):
a2 + b2 = c2
This relationship only holds for right triangles (one 90‑degree angle).
A quick visual proof
There are over 370 known proofs! A popular visual proof uses area: arrange four identical right triangles inside a square to show that the area of the central square is c² and equals the sum of the two smaller squares (a² and b²).
How to find the hypotenuse
If you know both legs (a & b), find c with:
c = sqrt(a2 + b2)
Example: A right triangle has legs 5 cm and 12 cm. Hypotenuse = sqrt(52 + 122) = 13 cm.
How to find a missing leg
If you know the hypotenuse c and one leg a, the other leg b is:
b = sqrt(c2 - a2)
Example: Hypotenuse 10 m, one leg 6 m. Other leg = sqrt(102 - 62) = 8 m.
Converse of the Pythagorean Theorem
If a triangle’s sides satisfy a² + b² = c², the triangle is right‑angled. You can test whether a triangle with sides 7, 24, 25 is right‑angled: 7² + 24² = 49 + 576 = 625 = 25² ⇒ right triangle.
Real‑life applications
- Construction & carpentry – ensuring walls are square.
- Navigation & GPS – calculating straight‑line distances.
- Computer graphics – distance between pixels/points.
Worked examples
Example 1 – Ladder against wall
A ladder 15 ft long reaches a wall at 12 ft above the ground. How far is the base from the wall?
Solution: b = sqrt(152 - 122) = 9 ft.
Example 2 – Roof truss
A roof truss forms a right triangle with a 20‑ft base and a 25‑ft hypotenuse. How tall is the truss?
Solution: b = sqrt(252 - 202) = 15 ft.
Practice questions
Click to reveal answers.
1. Find the hypotenuse of a right triangle with legs 9 cm and 40 cm.
sqrt(92 + 402) = 41 cm.
2. A triangle has sides 8 m, 15 m, 17 m. Is it a right triangle?
Yes. 82 + 152 = 64 + 225 = 289 = 172.
3. The diagonal of a rectangle is 26 cm and one side is 10 cm. Find the other side.
sqrt(262 - 102) = 24 cm.
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Master Geometry ConceptsFrequently Asked Questions
Does the theorem work for obtuse or acute triangles?
No. It only applies to right‑angled triangles (those with one 90° angle).
Can side lengths be decimals?
Yes. The theorem works for any real‑number lengths, including decimals and irrational numbers.
Who discovered the theorem?
It is named after the ancient Greek mathematician Pythagoras, who is traditionally credited with its proof. However, mathematical knowledge of this relationship existed in other cultures, like the Babylonians and Egyptians, centuries before Pythagoras.
What is the Pythagorean Theorem (and its formula)?
The Pythagorean Theorem states that in any right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle, typically denoted 'c') is equal to the sum of the squares of the lengths of the other two sides (the legs, typically denoted 'a' and 'b'). The formula is: a² + b² = c².
How do you do or use the Pythagorean Theorem?
To use the theorem, first identify the right angle in your triangle. The side opposite the right angle is the hypotenuse (c).
- **To find the hypotenuse (c)** when you know both legs (a and b): Square 'a', square 'b', add them together, then take the square root of the sum. Formula: c = √(a² + b²).
- **To find a missing leg (a or b)** when you know the hypotenuse (c) and one leg (say, a): Square 'c', square 'a', subtract a² from c², then take the square root of the difference. Formula: b = √(c² - a²).
Does the Pythagorean Theorem apply to all triangles?
No. The Pythagorean Theorem applies *only* to right-angled triangles (triangles that have one angle exactly 90 degrees). It does not work for acute or obtuse triangles.
Why are the side lengths squared in the Pythagorean Theorem?
The theorem is expressed with squares because it fundamentally relates to the areas of squares constructed on each side of the right triangle. Specifically, the area of the square built on the hypotenuse (c²) is equal to the sum of the areas of the squares built on the two legs (a² + b²).
What is a Pythagorean triple (or triad/triplet)?
A Pythagorean triple is a set of three positive integers (a, b, c) that perfectly satisfy the Pythagorean Theorem (a² + b² = c²). These sets represent the integer side lengths of a right-angled triangle. Common examples include (3, 4, 5) and (5, 12, 13).
What is the Converse of the Pythagorean Theorem?
The Converse of the Pythagorean Theorem states that if the square of the longest side of a triangle (c²) is equal to the sum of the squares of the other two sides (a² + b²), then the triangle must be a right-angled triangle. This is useful for determining if a given triangle is right-angled.
Who was Pythagoras, and who discovered the theorem?
Pythagoras was an influential ancient Greek mathematician and philosopher who lived around 570–495 BC. While the theorem is named after him and his school is credited with its formal proof, mathematical understanding of this relationship existed in other ancient civilizations, such as Babylonia and Egypt, centuries before Pythagoras.
What is the Pythagorean relationship?
The Pythagorean relationship refers to the fundamental mathematical connection between the three sides of a right-angled triangle, expressed by the equation a² + b² = c². It describes how the side lengths are related to each other in this specific type of triangle.
When and how is the Pythagorean Theorem used in real life?
The Pythagorean Theorem is widely used in various real-world applications:
- **Construction & Carpentry:** To ensure structures are square (e.g., walls, foundations), calculate diagonal braces, or determine ladder lengths.
- **Navigation & GPS:** To calculate the straight-line distance between two points (as the crow flies).
- **Computer Graphics:** Determining distances between points or rendering perspectives.
- **Art & Design:** For layout, perspective, and ensuring right angles.
- **Sports:** Calculating distances in fields or courts.
Can side lengths in the Pythagorean Theorem be decimals?
Yes. The theorem works for any real-number lengths, whether they are whole numbers, decimals, or irrational numbers.
How do you pronounce "Pythagorean Theorem"?
It is pronounced "Pih-THAG-uh-REE-un THEE-uh-rum".
Why is the Pythagorean Theorem called a "theorem" and not a "law"?
In mathematics, a "theorem" is a statement that has been rigorously proven true through logical deductions from a set of axioms or previously established theorems. A "law" (like a scientific law) typically describes observed phenomena and is often derived through experimentation and observation in the natural sciences. The Pythagorean relationship is a geometrically proven mathematical statement, fitting the definition of a theorem.
