Pythagoras Calculator
Find any side of a right-angled triangle using Pythagoras theorem a²+b²=c². Choose which side is unknown, enter the two known sides, and get the answer with full step-by-step working. Also includes a right-triangle checker and Pythagorean triples table.
Is This a Right-Angled Triangle?
Enter all three sides to check whether they form a right-angled triangle.
What is Pythagoras Theorem?
Pythagoras theorem states that in any right-angled 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. This is written as:
Where c is the hypotenuse (longest side) and a, b are the shorter sides
This deceptively simple equation is one of the most powerful relationships in all of mathematics. It lets you find any unknown side of a right-angled triangle if you know the other two sides.
History: Pythagoras and Beyond
Although the theorem bears the name of the Greek philosopher Pythagoras of Samos (c. 570–495 BC), the relationship was known thousands of years earlier. Babylonian clay tablets from around 1800 BC list what we now call Pythagorean triples (including the 3–4–5 triangle). Ancient Egyptian rope-stretchers used the 3–4–5 relationship to create right angles when building pyramids.
The Pythagoreans are credited with the first mathematical proof of the theorem. Over 370 different proofs have been discovered throughout history, including one by US President James Garfield in 1876. The theorem is foundational to Euclidean geometry and appears in virtually every branch of mathematics.
Finding the Hypotenuse: Step-by-Step
Example 1: Find c when a = 3, b = 4
Step 1: Write the formula: c² = a² + b²
Step 2: Take the square root:
This is a classic Pythagorean triple: (3, 4, 5).
Example 2: Find c when a = 5, b = 12
Another Pythagorean triple: (5, 12, 13).
Example 3: Find c when a = 7, b = 9 (non-triple)
The exact value is √130 (an irrational number). The decimal approximation is 11.4018 to 4 decimal places.
Finding a Shorter Side: Step-by-Step
To find side a when you know b and c, rearrange a² + b² = c² to get a² = c² − b²:
Example 4: Find a when b = 5, c = 13
Validate: c must be greater than b (13 > 5 ✓)
Simplifying Surds: Exact Values
When the result is not a perfect square, we express it as a simplified surd. For example, √50 simplifies to 5√2 because 50 = 25 × 2:
| Square Root | Simplified Form | Decimal (4 d.p.) |
|---|---|---|
| √2 | √2 | 1.4142 |
| √8 | 2√2 | 2.8284 |
| √12 | 2√3 | 3.4641 |
| √18 | 3√2 | 4.2426 |
| √50 | 5√2 | 7.0711 |
| √75 | 5√3 | 8.6603 |
Pythagorean Triples
A Pythagorean triple is a set of three positive integers (a, b, c) where a² + b² = c² exactly. These triangles have whole-number sides, making them extremely useful in construction and engineering.
| Triple (a, b, c) | Verification | Scaled from |
|---|---|---|
| (3, 4, 5) | 9 + 16 = 25 ✓ | Primitive |
| (5, 12, 13) | 25 + 144 = 169 ✓ | Primitive |
| (8, 15, 17) | 64 + 225 = 289 ✓ | Primitive |
| (7, 24, 25) | 49 + 576 = 625 ✓ | Primitive |
| (6, 8, 10) | 36 + 64 = 100 ✓ | (3,4,5) × 2 |
| (9, 12, 15) | 81 + 144 = 225 ✓ | (3,4,5) × 3 |
You can generate infinitely many Pythagorean triples. Every multiple of a primitive triple is also a triple. There are also formulas: for any integers m > n > 0, the values a = m²−n², b = 2mn, c = m²+n² produce a Pythagorean triple.
Real-World Applications
Pythagoras theorem appears across many practical situations:
| Application | How Pythagoras is Used |
|---|---|
| Construction | Checking corners are square using the 3–4–5 method; finding rafter lengths |
| Navigation | Calculating straight-line distance between two GPS coordinates |
| Screen sizes | A 55-inch TV screen is the diagonal measurement (from corner to corner) |
| Surveying | Measuring distances across rivers or obstacles |
| Architecture | Calculating staircase length and height to ensure legal compliance |
| Computer graphics | Calculating distances between pixels for collision detection |
Example 5: Screen diagonal — a TV is 48 inches wide and 27 inches tall. What is the screen size?
Screen sizes are measured diagonally — a "55-inch TV" has a diagonal of approximately 55 inches.
3D Pythagoras
In three dimensions, Pythagoras theorem extends naturally. For a cuboid with length l, width w, and height h, the space diagonal d (from one corner to the opposite corner) is:
3D Pythagoras: first apply in 2D across the base, then again vertically
Example 6: Diagonal of a room 5m × 4m × 3m (length × width × height)
Step 1: Base diagonal: d₁ = √(5² + 4²) = √41 ≈ 6.403m
Step 2: Space diagonal: d = √(41 + 9) = √50 = 5√2 ≈ 7.071m
The longest straight-line distance inside the room is approximately 7.07 metres.
Pythagoras in GCSE and A-Level Maths
At GCSE (Edexcel, AQA, OCR), Pythagoras theorem appears in both Foundation and Higher tiers. Common question types include:
- Finding the hypotenuse or a shorter side in a plain right-angled triangle
- 2D problems involving coordinates (find the distance between two points)
- Problem-solving with context (ladders against walls, triangles in rectangles)
At A-Level, Pythagoras is used implicitly in:
- Vectors: the magnitude of a vector (a, b) is √(a²+b²)
- Trigonometry: the Pythagorean identity sin²θ + cos²θ = 1 comes directly from a² + b² = c² in a unit circle
- Complex numbers: the modulus |z| = √(a²+b²) for z = a + bi
Example 7: Distance between two points A(1, 2) and B(7, 10)
The horizontal distance is 7−1 = 6; the vertical distance is 10−2 = 8.
Trigonometry Connection
Pythagoras and trigonometry are deeply linked. In a right-angled triangle with hypotenuse c, the three trigonometric ratios are:
- sin θ = opposite / hypotenuse = a/c
- cos θ = adjacent / hypotenuse = b/c
- tan θ = opposite / adjacent = a/b
From sin²θ + cos²θ = (a/c)² + (b/c)² = (a²+b²)/c² = c²/c² = 1. This is the fundamental Pythagorean identity in trigonometry.
Common Mistakes to Avoid
- Using Pythagoras on non-right triangles: The theorem only works when there is a 90° angle.
- Forgetting to take the square root: After calculating a², you must find √a² = a.
- Mixing up which side is the hypotenuse: It is always the longest side, opposite the right angle.
- Subtracting wrong: When finding a shorter side, use a² = c² − b² (not a² = c² + b²).
- Not checking units: All sides must be in the same units before applying the theorem.
Example 8: A ladder 10m long leans against a wall. The base is 4m from the wall. How high up the wall does it reach?
The ladder reaches approximately 9.17m up the wall.
Frequently Asked Questions
What is Pythagoras theorem?
Pythagoras theorem states that in any right-angled triangle, a² + b² = c², where c is the hypotenuse (the side opposite the right angle) and a and b are the two shorter sides. For example, a triangle with sides 3, 4, and 5 satisfies 9 + 16 = 25. The theorem allows you to find any unknown side when two sides of a right-angled triangle are known.
How do you find the hypotenuse?
To find the hypotenuse c: (1) Square both shorter sides: a² and b². (2) Add them together: a² + b². (3) Take the square root of the result: c = √(a² + b²). Example: a = 8, b = 15 → c = √(64 + 225) = √289 = 17. Always identify the hypotenuse as the side opposite the right angle — it is always the longest side.
What are Pythagorean triples?
Pythagorean triples are sets of three positive whole numbers (a, b, c) where a² + b² = c². The most common are (3,4,5), (5,12,13), (8,15,17), and (7,24,25). Multiples of any triple are also triples: (6,8,10), (9,12,15), (10,24,26), etc. They are called "primitive" triples when a, b, c share no common factor. These are extremely useful as they produce right-angled triangles with exact integer answers.
Can Pythagoras be used for any triangle?
No. Pythagoras theorem only applies to right-angled triangles (triangles with exactly one 90° angle). For other triangles, you need the cosine rule: c² = a² + b² − 2ab·cos(C). To check whether a triangle is right-angled, verify whether a² + b² = c² (where c is the longest side). If the equation holds, it is a right-angled triangle.
How do you do 3D Pythagoras?
For a cuboid with dimensions l, w, h, the space diagonal is found in two steps: (1) Find the 2D base diagonal: d = √(l² + w²). (2) Apply Pythagoras again with the height: space diagonal = √(d² + h²) = √(l² + w² + h²). For example, a box 3m × 4m × 12m has a space diagonal of √(9+16+144) = √169 = 13m.
What is the GCSE mark scheme for Pythagoras questions?
GCSE mark schemes (Edexcel, AQA, OCR) typically award marks as follows: M1 for stating or using the correct Pythagoras relationship (a² + b² = c² or a rearrangement); M1 for correct substitution of the given values; A1 for the correct final answer rounded to the required degree of accuracy (usually 1 or 2 decimal places). Always show the squaring step explicitly — e.g. write "c² = 9 + 16 = 25" before writing "c = 5".