What Does the Pythagorean Theorem Calculator Do?
Enter two sides of a right triangle - the calculator instantly finds the third. Formula: a2 + b2 = c2, where a and b are the legs, c is the hypotenuse (longest side). Fill in two fields, leave the third empty.
How to Use - Step by Step
1
Decide what you are looking for
Know a and b? The calculator will find c (hypotenuse). Know a and c (or b and c)? It will find the missing leg.
2
Enter two known sides
Type values into fields a, b or c. Leave the third field empty - the calculator automatically detects which side to compute.
3
Read the result
The calculator returns the missing side rounded to several decimal places. Remember: c (hypotenuse) is ALWAYS the longest side.
Formula and Math
a2 + b2 = c2
Finding c: c = sqrt(a2 + b2). E.g. a = 3, b = 4 => c = sqrt(9 + 16) = sqrt(25) = 5
Finding a: a = sqrt(c2 - b2). E.g. c = 13, b = 5 => a = sqrt(169 - 25) = sqrt(144) = 12
Finding b: b = sqrt(c2 - a2). E.g. c = 10, a = 6 => b = sqrt(100 - 36) = sqrt(64) = 8
Common Pythagorean Triples
Pythagorean triples are sets of three positive integers that satisfy the Pythagorean theorem. Worth memorizing - they appear in exams and standardized tests.
| a |
b |
c |
Verification |
| 3 |
4 |
5 |
9 + 16 = 25 |
| 5 |
12 |
13 |
25 + 144 = 169 |
| 6 |
8 |
10 |
36 + 64 = 100 |
| 7 |
24 |
25 |
49 + 576 = 625 |
| 8 |
15 |
17 |
64 + 225 = 289 |
| 9 |
40 |
41 |
81 + 1600 = 1681 |
Common Mistakes
Mixing up c with a or b. The hypotenuse c is always the LONGEST side and sits opposite the right angle. If you enter c smaller than a or b, the formula gives a negative number under the square root - the calculator will show an error.
Using on non-right triangles. The Pythagorean theorem only works for triangles with a 90-degree angle. For other triangles, use the Law of Cosines.
Forgetting to square. The formula is a2 + b2 = c2, not a + b = c. For a = 3, b = 4: the result is sqrt(9 + 16) = 5, not 3 + 4 = 7.
Practical Examples
Example 1: A ladder 5 m long leans against a wall. The base is 3 m from the wall. How high does the ladder reach?
b = sqrt(52 - 32) = sqrt(25 - 9) = sqrt(16) = 4 m
Example 2: A TV screen is 48 inches wide and 27 inches tall. What is the diagonal?
c = sqrt(482 + 272) = sqrt(2304 + 729) = sqrt(3033) = 55.07 inches
Example 3: A room is 4 m by 3 m. What is the diagonal distance corner to corner?
c = sqrt(42 + 32) = sqrt(16 + 9) = sqrt(25) = 5 m
Example 4: A ramp rises 1.2 m over a horizontal distance of 5 m. How long is the ramp?
c = sqrt(52 + 1.22) = sqrt(25 + 1.44) = sqrt(26.44) = 5.14 m
Example 5: Two ships leave port at the same time. One goes 8 km north, another 6 km east. Distance between them?
c = sqrt(82 + 62) = sqrt(64 + 36) = sqrt(100) = 10 km
FAQ - Frequently Asked Questions
What is a Pythagorean triple?
A set of three positive integers (a, b, c) satisfying a2 + b2 = c2. The most common is (3, 4, 5). Others include (5, 12, 13), (8, 15, 17), and (7, 24, 25). Any multiple of a triple is also a triple - e.g. (6, 8, 10) = 2 x (3, 4, 5).
Does the theorem work for non-right triangles?
No. The Pythagorean theorem only applies to right triangles (with a 90-degree angle). For other triangles use the Law of Cosines: c2 = a2 + b2 - 2ab cos(C).
Can I use decimal numbers?
Yes. The calculator handles decimals with precision up to several decimal places. Enter values like 3.5 and 4.2 - the result will be computed correctly.
How do I check if a triangle is right-angled?
Enter all three sides. If a2 + b2 = c2 (where c is the longest side), the triangle has a right angle. The calculator shows the verification formula automatically.
What real-world uses does the Pythagorean theorem have?
Construction (checking right angles with a 3-4-5 triangle), navigation (distance between two points), architecture (roof slopes), screen sizes (diagonal measurement), GPS calculations, and surveying.
Why does the calculator show an error when c is smaller than a or b?
Because the hypotenuse must be the longest side. If c < a, then c2 - a2 is negative, and you cannot take the square root of a negative number (in real numbers). Check which side is the hypotenuse.
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