Pythagorean Theorem with Side 25 - Find the Hypotenuse

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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Side 25 - Pythagorean Theorem Calculator

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