The Pythagorean Theorem
The Pythagorean Theorem is one of the most fundamental results in all of mathematics. It connects the three sides of a right triangle with a simple, elegant equation that has been known for over 2,500 years.
This guide covers the theorem itself, Pythagorean triples, the converse for classifying triangles, the distance formula, 3D extensions, worked examples, and a practice quiz.
1Introduction
Named after the ancient Greek mathematician Pythagoras (c. 570 - 495 BC), the Pythagorean Theorem describes the relationship between the sides of a right triangle. Although Pythagoras is credited with its proof, the relationship was known to Babylonian and Indian mathematicians centuries earlier.
Construction workers use 3-4-5 triangles to verify right angles. Pilots calculate direct distances between waypoints. Screen manufacturers specify diagonal measurements. GPS systems compute distances from coordinate differences.
Construction & Architecture
The 3-4-5 rule ensures walls are perfectly square. Carpenters, masons, and surveyors rely on it daily.
Navigation & Computing
The distance formula (derived from the theorem) powers everything from GPS to video game physics engines.
2Key Definitions
Right Triangle
A triangle with one angle that measures exactly 90°.
Hypotenuse
The side opposite the right angle; always the longest side of a right triangle.
Legs
The two sides that form the right angle (often labeled a and b).
Pythagorean Triple
A set of three positive integers (a, b, c) satisfying a² + b² = c².
Converse
If a² + b² = c², then the triangle is a right triangle.
Distance Formula
d = √((x&sub2;−x&sub1;)² + (y&sub2;−y&sub1;)²), derived from the Pythagorean Theorem.
3The Theorem
In any right triangle, the square of the hypotenuse equals the sum of the squares of the two legs.
a² + b² = c²
where a and b are legs, and c is the hypotenuse (longest side, opposite the right angle).
Interactive Triangle Explorer
Drag the ↕ and ↔ handles to change the legs (0–200). The hypotenuse updates in real time.
Three Ways to Use the Formula
Find hypotenuse: c = √(a² + b²)
Find leg a: a = √(c² − b²)
Find leg b: b = √(c² − a²)
Always identify the hypotenuse first. It is the side opposite the right angle and is always the longest side. The two legs form the right angle.
4Pythagorean Triples
Pythagorean triples are sets of three positive integers that satisfy a² + b² = c². Memorizing common triples allows you to solve problems instantly without a calculator.
Common Pythagorean Triples
| Triple (a, b, c) | Verification | Common Multiples |
|---|---|---|
| (3, 4, 5) | 9 + 16 = 25 | (6,8,10), (9,12,15), (12,16,20) |
| (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) |
Any multiple of a Pythagorean triple is also a triple. If (3, 4, 5) works, then so does (6, 8, 10), (30, 40, 50), etc. To check, divide all three sides by their GCF and see if you recognize a base triple.
5The Converse
The converse of the Pythagorean Theorem lets you classify a triangle by comparing a² + b² to c² (where c is the longest side).
a² + b² = c²
Right Triangle
a² + b² > c²
Acute Triangle
a² + b² < c²
Obtuse Triangle
Example: Classify the triangle with sides 6, 9, 11
Step 1: Identify longest side: c = 11
Step 2: Compute a² + b² = 6² + 9² = 36 + 81 = 117
Step 3: Compute c² = 11² = 121
Step 4: Compare: 117 < 121, so a² + b² < c²
Answer: The triangle is obtuse.
6Distance Formula
The distance formula is a direct application of the Pythagorean Theorem on the coordinate plane. The horizontal and vertical differences between two points form the legs of a right triangle, and the distance between the points is the hypotenuse.
2D Distance Formula
d = √((x&sub2; − x&sub1;)² + (y&sub2; − y&sub1;)²)
3D Extension
d = √((x&sub2; − x&sub1;)² + (y&sub2; − y&sub1;)² + (z&sub2; − z&sub1;)²)
The distance between A(1,1) and B(7,7) is the hypotenuse of the right triangle formed by the horizontal and vertical legs.
The midpoint formula and the distance formula are a powerful pair. Use the distance formula to find lengths and the midpoint formula to find centers. Both are rooted in the Pythagorean Theorem.
7Worked Examples
Example 1: Finding the Hypotenuse
A right triangle has legs of length 5 and 12. Find the hypotenuse.
Step 1: Write the formula: a² + b² = c²
Step 2: Substitute: 5² + 12² = c²
Step 3: Calculate: 25 + 144 = 169
Step 4: Solve: c = √169 = 13
Answer: The hypotenuse is 13 units. This is the (5, 12, 13) Pythagorean triple.
Example 2: Finding a Missing Leg
A right triangle has a hypotenuse of 25 and one leg of 7. Find the other leg.
Step 1: Write: a² + b² = c²
Step 2: Substitute: 7² + b² = 25²
Step 3: Calculate: 49 + b² = 625
Step 4: Solve: b² = 576, so b = √576 = 24
Answer: The missing leg is 24 units. (7, 24, 25) is a Pythagorean triple.
Example 3: Ladder Problem
A 13-foot ladder leans against a wall with its base 5 feet from the wall. How high up the wall does it reach?
Step 1: The ladder is the hypotenuse (c = 13), the base distance is a leg (a = 5)
Step 2: 5² + h² = 13²
Step 3: 25 + h² = 169
Step 4: h² = 144, so h = 12
Answer: The ladder reaches 12 feet up the wall.
Example 4: Distance Between Two Points
Find the distance between A(2, 3) and B(8, 11).
Step 1: d = √((x&sub2;−x&sub1;)² + (y&sub2;−y&sub1;)²)
Step 2: d = √((8−2)² + (11−3)²)
Step 3: d = √(36 + 64) = √100
Step 4: d = 10
Answer: The distance is 10 units.
Example 5: 3D Diagonal
Find the space diagonal of a rectangular box with dimensions 3 × 4 × 12.
Step 1: d = √(3² + 4² + 12²)
Step 2: d = √(9 + 16 + 144)
Step 3: d = √169 = 13
Answer: The space diagonal is 13 units.
8Memory Aids
"a squared plus b squared equals c squared"
The most famous equation in geometry. c is always the hypotenuse (longest side, opposite the right angle).
"3-4-5, 5-12-13, 8-15-17, 7-24-25"
Memorize these four base triples. Any multiple works too (e.g., 6-8-10 is 2 × 3-4-5).
"Equal = Right, Greater = Acute, Less = Obtuse"
Compare a² + b² to c². If they're equal, it's right. If the legs' sum is bigger, the angle is smaller (acute). If less, it's obtuse.
"Measure 3-4-5 to check a right angle"
Construction workers have used this for centuries. Mark 3 units on one side, 4 on the other. If the diagonal is exactly 5, the corner is 90°.
9Common Mistakes
Using the theorem on non-right triangles
a² + b² = c² only works for right triangles. For non-right triangles, you need the Law of Cosines.
Putting the hypotenuse on the wrong side of the equation
The hypotenuse (c) must be alone on one side: a² + b² = c². Writing a² + c² = b² gives a wrong answer if b is not the hypotenuse.
Adding instead of subtracting when finding a leg
To find a leg: a = √(c² − b²). Don't add the squares — you need to subtract the known leg's square from the hypotenuse's square.
Forgetting to take the square root
After computing a² + b² = c², you get c², not c. Don't forget the final step: c = √(a² + b²).
Thinking √(a² + b²) = a + b
This is a common algebra error. √(9 + 16) = √25 = 5, NOT √9 + √16 = 3 + 4 = 7. You cannot split a square root across addition.
Not identifying the longest side for the converse
When using the converse, c must be the longest side. Always sort the sides first and assign the largest to c before comparing.
Quick Revision Summary
- ✓Pythagorean Theorem: a² + b² = c² for right triangles only.
- ✓The hypotenuse (c) is always opposite the right angle and is the longest side.
- ✓Find hypotenuse: c = √(a² + b²). Find leg: a = √(c² − b²).
- ✓Common Pythagorean triples: (3,4,5), (5,12,13), (8,15,17), (7,24,25). Multiples also work.
- ✓Converse: a² + b² = c² → right; > c² → acute; < c² → obtuse.
- ✓Distance formula d = √((x&sub2;−x&sub1;)² + (y&sub2;−y&sub1;)²) is derived from the theorem.
- ✓3D extension: d = √(x² + y² + z²) for space diagonals.
- ✓Remember: √(a² + b²) ≠ a + b. You cannot split a square root over addition.
Frequently Asked Questions
- What is the Pythagorean Theorem?
- The Pythagorean Theorem states that in a right triangle, the square of the hypotenuse (c) equals the sum of the squares of the two legs (a and b): a² + b² = c². It only applies to right triangles.
- What are Pythagorean triples?
- Pythagorean triples are sets of three positive integers (a, b, c) that satisfy a² + b² = c². Common examples: (3,4,5), (5,12,13), (8,15,17), (7,24,25). Any multiple of a triple is also a triple, e.g., (6,8,10) = 2×(3,4,5).
- How is the distance formula related to the Pythagorean Theorem?
- The distance formula d = √((x₂−x₁)² + (y₂−y₁)²) is derived directly from the Pythagorean Theorem. The horizontal and vertical differences form the legs of a right triangle, and the distance is the hypotenuse.
- How do you use the converse of the Pythagorean Theorem?
- The converse classifies triangles: if a² + b² = c², it’s right; if a² + b² > c², it’s acute; if a² + b² < c², it’s obtuse (where c is the longest side).
- Can the Pythagorean Theorem be used in 3D?
- Yes. The 3D distance formula d = √(x² + y² + z²) extends the theorem to three dimensions, finding the space diagonal through a rectangular prism.
Practice Quiz
Test your understanding of the Pythagorean Theorem, triples, the converse, and real-world applications.
1.In a right triangle, which side is always the longest?
2.If the legs of a right triangle are 9 units and 12 units, what is the length of the hypotenuse?
3.A triangle has side lengths of 8, 15, and 17. What type of triangle is it?
4.Which of the following is NOT a Pythagorean Triple?
5.A ladder is 20 feet long and its base is 12 feet from a wall. How high does the ladder reach on the wall?
6.The distance formula is derived directly from which theorem?
7.If a² + b² > c² (where c is the longest side), what type of triangle is it?
8.A rectangle has a length of 24 cm and a width of 10 cm. What is the length of its diagonal?
9.Which of these side lengths could form a right triangle?
10.What is the missing leg length if the hypotenuse is 25 and one leg is 7?
Final Study Advice
- 1.Always draw and label the triangle before solving. Mark the right angle and identify the hypotenuse.
- 2.Memorize the four common Pythagorean triples to spot instant answers on tests.
- 3.When using the converse, always sort the sides and assign the largest to c.
- 4.Practice real-world problems (ladders, diagonals, distances) to build intuition.
- 5.Remember that the distance formula is just the Pythagorean Theorem in disguise.