Enter two sides of a right triangle to calculate the missing side using the Pythagorean theorem. Choose whether you are solving for a leg or the hypotenuse.
The Pythagorean theorem is one of the most famous results in mathematics: in any right triangle, the square of the hypotenuse equals the sum of squares of the other two sides. Written algebraically: a² + b² = c², where c is the hypotenuse (longest side, opposite the right angle) and a, b are the two legs. This 2500-year-old theorem, attributed to Pythagoras (c. 570-495 BC), underlies coordinate geometry, distance calculations, trigonometry, and countless practical applications.
The theorem works in only one direction: any right triangle satisfies the equation. Conversely, any three positive numbers a, b, c satisfying a² + b² = c² can be the sides of a right triangle (the converse). This bidirectional truth is what makes the theorem so powerful.
Famous "Pythagorean triples" — integer solutions to a² + b² = c² — appear throughout geometry: 3-4-5, 5-12-13, 8-15-17, 7-24-25, 20-21-29. These triples have been studied for thousands of years and remain useful for clean-number examples.
The theorem extends to: - **Distance formula**: distance between two points = √((x₂-x₁)² + (y₂-y₁)²) (Pythagorean applied to coordinates). - **3D space**: space diagonal = √(a² + b² + c²). - **Vectors**: magnitude = √(x² + y² + ...). - **Complex numbers**: modulus |a + bi| = √(a² + b²).
Common applications: construction (squaring corners), navigation, surveying, computer graphics, physics (vector magnitudes), trigonometry (foundation), and any problem involving right-triangle geometry.
**Scenario:** 16:9 aspect ratio TV with width 32" and height 18". What's the diagonal (TV size)? **Calculation:** diagonal = √(32² + 18²) = √(1024 + 324) = √1348 ≈ 36.7". **Result:** ~37" diagonal TV. TV sizes are diagonal measurements. For exact 32" TV: width = 32 × 16/√(16² + 9²) ≈ 27.9"; height ≈ 15.7".
**Scenario:** 12 ft ladder. Base 4 ft from wall. How high does it reach? **Calculation:** height = √(12² - 4²) = √(144 - 16) = √128 ≈ 11.3 ft. **Result:** ~11.3 ft. Safe placement check: OSHA recommends ladder angle ~75° (ratio 1:4, base:height). Ratio here: 4:11.3 ≈ 1:2.8, much steeper. With 4 ft base, ladder is fairly steep — could be safer with bigger base distance.
**Scenario:** Building a deck corner. Want to verify 90° angle. Measure 3 ft along one wall, 4 ft along other. Diagonal should be? **Calculation:** Diagonal = √(3² + 4²) = √25 = 5 ft. **Result:** If diagonal is exactly 5 ft, corner is 90°. If not, adjust. "3-4-5 method" — used by carpenters for centuries to verify square corners without complex tools.
**Use the Pythagorean theorem for:**
- **Finding missing sides** of right triangles. - **Distance calculations** between points (uses formula derived from theorem). - **Construction**: verifying right angles, sizing diagonal braces. - **Navigation**: straight-line distances. - **Computer graphics**: pixel distances, collision detection. - **Physics**: vector magnitudes (velocity, force). - **Roof and stair design**: rise/run/diagonal calculations. - **Trigonometry**: foundation for sin, cos, tan in right triangles.
**Only for right triangles:**
Pythagorean theorem works ONLY for right triangles. For other triangles: - **Law of Cosines**: c² = a² + b² - 2ab cos(C). - **Law of Sines**: a/sin(A) = b/sin(B) = c/sin(C).
Pythagorean is the special case when C = 90°.
**Identifying the hypotenuse:**
The hypotenuse is: - Always opposite the right angle. - Always the longest side. - Labeled "c" in standard notation.
The other two sides (legs, a and b) are adjacent to the right angle.
**Common Pythagorean triples to recognize:**
- 3-4-5 (and multiples: 6-8-10, 9-12-15, ...) - 5-12-13 (and multiples) - 7-24-25 - 8-15-17
Recognizing these saves calculation time.
**Generating new triples:**
For positive integers m > n: - a = m² - n² - b = 2mn - c = m² + n²
Examples: (m,n) = (2,1) → (3,4,5). (3,2) → (5,12,13). (4,1) → (15,8,17).
**Common applications:**
- **Construction**: verifying square corners, roof rafters, stair stringers. - **Carpentry**: diagonal bracing, frame squareness. - **Surveying**: indirect distance measurement. - **Architecture**: blueprint distances, room diagonals. - **Navigation**: straight-line ("as the crow flies") distances. - **Computer graphics**: pixel-distance calculations. - **Physics**: 2D vector resultant magnitudes. - **GPS/Mapping**: distance approximations. - **Sports**: trajectory analysis (e.g., baseball, golf).
**3D extension:**
For 3D coordinates: d = √((x₂-x₁)² + (y₂-y₁)² + (z₂-z₁)²)
Used in: - 3D graphics. - Astronomy. - Robotics. - Aerospace.
**Connection to other math:**
- **Trigonometry**: sin² + cos² = 1 (special case). - **Law of Cosines**: generalization to any triangle. - **Distance formula**: Pythagorean in coordinate plane. - **Vector magnitude**: |v| = √(sum of squares). - **Complex modulus**: |a + bi| = √(a² + b²).
**Software:**
- **Calculator**: simple formula. - **Excel**: =SQRT(A1^2 + B1^2). - **Python**: math.sqrt(a**2 + b**2) or math.hypot(a, b). - **CAD**: built into distance and dimension tools. - **GIS**: distance calculations.
**Why "Pythagorean"?**
Attributed to Pythagoras (Greek mathematician, 570-495 BC) who reportedly proved the theorem. But it was known earlier — Babylonian tablets (c. 1900 BC) list Pythagorean triples. Several Chinese, Indian, and Egyptian texts include proofs. Pythagoras gets credit for first deductive proof in Western mathematics.
**Number of proofs:**
Over 370 distinct proofs of Pythagorean theorem exist — geometric, algebraic, trigonometric. Includes proofs by President James Garfield (1881) and Einstein (as a teenager).
**Pitfalls:**
- **Using on non-right triangles**: only works for right triangles. - **Wrong identification of hypotenuse**: must be opposite right angle. - **Forgetting to take square root**: result is c², not c. - **Subtracting wrong side**: for leg, subtract from hypotenuse squared. - **For decimals**: most triangles have irrational hypotenuse. - **3D vs 2D**: include z-coordinate when needed. - **Confusing with distance formula**: same idea, different setup.
Calculate the distance between two points on a coordinate plane.
Calculate the area of common shapes: circle, rectangle, triangle, and trapezoid.
Calculate the slope of a line between two points.
Find the greatest common factor (GCF) of two numbers.
Find the least common multiple (LCM) of two numbers.
Find the prime factors of any positive integer.
Missing Side
5
Hypotenuse (c)
5
Leg a
3
Leg b
4
Triangle Area
6
Perimeter
12