04 // Interactive Formula Compendium // GEOMETRY-TRIGc = √(a² + b²) (2D), d = √(a² + b² + h²) (3D Space Diagonal)
Pythagorean Theorem in 2D & 3D Space — Hypotenuse & Space Diagonal
The Pythagorean Theorem states that in a right triangle with perpendicular legs a and b, the hypotenuse c satisfies c² = a² + b²; in three orthogonal dimensions (a, b, h), the space diagonal is d = √(a² + b² + h²).
Formula Reference
Main Formula
c = √(a² + b²) (2D), d = √(a² + b² + h²) (3D Space Diagonal)
Rearranged Forms
a = √(c² − b²) (Leg Recovery)
sin²(θ) + cos²(θ) = 1 (Trigonometric Pythagorean Identity)
(m² − n²)² + (2mn)² = (m² + n²)² (Euclid's Pythagorean Triple Generator)
Variables & Units
| Symbol | Name | Unit |
|---|---|---|
| d | Diagonal Length √(a² + b² + h²) | m |
| a | Length / Leg a | m |
| b | Width / Leg b | m |
| h | Height h (set 0 for 2D right triangle) | m |
Inputs & Parameters
[m]
[m]
[m]
Calculated Result (Diagonal Length √(a² + b² + h²))
d =13m
Formula Notes & Background
- Arrange four congruent right triangles of legs a, b and hypotenuse c inside a square of side (a + b).
- The total area (a + b)² = a² + 2ab + b² equals the central tilted square c² plus four triangles 4(½ab) = 2ab, leaving a² + b² = c².
- In 3D, the floor diagonal is f = √(a² + b²), which forms a right triangle with vertical height h so d = √(f² + h²) = √(a² + b² + h²).
Step-by-Step Examples
#1: Find the 3D space diagonal of a room measuring 3 m × 4 m × 12 m.
a = 3 mb = 4 mh = 12 m
d = √(3² + 4² + 12²) = √(9 + 16 + 144) = √169
→ d = 13 m (exact integer Pythagorean quadruple)
#2: Generate a primitive Pythagorean triple using Euclid's formula with m = 4, n = 1.
m = 4n = 1
a = 4² − 1² = 15, b = 2(4)(1) = 8, c = 4² + 1² = 17
→ (8, 15, 17) since 64 + 225 = 289 = 17²
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Frequently Asked Questions
What are the most common integer Pythagorean triples to memorize?+
(3, 4, 5), (5, 12, 13), (8, 15, 17), (7, 24, 25), (20, 21, 29), and their integer multiples like (6, 8, 10).
How do carpenters use the 3-4-5 rule to square building corners?+
Measure 3 ft along one wall and 4 ft along the perpendicular wall; when the diagonal tape measure between those marks reads exactly 5 ft, the corner is 90°.
How does the Pythagorean theorem generalize to inner product spaces?+
For any two orthogonal vectors u ⊥ v (where u · v = 0), ||u + v||² = ||u||² + ||v||².