#G01 · Geometry & Trigonometry · Lv.1View in 3D Mind-Map

Pythagorean Theorem

Pythagoras & Euclid · 530 BCE

a^2 + b^2 = c^2 \implies c = \sqrt{a^2 + b^2}

\cos^2\theta + \sin^2\theta = 1

In any right triangle, the square of the hypotenuse equals the sum of squares of the other two sides—the root of Euclidean distance, trigonometry, and vector L2 norms.

🧮 Live Formula Calculator & Plotter

Base a

Height b

c = \sqrt{3^2 + 4^2} = \sqrt{25.0} = 5.000

Hypotenuse c

5.0000

Area S

6.00

Angle θ

53.1°

Prerequisite Path (4 STEP)

Full Path Analyzer

Click steps in order to follow the 3D neon path from basics to target.

STEP 1#A01Linear Equation & Balance Axiom

ax + b = 0 \implies x = -\frac{b}{a}

STEP 2#A02Polynomial Expansion & Factoring

(a+b)^2 = a^2 + 2ab + b^2

STEP 3#A03Laws of Exponents & Radicals

a^m \cdot a^n = a^{m+n}

STEP 4#G01Pythagorean Theorem

a^2 + b^2 = c^2 \implies c = \sqrt{a^2 + b^2}

🚀 Next Unlocks (Derived Formulas) (8)

#A09Complex Numbers & Polar Plane #G022D/3D Euclidean Distance Formula #G03Pi, Radian Measure & Circular Sector #G04Unit Circle Trigonometry & Identities #G05Law of Sines & Law of Cosines #G07Conic Sections (Parabola, Ellipse, Hyperbola)#G09Euler's Polyhedron Formula & Topology #L01Vector Operations, L1/L2 Norms & Normalization

Geometric & Intuitive Insight

한 변이 (a+b)인 큰 정사각형 안에 합동인 직각삼각형 4개를 배치하면, 남는 빈 공간의 넓이는 위치에 따라 a² + b² 이 되기도 하고 한 변이 c인 정사각형 c² 이 되기도 합니다.

Step-by-Step Derivation

Step 1.1단계: 한 변의 길이가 a+b 인 정사각형의 전체 넓이는 (a+b)² = a² + 2ab + b² 입니다.
Step 2.2단계: 이 정사각형은 넓이가 ½ab 인 직각삼각형 4개(총 2ab)와 가운데 정사각형 c² 으로 분할됩니다.
Step 3.3단계: a² + 2ab + b² = 2ab + c² 에서 양변의 2ab를 소거하면 a² + b² = c² 이 증명됩니다.

AI & Coding Real-World Use

2D/3D game collision distance (Math.hypot) and foundational L2 metric in AI vector spaces

#Game Physics Distance#L2 Metric Foundation#CAD Geometry

const hypotenuse = (a: number, b: number) => Math.hypot(a, b);

Geometry & Trigonometry — Related Formulas in This Cluster

2D/3D Euclidean Distance Formula

Lv.2 · K-Means & KNN ML

#G02

Pi, Radian Measure & Circular Sector

Lv.1 · Three.js Radians

#G03

Unit Circle Trigonometry & Identities

Lv.2 · LLM RoPE Encoding

#G04

Law of Sines & Law of Cosines

Lv.2 · Inverse Kinematics IK

#G05

Trigonometric Addition & Double-Angle Formulas

Lv.3 · Rotation Matrix Proof

#G06

Conic Sections (Parabola, Ellipse, Hyperbola)

Lv.2 · Kepler Orbit Sim

#G07