#G03 · Geometry & Trigonometry · Lv.1View in 3D Mind-Map

Pi, Radian Measure & Circular Sector

Archimedes & Roger Cotes · 250 BCE

l = r\theta,\quad S = \frac{1}{2}r^2\theta = \frac{1}{2}rl

\pi\text{ rad} = 180^\circ,\quad C = 2\pi r,\quad A = \pi r^2

Defines angle θ in radians as arc length divided by radius (l/r), making angles dimensionless real numbers essential for clean calculus derivatives like (sin x)' = cos x.

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Radius r

Angle (deg)

l = 6\times 1.05 = 6.28, \quad S = \frac{1}{2}(6)^2(1.05) = 18.85

Arc Length l

6.283

Sector Area S

18.850

Full Circle πr²

113.10

Prerequisite Path (5 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}

STEP 5#G03Pi, Radian Measure & Circular Sector

l = r\theta

🚀 Next Unlocks (Derived Formulas) (2)

#G04Unit Circle Trigonometry & Identities #G08Spherical Coordinates & Sphere Volume/Area

Geometric & Intuitive Insight

반지름 r만큼의 실을 원 둘레를 따라 얹었을 때 벌어지는 중심각이 정확히 1 라디안(약 57.3°)입니다. 반원을 덮으려면 반지름이 π(3.14159…)개 필요하므로 180° = π rad 입니다.

Step-by-Step Derivation

Step 1.1단계: 원 전체 둘레는 2πr 이므로 중심각 θ(rad)에 비례하는 호의 길이는 l = (θ / 2π) · 2πr = rθ
Step 2.2단계: 원 전체 넓이는 πr² 이므로 부채꼴 넓이는 S = (θ / 2π) · πr² = ½r²θ
Step 3.3단계: l = rθ 를 대입하면 삼각형 넓이 공식과 닮은 S = ½rl 이 도출됩니다.

AI & Coding Real-World Use

Standard angle unit for Three.js/WebGL shaders, Math.sin/cos, and robotics wheel odometry (l = rθ)

#Three.js Radians#Robotics Odometry#Shader Angles

const degToRad = (deg: number) => (deg * Math.PI) / 180;

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