#G08 · Geometry & Trigonometry · Lv.3View in 3D Mind-Map

Spherical Coordinates & Sphere Volume/Area

Archimedes & Leonhard Euler · 225 BCE

V = \frac{4}{3}\pi r^3,\quad A = 4\pi r^2 = \frac{dV}{dr}

(x, y, z) = (r\sin\phi\cos\theta,\; r\sin\phi\sin\theta,\; r\cos\phi)

Converts (radius, longitude θ, colatitude φ) into 3D (x, y, z) coordinates, while linking sphere volume 4/3 πr³ and surface area 4πr² via differentiation.

🧮 Live Formula Calculator & Plotter

Radius r

Azimuth θ°

Polar φ°

V = \frac{4}{3}\pi (5)^3 = 523.6, \; P(x,y,z) = (3.06, 3.06, 2.50)

Volume V

523.60

Surface Area A

314.16

3D Coord (x,y,z)

(3.1, 3.1, 2.5)

Prerequisite Path (10 STEP)

Full Path Analyzer

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

STEP 1#A02Polynomial Expansion & Factoring

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

STEP 2#A03Laws of Exponents & Radicals

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

STEP 3#A01Linear Equation & Balance Axiom

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

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

STEP 6#A04Quadratic Formula & Discriminant

x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

STEP 7#G022D/3D Euclidean Distance Formula

d(P

STEP 8#G04Unit Circle Trigonometry & Identities

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

STEP 9#G07Conic Sections (Parabola, Ellipse, Hyperbola)

\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1

STEP 10#G08Spherical Coordinates & Sphere Volume/Area

V = \frac{4}{3}\pi r^3

🚀 Next Unlocks (Derived Formulas) (1)

#G09Euler's Polyhedron Formula & Topology

Geometric & Intuitive Insight

양파 껍질처럼 두께 dr 인 얇은 구면 껍질(넓이 4πr²)들을 r = 0 부터 R 까지 겹겹이 적분하여 쌓으면 구의 부피 ∫ 4πr² dr = (4/3)πR³ 이 됩니다.

Step-by-Step Derivation

Step 1.1단계: 천정각 φ에 대해 z축 높이는 z = r cos φ 이고, xy평면에 투영된 반지름은 ρ = r sin φ 입니다.
Step 2.2단계: 방위각 θ에 대해 x = ρ cos θ = r sin φ cos θ, y = ρ sin θ = r sin φ sin θ 로 분해됩니다.
Step 3.3단계: 구의 단면적 π(r² - z²) 을 z = -r 부터 r 까지 적분하면 V = (4/3)πr³ 이 도출됩니다.

AI & Coding Real-World Use

3D globe 360° UV texture mapping, Three.js OrbitControls camera, and quantum spherical harmonics

#3D Globe Mapping#OrbitControls Camera#Spherical Harmonics

const toXYZ = (r:number, th:number, ph:number) => [r*Math.sin(ph)*Math.cos(th), r*Math.cos(ph), r*Math.sin(ph)*Math.sin(th)];

Geometry & Trigonometry — Related Formulas in This Cluster

Pythagorean Theorem

Lv.1 · Game Physics Distance

#G01

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