3D 입체 기하 엔진d = √(x² + y² + z²)

3D 공간 대각선 계산기

Calculate the longest straight rod, beam, or internal vector fitting inside a 3D rectangular prism with exact simplified radicals and cuboid surface metrics. Working purely in a flat 2D plane instead? Use our flagship Pythagoras equation solver.

Interactive 3D Wireframe Box2-Stage KaTeX DerivationVolume & Surface Area

3D 공간 대각선 계산기

d_base = √(x² + y²) · d_space = √(x² + y² + z²)

가로 (x)

세로 (y)

높이 (z)

3D 공간 대각선 (d_space)

≈ 13

밑면 대각선 (d_base)

≈ 5

부피

144 units³

겉넓이

192 units²

2단계 3D 유도 과정 (KaTeX)

Stage 1: Solve the 2D Base Face Diagonal (XY Plane)

d_base = √(3² + 4²)

d_base = 5

d_{\text{base}} = \sqrt{3^2 + 4^2} = \sqrt{25} = 5

Stage 2: Solve the 3D Internal Space Diagonal (Vertical Right Triangle)

d_space = √((√25)² + 12²) = √(25 + 144)

d_space = 13

d_{\text{space}} = \sqrt{d_{\text{base}}^2 + 12^2} = \sqrt{25 + 144} = \sqrt{169} = 13

3D 대각선에서 피타고라스 정리를 두 번 쓰는 이유

First, apply 2D Pythagoras across the floor rectangle (x by y) to find the base face diagonal d_base² = x² + y². Because the vertical height edge z stands strictly perpendicular (90°) to every line on the floor, d_base and z form a second vertical right triangle whose hypotenuse is the 3D space diagonal.

d_space² = d_base² + z² = (x² + y²) + z² ⟹ d_space = √(x² + y² + z²)