Solids + 3D coordinate geometry: direction cosines, lines, planes
Reviewed by [email protected], Geometry Calculator Developer & Online Math Educator Last updated May 14, 2026
3D geometry covers two related topics in standard school curricula: (1) solid shapes — volumes and surface areas of cube, cylinder, sphere, cone, pyramid, and prism; and (2) 3D coordinate geometry (NCERT Class 12 in India, A-level equivalent elsewhere) — direction cosines, lines and planes in 3-space. This page collects every formula you need for both, with worked examples.
| Name | Formula | Notes |
|---|---|---|
| Cube — Volume | V = s³ |
s = edge length. SA = 6s², diagonal d = s√3. |
| Rectangular Prism — Volume | V = l × w × h |
SA = 2(lw + lh + wh); space diagonal d = √(l² + w² + h²). |
| Cylinder — Volume | V = π × r² × h |
SA = 2πr(r + h); lateral SA = 2πrh. |
| Sphere — Volume | V = (4/3) × π × r³ |
SA = 4πr². Only shape with one parameter. |
| Cone — Volume | V = (1/3) × π × r² × h |
Exactly 1/3 of equivalent cylinder. Slant l = √(r²+h²); SA = πr(r + l). |
| Square Pyramid — Volume | V = (1/3) × b² × h |
b = base side. 1/3 of cube with same base + height. |
| Distance in 3D | d = √[(x₂−x₁)² + (y₂−y₁)² + (z₂−z₁)²] |
3D Pythagorean theorem. Extension of the 2D distance formula. |
| Midpoint in 3D | M = ((x₁+x₂)/2, (y₁+y₂)/2, (z₁+z₂)/2) |
Component-wise average — exactly the same idea as 2D. |
| Direction Cosines | l = cos α, m = cos β, n = cos γ |
α, β, γ = angles a line makes with x-, y-, z-axes. Identity: l² + m² + n² = 1. |
| Direction Ratios → Direction Cosines | l = a/√(a²+b²+c²), m = b/√(...), n = c/√(...) |
Normalize direction ratios (a,b,c) to get the unit vector (l,m,n). |
| Line — Vector Form | ⃗r = ⃗a + λ⃗b |
⃗a = position vector of a point on the line; ⃗b = direction vector; λ = parameter (any real). |
| Line — Cartesian (Symmetric) Form | (x−x₁)/a = (y−y₁)/b = (z−z₁)/c |
(x₁,y₁,z₁) = point on line; (a,b,c) = direction ratios. |
| Plane — Vector Normal Form | ⃗r · ⃗n = d |
⃗n = normal vector; d = distance from origin. |
| Plane — Cartesian Form | Ax + By + Cz + D = 0 |
(A, B, C) is the normal vector to the plane. |
| Plane — Intercept Form | x/a + y/b + z/c = 1 |
a, b, c = x-, y-, z-intercepts of the plane. |
| Distance from Point to Plane | d = |Ax₀ + By₀ + Cz₀ + D| / √(A² + B² + C²) |
Point (x₀, y₀, z₀); plane Ax+By+Cz+D=0. Pure 3D analog of point-to-line distance. |
| Angle Between Two Lines | cos θ = |⃗b₁ · ⃗b₂| / (|⃗b₁| × |⃗b₂|) |
Dot product of direction vectors, normalized. θ ∈ [0°, 90°]. |
| Angle Between Two Planes | cos θ = |⃗n₁ · ⃗n₂| / (|⃗n₁| × |⃗n₂|) |
Dot product of normal vectors. Parallel planes → θ = 0; perpendicular → θ = 90°. |
| Skew Lines — Shortest Distance | d = |(⃗a₂ − ⃗a₁) · (⃗b₁ × ⃗b₂)| / |⃗b₁ × ⃗b₂| |
Cross product gives common perpendicular direction; project the connecting vector onto it. |
Plug in your numbers and get instant step-by-step results.