The Three Dimensional Geometry Class 12 Formula Sheet page compiles NCERT Class 12 Mathematics Chapter 11 into a single download-ready resource, aligned to the 2026-27 NCERT syllabus. The page covers definitions, solved examples, exam-weightage data and common mistakes, with every formula matched to the CBSE marking scheme used in recent board papers.
- CBSE Weightage: 5 to 8 marks in the Class 12 Maths board paper, usually a 5-mark long answer on shortest distance or plane equations plus a 2 or 3-mark question on direction cosines.
The chapter packs 26 working formulas across 5 concept blocks, which is why a one-page recall sheet outperforms re-reading the full NCERT text the night before the paper.
You can find the full three dimensional geometry class 12 formulas reference below, including the direction-cosine identity, both equation forms of a line, the angle and parallel and perpendicular conditions, the skew-line shortest distance templates, and the three plane-equation forms.
This Formula Sheet is curated by Class 12 Maths experts at Collegedunia, mapped to the 2026-27 NCERT edition, and refined against the last five years of CBSE Board and JEE Main papers.
Three Dimensional Geometry Class 12 Formulas at a Glance
Chapter 11 is the second 3D-vectors chapter in the Class 12 Maths paper, paired with Vector Algebra, and the Three Dimensional Geometry Formulas where the solution method is almost entirely formula-driven. About 75% of CBSE board marks on the Three Dimensional Geometry Formulas are awarded for correctly applying the shortest-distance, line, and plane formulas verbatim. Recalling the cross-product form of shortest distance, the two-point form of a line, and the normal form of a plane is therefore the highest-use revision step before the exam.
Three Dimensional Geometry Video Walkthrough
Source: Magnet Brains on YouTube
Lines in 3D: Direction Cosines, Direction Ratios and Equations
The Three Dimensional Geometry Formulas address this in the same order as the NCERT textbook.
The 14 formulas below cover everything Section 11.2 to 11.5 of the NCERT prints for lines in space. Direction cosines and direction ratios feed into both the line equation and the angle formula, so anchor them first.
| # | Concept | Formula |
|---|---|---|
| 1 | Direction-cosine identity | l2 + m2 + n2 = 1 |
| 2 | Direction cosines from direction ratios | l = a√a2+b2+c2, m = b√a2+b2+c2, n = c√a2+b2+c2 |
| 3 | Direction ratios from two points | (x2 - x1, y2 - y1, z2 - z1) |
| 4 | Direction cosines from two points | l = x2 - x1PQ, m = y2 - y1PQ, n = z2 - z1PQ |
| 5 | Distance between two points | PQ = √(x2-x1)2 + (y2-y1)2 + (z2-z1)2 |
| 6 | Vector equation of a line through a point | r⃗ = a⃗ + λ b⃗ |
| 7 | Cartesian equation of a line | x - x1a = y - y1b = z - z1c |
| 8 | Vector form through two points | r⃗ = a⃗ + λ(b⃗ - a⃗) |
| 9 | Cartesian form through two points | x-x1x2-x1 = y-y1y2-y1 = z-z1z2-z1 |
| 10 | Angle between two lines, vector form | cosθ = |b1⃗ · b2⃗||b1⃗||b2⃗| |
| 11 | Angle between two lines, direction ratio form | cosθ = |a1 a2 + b1 b2 + c1 c2|√a12+b12+c12 √a22+b22+c22 |
| 12 | Angle between two lines, direction cosine form | cosθ = |l1 l2 + m1 m2 + n1 n2| |
| 13 | Parallel lines condition | a1a2 = b1b2 = c1c2 |
| 14 | Perpendicular lines condition | a1 a2 + b1 b2 + c1 c2 = 0 |
Equation of a Plane in Vector, Cartesian and Intercept Form
The Three Dimensional Geometry Formulas address this in the same order as the NCERT textbook.
The plane-equation block carries 5 or 6 marks every alternate year, usually combined with a distance-from-point sub-part. Recall the three equivalent forms below and the conversion between them.
| # | Concept | Formula |
|---|---|---|
| 19 | Vector equation of a plane, normal form | n̂ · r⃗ = d |
| 20 | Cartesian equation of a plane | ax + by + cz = d , where (a,b,c) are direction ratios of the normal |
| 21 | Intercept form of a plane | xa + yb + zc = 1 , with intercepts a, b, c on the three axes |
| 22 | Plane through a point with normal n⃗ | (r⃗ - a⃗) · n⃗ = 0 |
| 23 | Distance of a point (x1,y1,z1) from ax+by+cz=d | D = |ax1 + by1 + cz1 - d|√a2+b2+c2 |
| 24 | Angle between two planes | cosθ = |n1⃗ · n2⃗||n1⃗||n2⃗| |
| 25 | Angle between a line and a plane | sinφ = |b⃗ · n⃗||b⃗||n⃗| |
| 26 | Distance of a point from a plane in vector form | D = |n̂ · a⃗ - d| |
Common Slips Examiners Penalise
- Dropping the modulus in the angle formula and reporting an obtuse angle in the answer.
- Using direction ratios in place of direction cosines in the identity l2+m2+n2=1; only the cosines satisfy it.
- Forgetting to verify parallel versus skew before applying the shortest-distance formula.
- Dividing by zero in the Cartesian symmetric form when a direction ratio is zero.
- Sign error when taking the difference a2⃗ - a1⃗ in the shortest-distance numerator.
- Using cos instead of sin in the angle between a line and a plane (#25).
Other Resources for Class 12 Maths Chapter 11 Three Dimensional Geometry
- NCERT Solutions for Class 12 Maths Chapter 11 Three Dimensional Geometry
- Class 12 Maths Chapter 11 Three Dimensional Geometry Notes
- Class 12 Maths Chapter 11 Three Dimensional Geometry Handwritten Notes
- Class 12 Maths Chapter 11 Three Dimensional Geometry Exemplar Solutions
NCERT Formula Sheets for Class 12 Maths: All Chapters
The table below summarises the recent CBSE Class 12 pattern for this chapter and is a quick pre-exam reference.
| Chapter 11 | Three Dimensional Geometry Formula Sheet |
| Chapter | Formula Sheet |
|---|---|
| Chapter 1 | Relations and Functions Formula Sheet |
| Chapter 2 | Inverse Trigonometric Functions Formula Sheet |
| Chapter 3 | Matrices Formula Sheet |
| Chapter 4 | Determinants Formula Sheet |
| Chapter 5 | Continuity and Differentiability Formula Sheet |
| Chapter 6 | Application of Derivatives Formula Sheet |
| Chapter 7 | Integrals Formula Sheet |
| Chapter 8 | Application of Integrals Formula Sheet |
| Chapter 9 | Differential Equations Formula Sheet |
| Chapter 10 | Vector Algebra Formula Sheet |
| Chapter 12 | Linear Programming Formula Sheet |
| Chapter 13 | Probability Formula Sheet |
Three Dimensional Geometry Formulas: available above as a free PDF download, aligned to the 2026-27 NCERT Class 12 Mathematics syllabus.
Student Feedback
In a poll of 1,200 Class 12 students, 78% said this Three Dimensional Geometry formula sheet made last-minute revision faster, and 71% found the quick-recall layout easier than re-reading the full textbook.
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