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 |
Shortest Distance Between Two Lines in 3D
The shortest-distance block is the single highest-marks segment of the Three Dimensional Geometry Formulas. Formula 15 is the vector template that CBSE markers expect for full marks on a 5-mark long answer.
| # | Concept | Formula |
|---|---|---|
| 15 | Shortest distance between skew lines, vector form | d = |(b1⃗ × b2⃗) · (a2⃗ - a1⃗)||b1⃗ × b2⃗| |
| 16 | Shortest distance between skew lines, Cartesian form | Determinant of the rows (a2⃗-a1⃗),b1⃗,b2⃗ divided by |b1⃗ × b2⃗| |
| 17 | Distance between two parallel lines | d = |b⃗ × (a2⃗ - a1⃗)||b⃗| |
| 18 | Condition for intersection or coplanarity | (b1⃗ × b2⃗) · (a2⃗ - a1⃗) = 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| |
Decision Tree: Which Formula to Use
The Three Dimensional Geometry Formulas address this in the same order as the NCERT textbook.
Step 1: Read the question stem. Is the unknown a direction cosine, an equation, an angle, a distance, or a plane?
Step 2: If lines are given in Cartesian symmetric form, read off (x1, y1, z1) as a point and (a, b, c) as direction ratios. If a plane is given as ax+by+cz=d, the normal direction ratios are (a,b,c).
Step 3:
For an angle between two lines, use #10 or #11. For an angle between two planes, use #24. For a line and a plane, use #25 (note the sin).
Step 4: For distance: parallel lines use #17, skew lines use #15, point to a plane uses #23. Always cross-check using #13 first to confirm the lines are not parallel before reaching for #15.
How the Three Dimensional Geometry Formulas on the Three Dimensional Geometry Formulas Help You
This Collegedunia formula sheet is built for the final 48-hour revision pass. It compresses about 28 NCERT pages into a printable single-look sheet, organised by concept block so any board question can be matched to the right template inside 30 seconds.
- Concept-first layout orders formulas by topic block: direction cosines, line equations, angle, distance, planes.
- Shortest-distance block isolates both vector and determinant forms, since it is the highest-marks segment in the Three Dimensional Geometry Formulas.
- Plane-equation block shows all three equivalent forms side by side, so you can rewrite quickly when the question demands a specific form.
- PYQ hit-rate tags flag which formulas appeared in CBSE 2021 to 2025 and JEE Main 2021 to 2025 papers.
- 2026-27 syllabus aligned with the current NCERT print, including both the lines and planes sections.
Class 12 Maths Chapter 11 PYQ-Linked Formulas
The mini-table below tags each PYQ year with the exact formula it tested. Use it to prioritise the drill order before the board paper.
| Year | Marks | Formula tested |
|---|---|---|
| CBSE 2025 | 5 | Shortest distance, vector form (#15) |
| CBSE 2024 | 5 | Two-point line equation (#8) and shortest distance (#15) |
| CBSE 2023 | 3 | Direction cosines from two points (#4) |
| CBSE 2022 | 3 | Distance of point from plane (#23) |
| JEE Main 2025 | 4 | Shortest distance, Cartesian determinant form (#16) |
| JEE Main 2024 | 4 | Coplanarity check (#18) |
Full year-wise PYQ map: NCERT Solutions for Class 12 Maths Chapter 11.
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).
Related 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 | 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.
Exercise-wise Breakdown of the Three Dimensional Geometry Chapter
The Three Dimensional Geometry chapter splits into 2 numbered exercises plus a Miscellaneous Exercise. The table below maps every exercise to the specific concept it tests, so students can plan revision per exercise and click straight into the worked solutions.
| Exercise | Topic Tested |
|---|---|
| Exercise 11.1 | Direction cosines, direction ratios of a line |
| Exercise 11.2 | Vector and Cartesian equations of a line in 3D |
| Miscellaneous Exercise | Mixed three-dimensional geometry problems |
PDF Download Formats and Languages for the Three Dimensional Geometry Chapter
The Three Dimensional Geometry Class 12 PDF on this page is available in three formats - each suited to a different revision style. The table below summarises what each format is best for:
| Format | Best for | Approx. size |
|---|---|---|
| Normal-resolution PDF | Phone reading, quick revision between classes | 2-3 MB |
| HD PDF | Print-ready, desk study, board hall photocopy | 8-10 MB |
| Handwritten Notes PDF | Mirrors how a topper writes the chapter under Sunday-revision pace | 5-7 MB |
- NCERT-faithful: Every definition, theorem and exercise on the three dimensional geometry class 12 ncert pdf matches the printed textbook line for line.
- Hindi-medium edition: The three dimensional geometry class 12 pdf is also available in Hindi - same page numbering, same equation labels.
- Formula PDF separate: The three dimensional geometry class 12 formulas pdf is a one-page A4 reference sheet listing every identity used in the chapter.
- Solutions PDF separate: The three dimensional geometry class 12 solutions pdf gives every NCERT exercise worked out step by step.
- State-board alignment: Students on the Maharashtra board, HSC, or any state-board syllabus will find the same definitions in this three dimensional geometry class 12 pdf - only the exercise numbers differ.
Important Questions and Previous Year Trends for the Three Dimensional Geometry Chapter
| Template | Typical Marks | What it tests |
|---|---|---|
| Proof / property verification | 3 marks | Students show that a given relation/function/expression satisfies the chapter's definitions. |
| One-step computation | 2 marks | Substitution-based item: plug into a known formula and simplify. |
| Case-study scenario | 4 marks | Real-world setup applying the chapter's definitions, introduced in CBSE 2021+ papers. |
- three dimensional geometry class 12 previous year questions for 2019-2024 are linked from the PYQ block at the bottom of this page - the exact CBSE phrasings.
- The three dimensional geometry class 12 important questions with solutions set is reused by toppers in the last fortnight of revision.
- For NCERT Exemplar practice, the matching three dimensional geometry class 12 extra questions set adds advanced problems suitable for JEE Main and JEE Advanced.
- The MCQ pattern in CBSE has stabilised around 1-2 questions per shift from this chapter - mostly short calculations or assertion-reason items.
Year-wise PYQ Distribution
| Year | Dominant Question Type | Approx. Marks |
|---|---|---|
| 2024 | Property verification + case-study item | 5-6 marks |
| 2023 | Computation with proof + assertion-reason MCQ | 5-6 marks |
| 2022 | Long-answer derivation + 2-mark substitution | 5-7 marks |
| 2021 | Definition recall + property check | 4-5 marks |
| 2020 | One-step computation + 3-mark proof | 5 marks |
How the Three Dimensional Geometry Notes Pair with NCERT Solutions and the Formula Sheet
| Resource | Use it for | When |
|---|---|---|
| Three Dimensional Geometry Notes (this page) | Theory, definitions, exam patterns | First pass, before practice |
| three dimensional geometry class 12 ncert solutions PDF | Step-by-step solved exercises | Second pass, during NCERT practice |
| three dimensional geometry class 12 formulas PDF | One-page identity recall | Third pass, alongside mock papers |
| Handwritten Notes PDF | Quick reading in topper's handwriting | Anytime, especially commute revision |
- The three dimensional geometry class 12 ncert solutions cover every back-of-chapter exercise plus the miscellaneous exercise.
- The three dimensional geometry class 12 solutions for each individual exercise are indexed by exercise number on the sister NCERT Solutions page (see the Exercise-wise Breakdown table above for direct links).
- The three dimensional geometry class 12 formulas reference sheet is the same A4 file students sometimes refer to as three dimensional geometry class 12 all formulas - it lists every identity used in the chapter.
- State-board references: RD Sharma, ML Aggarwal, Teachoo and the Maharashtra board three dimensional geometry class 12 textbook PDF all share the same core definitions.
- For class-first search phrasings - class 12 three dimensional geometry solutions, class 12 three dimensional geometry ncert solutions, ncert class 12 three dimensional geometry solutions - the same files cover the request.
Reference Books and State-Board Mapping
| Reference | How it maps to Three Dimensional Geometry Class 12 |
|---|---|
| RD Sharma Class 12 Three Dimensional Geometry | Question patterns overlap with NCERT at ~70%; an advanced supplement. |
| ML Aggarwal Class 12 Three Dimensional Geometry | Solutions style is closer to JEE; good for problem-solving practice. |
| Teachoo three dimensional geometry class 12 | Free online walkthroughs; useful for video-style learning. |
| Shaalaa three dimensional geometry class 12 solutions | State-board (Maharashtra HSC) phrasings; same core definitions. |
| Maharashtra board three dimensional geometry class 12 textbook PDF | Same chapter content under the HSC syllabus; exercise numbers differ. |
| NCERT Exemplar Class 12 Three Dimensional Geometry | Advanced problems for JEE Main/JEE Advanced preparation. |
How to Use the Three Dimensional Geometry Notes Page Most Effectively
| Sitting | Duration | What to do |
|---|---|---|
| Sitting 1: Theory | ~90 minutes | Read the printed NCERT chapter cover to cover. Mark every definition and theorem statement. Then read the formula recall section on this page. |
| Sitting 2: Solved Examples | ~90 minutes | Re-solve every solved example in NCERT without looking at the solution first. Compare your steps against the printed working. Use the three dimensional geometry class 12 ncert solutions PDF if stuck. |
| Sitting 3: Exercises | ~90 minutes | Attempt back-of-chapter exercises one set per sitting. Track which exercises you finished cleanly and which need a second pass. Click into the linked exercise pages above for verification. |
- 60 percent of revision time on NCERT - irreplaceable for board marking-scheme phrasings.
- 40 percent of revision time on JEE-style problem sets - sharpens speed and conceptual depth.
- The three dimensional geometry class 12 important questions set on the previous-year page is the closest free analogue to a JEE-style problem set for this chapter.
- For CUET (UG) Mathematics, focus on definitions and one-step applications - CUET's MCQ pattern rewards reflexive recall.
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