NCERT Solutions for Class 11 Maths Chapter 12 – Introduction to Three-Dimensional Geometry
Subject: Mathematics | Class: 11 | Chapter: 12 | Board: CBSE | Curriculum: New NCERT
What Are NCERT Solutions for Class 11 Maths Chapter 12 – Introduction to Three-Dimensional Geometry?
Two-dimensional geometry lives on a flat page. Three-dimensional geometry lives in the world you actually inhabit. The moment you move from a plane to space, you need a third coordinate — and a third axis. Chapter 12 of Class 11 Mathematics introduces this extension: the three-dimensional coordinate system, how to locate points in space, calculate distances between them, find midpoints, and understand how space is divided by coordinate planes.
This is a relatively short chapter, with three exercises and a modest number of problems — but it lays the conceptual groundwork for the three-dimensional geometry studied in Class 12, which involves lines and planes in space. Students who build intuition here (by visualising where points like (2, –3, 4) sit relative to the coordinate planes) will find Class 12's 3D geometry significantly easier. For all Chapters must, read NCERT Solutions for Class 11 Maths and subject-wise NCERT Solutions for Class 11.
The three coordinates of a point in space — x, y, z — represent its signed distance from the yz-plane, xz-plane, and xy-plane respectively. This is a direct and natural extension of 2D coordinates, and Chapter 12 uses that familiarity to build confidence quickly. NCERT solutions keeps the problems focused and manageable, making this one of the more student-friendly chapters to score well in — provided you are thorough with the distance formula extension and the section formula in 3D.
Download PDF – All Exercises of NCERT Solutions for Class 11 Maths Chapter 12 Three-Dimensional Geometry
| Exercise | Topic Covered | Number of Questions |
|---|---|---|
| Exercise 12.1 | Coordinate Axes, Planes, Octants, and Point Location | 4 Questions |
| Exercise 12.2 | Distance Formula in 3D and Its Applications | 5 Questions |
| Exercise 12.3 | Section Formula in 3D; Midpoint Formula | 5 Questions |
| Miscellaneous | Mixed 3D Problems | 6 Questions |
Chapter 12 – Three-Dimensional Geometry: Concepts, Explanation and Key Tables
The Three-Dimensional Coordinate System
In 3D geometry, three mutually perpendicular axes — x, y, z — pass through a common point called the origin. They define three coordinate planes:
| Coordinate Plane | Formed By | Equation |
|---|---|---|
| XY-plane | x-axis and y-axis | z = 0 |
| YZ-plane | y-axis and z-axis | x = 0 |
| XZ-plane | x-axis and z-axis | y = 0 |
These three planes divide all of space into 8 regions called octants.
The Eight Octants and Their Sign Patterns
| Octant | x | y | z |
|---|---|---|---|
| I (XOYZ) | + | + | + |
| II | – | + | + |
| III | – | – | + |
| IV | + | – | + |
| V | + | + | – |
| VI | – | + | – |
| VII | – | – | – |
| VIII | + | – | – |
Location of a Point Relative to Coordinate Planes
| Point Lies On | Condition |
|---|---|
| x-axis | y = 0 and z = 0 |
| y-axis | x = 0 and z = 0 |
| z-axis | x = 0 and y = 0 |
| XY-plane | z = 0 |
| YZ-plane | x = 0 |
| XZ-plane | y = 0 |
| Origin | x = 0, y = 0, z = 0 |
Distance and Section Formulas in 3D
| Formula | Expression | Notes |
|---|---|---|
| Distance between P(x₁,y₁,z₁) and Q(x₂,y₂,z₂) | PQ = √[(x₂–x₁)² + (y₂–y₁)² + (z₂–z₁)²] | Direct extension of 2D formula; add z-component |
| Distance from origin to P(x,y,z) | OP = √(x² + y² + z²) | Set (x₁,y₁,z₁) = (0,0,0) |
| Section formula (internal division) | Point = [(mx₂+nx₁)/(m+n), (my₂+ny₁)/(m+n), (mz₂+nz₁)/(m+n)] | Divides PQ in ratio m:n internally |
| Section formula (external division) | Point = [(mx₂–nx₁)/(m–n), (my₂–ny₁)/(m–n), (mz₂–nz₁)/(m–n)] | Divides PQ in ratio m:n externally |
| Midpoint formula | M = [(x₁+x₂)/2, (y₁+y₂)/2, (z₁+z₂)/2] | Special case of section formula with m = n |
Centroid of a Triangle in 3D
If the vertices of a triangle are A(x₁,y₁,z₁), B(x₂,y₂,z₂), C(x₃,y₃,z₃), then:
Centroid G = [(x₁+x₂+x₃)/3, (y₁+y₂+y₃)/3, (z₁+z₂+z₃)/3]
This is the same structure as in 2D — simply include the z-coordinates.
Study Tips for Chapter 12
- This chapter is short but has high accuracy-to-effort ratio. Get all the octant sign patterns right — they appear in Exercise 12.1 and MCQs.
- The distance formula in 3D is structurally identical to the 2D version with a z-term added. If you know it in 2D, do not second-guess it in 3D.
- Always label the points clearly before substituting into the section formula — mixing m and n ratios or substituting coordinates in the wrong order is the most frequent error.
- Verify collinearity in 3D by checking that AB + BC = AC using the distance formula — the same logic as in 2D applies here.
FAQs on NCERT Solutions Class 11 Maths Chapter 12 Three-Dimensional Geometry
Three-Dimensional Geometry is the branch of mathematics that studies points, lines, and shapes in space. Unlike two-dimensional geometry, which deals only with length and width, three-dimensional geometry includes height as an additional dimension. In this chapter, students learn about the coordinate axes, coordinate planes, octants, and the position of points in space. Understanding these concepts helps students visualize objects and solve spatial problems more effectively. NCERT Solutions provide detailed explanations and examples that make the concepts easy to understand. This chapter serves as the foundation for advanced geometry topics and has applications in engineering, architecture, physics, and computer graphics.
In three-dimensional geometry, three mutually perpendicular axes known as the x-axis, y-axis, and z-axis intersect at a common point called the origin. These axes divide space into eight regions known as octants. Every point in space is represented by three coordinates that indicate its position relative to these axes. Understanding coordinate axes and octants is essential for locating points accurately and solving geometry problems. NCERT Solutions explain these concepts with diagrams and solved examples, helping students develop a clear understanding of spatial representation and coordinate systems used in higher mathematics.
Three-Dimensional Geometry is important because it helps students understand the mathematical representation of objects in space. It forms the basis for advanced topics such as vectors, analytical geometry, engineering drawing, and computer-aided design. Many scientific and technical fields use three-dimensional concepts to model real-world objects and structures. NCERT Solutions provide step-by-step explanations that strengthen conceptual understanding and help students develop visualization skills. Learning this chapter thoroughly prepares students for higher-level mathematics and competitive examinations where spatial reasoning and geometry-based problem-solving are frequently tested.
NCERT Solutions provide accurate and detailed answers to all textbook questions, making complex concepts easier to understand. The solutions explain coordinate systems, axes, planes, octants, and point representation through simple methods and examples. Students can use them for classroom learning, homework assistance, revision, and examination preparation. Regular practice improves accuracy, confidence, and conceptual clarity. Since this chapter introduces important spatial concepts that are used in higher mathematics, mastering NCERT Solutions helps students build a strong foundation and perform better in school and competitive examinations.