NCERT Solutions Class 11 Maths Chapter 11 introduces you to Three Dimensional Geometry through detailed solutions to all 13 questions across 3 exercises. You’ll learn how to plot points in 3D space using coordinates (x, y, z), calculate distances between two points using the distance formula, find section formulas for internal and external divisions, and determine the coordinates of centroids. These fundamental concepts are essential for understanding vectors, planes, and lines in 3D space in Chapter 12 and for JEE preparation.
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All exercises with step-by-step solutions | Updated 2025-26 | Free Download
Download PDF (Free)NCERT Solutions Class 11 Maths Chapter 11 Introduction to Three Dimensional Geometry – Complete Guide
NCERT Class 11 Chapter 11 – Introduction to Three Dimensional Geometry marks your exciting transition from the flat plane to the spatial world around you. You’ll explore how to represent any point in space using three coordinates (x, y, z) and understand the rectangular coordinate system that forms the foundation of 3D geometry. This chapter introduces you to concepts that are essential not just for CBSE board exams but also for competitive examinations and higher studies in mathematics, physics, and engineering.
📊 CBSE Class 11 Maths Chapter 11 – Exam Weightage & Marking Scheme
| CBSE Board Marks | Varies Marks |
| Unit Name | Unit 11 |
| Difficulty Level | Medium |
| Importance | Medium |
| Exam Types | CBSE Board, State Boards |
| Typical Questions | 1-2 questions |
You will learn to calculate the distance between two points in 3D space using the distance formula, which is a natural extension of the 2D distance formula you already know. The chapter covers section formulas (both internal and external division) that help you find coordinates of points dividing a line segment in a given ratio. You’ll also discover how to find the coordinates of the midpoint of a line segment and the centroid of a triangle in three-dimensional space.
For CBSE board exams, this chapter typically carries 4-6 marks and questions appear as short answer type (2-3 marks) or as part of longer problems. You can expect straightforward numerical problems on distance calculation, section formula applications, and coordinate-based proofs. The concepts you learn here form the building blocks for advanced topics like vectors, 3D lines, and planes in later chapters.
Quick Facts – Class 11 Chapter 11
| 📖 Chapter Number | Chapter 11 |
| 📚 Chapter Name | Introduction to Three Dimensional Geometry |
| ✏️ Total Exercises | 3 Exercises |
| ❓ Total Questions | 13 Questions |
| 📅 Updated For | CBSE Session 2025-26 |
Mastering this chapter opens doors to understanding the geometry of the physical world around you—from architecture and computer graphics to physics and engineering applications. With consistent practice of NCERT solutions and previous year CBSE questions, you’ll develop strong spatial visualization skills and confidence in handling three-dimensional problems effectively.
NCERT Solutions Class 11 Maths Chapter 11 – All Exercises PDF Download
Download exercise-wise NCERT Solutions PDFs for offline study
| Exercise No. | Topics Covered | Download PDF |
|---|---|---|
| EXERCISE 11.1 | Complete step-by-step solutions for 4 questions | 📥 Download PDF |
| EXERCISE 11.2 | Complete step-by-step solutions for 5 questions | 📥 Download PDF |
| Miscellaneous Exercise on Chapter 11 | Complete step-by-step solutions for 4 questions | 📥 Download PDF |
Introduction to Three Dimensional Geometry – Key Formulas & Concepts
Quick reference for CBSE exams
| Formula | Description | When to Use |
|---|---|---|
| Distance Formula (2 points) \(\sqrt{(x_2 – x_1)^2 + (y_2 – y_1)^2 + (z_2 – z_1)^2}\) | Calculates the distance between two points (x₁, y₁, z₁) and (x₂, y₂, z₂) in 3D space. Note: The order of subtraction doesn’t matter because you’re squaring the result. Make sure to include the z-coordinate difference. | Finding the distance between any two points, determining if points are equidistant from another point, or proving properties of shapes (e.g., showing a triangle is isosceles). |
| Section Formula (Internal Division) \(\left(\frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n}, \frac{mz_2 + nz_1}{m+n}\right)\) | Finds the coordinates of a point that divides the line segment joining (x₁, y₁, z₁) and (x₂, y₂, z₂) internally in the ratio m:n. Note: Remember to multiply each coordinate by the *opposite* ratio value. Ensure you are dividing *internally*. | When a question asks for the coordinates of a point dividing a line segment in a given ratio. Also used to find the centroid of a tetrahedron. |
| Section Formula (External Division) \(\left(\frac{mx_2 – nx_1}{m-n}, \frac{my_2 – ny_1}{m-n}, \frac{mz_2 – nz_1}{m-n}\right)\) | Finds the coordinates of a point that divides the line segment joining (x₁, y₁, z₁) and (x₂, y₂, z₂) externally in the ratio m:n. Note: The only difference from the internal division formula is the minus sign instead of plus. Be careful with the order of subtraction if m < n. | Similar to internal division, but when the point lies *outside* the line segment. Recognize keywords like ‘extended’ or ‘produced’. |
| Midpoint Formula \(\left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}, \frac{z_1 + z_2}{2}\right)\) | Finds the coordinates of the midpoint of the line segment joining (x₁, y₁, z₁) and (x₂, y₂, z₂). Note: It’s just the average of the coordinates. A special case of the section formula where m = n = 1. | When you need the middle point of a line segment, or to find the center of a sphere given two endpoints of a diameter. |
| Coordinates of Centroid of a Triangle \(\left(\frac{x_1 + x_2 + x_3}{3}, \frac{y_1 + y_2 + y_3}{3}, \frac{z_1 + z_2 + z_3}{3}\right)\) | Finds the coordinates of the centroid of a triangle with vertices (x₁, y₁, z₁), (x₂, y₂, z₂), and (x₃, y₃, z₃). Note: Average of the x-coordinates, y-coordinates, and z-coordinates. Easy to remember! | Finding the centroid (point of intersection of medians) of a triangle in 3D space. |
| Equation of XY-plane \(z = 0\) | Represents the XY-plane in 3D space. Note: Any point on the XY-plane has a z-coordinate of 0. Similarly, XZ-plane is y=0, YZ-plane is x=0. | When asked for the equation of the XY-plane. Also useful when finding the foot of the perpendicular from a point to the XY-plane (the z-coordinate will be 0). |
| Equation of a Line Parallel to X-axis \(y = b, z = c\) | Represents a line parallel to the x-axis passing through the point (a, b, c) Note: x can be any value, but y and z are fixed. | To represent a line parallel to X-axis. Use similar logic for lines parallel to Y or Z axes |
| Distance of a Point from XY-plane \(|z|\) | Represents the perpendicular distance of a point (x, y, z) from the XY-plane. Note: Take the absolute value of the z-coordinate. Distance is always positive! Similarly, distance from XZ-plane is |y|, and from YZ-plane is |x|. | When a question asks for the distance of a point from any of the coordinate planes. |
| Octants Sign Convention N/A | Memorize the sign conventions for x, y, and z in each of the eight octants. Note: A good way to remember it is to start with the four quadrants in the XY-plane and then add the z-coordinate (either positive or negative) for each to get the octants. | Identifying which octant a point lies in. This helps in visualizing the position of points. |
Frequently Asked Questions – NCERT Class 11 Maths Chapter 11
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