Class 11 Maths NCERT Solutions Chapter 11 Introduction to Three Dimensional Geometry | Boundless Maths
Chapter 11 Class 11 Maths NCERT Solutions · Unit III · Coordinate Geometry

Class 11 Maths NCERT Solutions Chapter 11: Introduction to Three Dimensional Geometry

Free, step-by-step NCERT Solutions for both exercises of this chapter — the coordinate axes, coordinate planes and octants of three-dimensional space, and the distance formula between two points in space — plus the Miscellaneous Exercise on the fourth vertex of a parallelogram, medians of a triangle, the centroid, and locus problems. Solved the way CBSE awards marks, with the key formulas and the mistakes that cost students marks every year, right on this page.

Every point studied so far in coordinate geometry has lived on a flat plane, located by just two numbers. This chapter takes the natural next step: locating a point in space using three numbers instead of two, measured as perpendicular distances from three mutually perpendicular coordinate planes. Once a point can be written as an ordered triplet (x, y, z), almost every idea from plane coordinate geometry — the distance formula, collinearity, the centroid of a triangle — extends into three dimensions with one extra term added for the new z-coordinate.

3Exercises (incl. Misc.)
13Total Questions
2026-27CBSE Syllabus
100%Solved

Class 11 Maths NCERT Solutions Chapter 11 — Overview

This Class 11 Maths NCERT Solutions Chapter 11 hub covers Introduction to Three Dimensional Geometry, the chapter that extends coordinate geometry from the flat xy-plane into full three-dimensional space. Three mutually perpendicular planes — the XY, YZ and ZX planes — meet at the origin and divide all of space into eight regions called octants. Every point in space corresponds to a unique ordered triplet (x, y, z), where each coordinate is the point's perpendicular distance from one of the three coordinate planes, and the sign of each coordinate tells you exactly which octant the point lies in.

Exercise 11.2 introduces the distance formula for two points in space — a direct extension of the familiar 2D formula, with one extra squared term for the difference in z-coordinates — and uses it to test whether points are collinear, or form an isosceles, right-angled, or parallelogram-shaped set of vertices. The Miscellaneous Exercise applies these ideas further: finding a missing vertex of a parallelogram, computing the lengths of a triangle's medians, using the centroid formula to recover unknown coordinates, and setting up the equation of a locus from a distance condition.

How the Chapter Builds

One Idea Leads to the Next

1

Coordinate Axes & Planes

The X, Y and Z axes, and the three coordinate planes they define. Exercise 11.1.

→
2

Octants & the Coordinates of a Point

Locating a point as an ordered triplet (x, y, z), and identifying its octant from the signs. Exercise 11.1.

→
3

Distance between Two Points

The 3D distance formula, and using it to test collinearity and identify triangles/parallelograms. Exercise 11.2.

→
4

Applications

Missing vertices, medians, the centroid formula, and locus problems. Misc. Exercise.

Quick Reference

Important Formulas — Chapter 11

Everything you need before you start solving. This is a summary for quick recall — the Formula Cards below has the full printable version for all of Introduction to Three Dimensional Geometry.

Coordinate Axes, Planes & Octants (§11.2–11.3)

Points on an axis

(x,0,0),\quad(0,y,0),\quad(0,0,z)

A point on the x-axis, y-axis, or z-axis has zero for the other two coordinates.

Points in a coordinate plane

(x,y,0),\quad(0,y,z),\quad(x,0,z)

A point in the XY, YZ, or ZX plane has zero for whichever coordinate that plane doesn't contain.

Number of octants

2^3=8\text{ octants (I-VIII)}

Each octant corresponds to one unique combination of signs for x, y and z.

Distance between Two Points (§11.4)

Distance formula in 3D

PQ=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2+(z_2-z_1)^2}

The same 2D distance formula, extended with one extra squared term for the z-coordinate.

Collinearity test

PQ+QR=PR

Three points are collinear exactly when the distance between the outer two equals the sum of the distances to the middle one.

Right-angle test (no Pythagoras needed)

AB^2+BC^2=CA^2

If the squared side lengths satisfy this relation, the triangle has a right angle at the vertex between the two shorter sides.

Applications (Miscellaneous Exercise)

Centroid of a triangle

\left(\dfrac{x_1+x_2+x_3}{3},\dfrac{y_1+y_2+y_3}{3},\dfrac{z_1+z_2+z_3}{3}\right)

The average of the three vertices' coordinates in each direction.

Fourth vertex of a parallelogram

\text{diagonals share the same midpoint}

The diagonals of a parallelogram bisect each other — the key fact used to find a missing vertex.

Locus from a distance condition

PA^2+PB^2=k^2

Expand each squared distance in terms of (x, y, z) and simplify to get the equation of the locus.

Decision Guide

Which 3D Geometry Technique Applies?

A quick way to decide, once you know what the question is actually asking for.

What the question is askingUse thisWhy
Name the octant a point lies inCheck the sign of each coordinateMatch the +/− pattern of (x, y, z) against the standard octant sign table (§11.3).
Find the distance between two points in space3D distance formulaAdd one extra squared term, (z2 − z1)², to the familiar 2D formula (§11.4).
Show three points lie on a straight lineCollinearity via distancesCompute all three pairwise distances and check that the two shorter ones add up to the longest (§11.4).
Verify a triangle is isosceles or right-angledCompare squared side lengthsIsosceles: two sides equal; right-angled: sum of two squared sides equals the third squared side (§11.4).
Show four points form a parallelogramOpposite sides equal, or diagonals bisect each otherEither check AB = CD and BC = DA, or confirm both diagonals share the same midpoint (§11.4).
Find a missing vertex or use the centroidSection/midpoint or centroid formulaA parallelogram's diagonals bisect each other; a centroid is the average of all three vertices (Misc. Exercise).
Find the equation satisfied by a moving point PSet up and simplify the locus equationWrite the given distance condition in terms of (x, y, z) using the distance formula, then expand and simplify (Misc. Exercise).
Avoid These

Common Mistakes to Avoid in This Chapter

Drawn from where students actually lose marks across both exercises.

  • Misreading the octant sign table — a point's octant depends on the signs of all three coordinates together, not just one; a careless sign check leads to the wrong octant number.
  • Forgetting the z-coordinate term — when adapting a 2D distance/section/midpoint formula to 3D, it's easy to drop the extra z-term out of habit; every formula needs one more piece for the third dimension.
  • Using the Pythagoras theorem when the question forbids it — several questions specifically ask to verify a right angle "without using the Pythagoras theorem"; use the squared-side-length relation instead of assuming it.
  • Only checking one pair of opposite sides for a parallelogram — both pairs of opposite sides must be shown equal (or both diagonals shown to share a midpoint); checking just one pair isn't sufficient proof.
  • Sign errors when squaring a distance condition — locus problems that involve PA² + PB² = k² require careful expansion of each squared bracket; a dropped negative sign early on throws off the entire simplified equation.
  • Averaging incorrectly for the centroid — the centroid formula divides each coordinate sum by 3 (for a triangle); forgetting this and just adding the coordinates is a simple but common slip.
Solve Chapter-Wise

Class 11 Maths NCERT Solutions Chapter 11 — Choose an Exercise

11.1

Exercise 11.1

Coordinate Axes, Coordinate Planes and Octants — locating points and identifying octants · 4 questions

Solve Exercise 11.1 →
11.2

Exercise 11.2

Distance between Two Points — collinearity, triangles, parallelograms, and locus equations · 5 questions

Solve Exercise 11.2 →
M

Miscellaneous Exercise

Missing vertices, medians of a triangle, the centroid, and locus problems · 4 questions

Solve Miscellaneous →

📐 Keep the Formulas Handy

Every formula for Introduction to Three Dimensional Geometry — plus every other Class 11 Maths chapter — in one printable PDF.

Get Formula Cards →
Common Questions

Frequently Asked Questions

Quick answers from Class 11 Maths NCERT Solutions Chapter 11, Introduction to Three Dimensional Geometry.

How many exercises are there in Chapter 11, Introduction to Three Dimensional Geometry?
There are two main exercises — 11.1 (Coordinate Axes, Coordinate Planes and Octants, 4 questions) and 11.2 (Distance between Two Points, 5 questions) — plus a Miscellaneous Exercise of 4 questions, totalling 13 questions.
How many octants does 3D space have, and how do you find which one a point lies in?
The three coordinate planes divide space into eight octants, numbered I through VIII. Which octant a point (x, y, z) lies in is decided entirely by the signs of its coordinates — each octant corresponds to one particular combination of positive and negative signs for x, y and z.
What is the distance formula in three dimensions?
The distance between two points P(x1, y1, z1) and Q(x2, y2, z2) in space is PQ = √[(x2 − x1)² + (y2 − y1)² + (z2 − z1)²] — the same distance formula from two-dimensional coordinate geometry, with one extra squared term added for the z-coordinate.
Where can I find the official NCERT textbook for this chapter?
Introduction to Three Dimensional Geometry is Chapter 11 of the NCERT Class 11 Mathematics textbook, published by the National Council of Educational Research and Training (NCERT) and prescribed by CBSE. You can download the official textbook PDF directly from ncert.nic.in, NCERT's official website — the solutions on this page follow the exercises exactly as they appear there.
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