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Introduction to Three Dimensional Geometry

NCERT Class 11 Mathematics Chapter 11

Introduction to Three Dimensional Geometry for Class 11 Maths and JEE: axes and planes, coordinates of a point and the distance between two points.

4 topics Updated 8 Nov 2026

For: Class 11 and JEE, unless a topic says otherwise

In the syllabus: JEE Class 11 syllabus

With your textbook

NCERT Mathematics (Class 11), Chapter 11 “Introduction to Three Dimensional Geometry”, reprint 2026-27.

Open the official NCERT chapter (PDF, 1.3 MB, ncert.nic.in)

Checked on 19 Oct 2026.

Learn in this order

  1. Coordinate axes and planes in space

    The x, y and z axes, the XY, YZ and ZX planes with their equations, and the signs of all eight octants.

    NCERT Sections 11.1–11.2 11 min read 6 solved examples 12 practice questions

  2. Coordinates of a point in space

    The triplet (x, y, z), points on the x-axis or in the XY-plane, and distances from the planes.

    NCERT Section 11.3 12 min read 6 solved examples 12 practice questions

  3. Distance between two points in space

    The distance between two points in space, its proof by Pythagoras, distance from the origin, solved examples.

    NCERT Section 11.4 15 min read 6 solved examples 12 practice questions

  4. Section formula in three dimensions

    Section formula in 3D for JEE Main: internal and external division, the mid-point and centroid, and the ratio in which a coordinate plane cuts a segment.

    JEE only 15 min read 6 solved examples 12 practice questions

Key formulas in this chapter

Coordinate axes and planes

  • XY-plane: z = 0; YZ-plane: x = 0; ZX-plane: y = 0
  • x-axis: y = 0 and z = 0
  • octants I to IV: (+, +, +), (−, +, +), (−, −, +), (+, −, +)
  • octants V to VIII: (+, +, −), (−, +, −), (−, −, −), (+, −, −)

Coordinates of a point in space

  • origin (0, 0, 0); x-axis (x, 0, 0); y-axis (0, y, 0); z-axis (0, 0, z)
  • XY-plane (x, y, 0); YZ-plane (0, y, z); ZX-plane (x, 0, z)
  • foot of P(x, y, z) on the XY-plane: (x, y, 0); on the x-axis: (x, 0, 0)
  • distances from the YZ-, ZX- and XY-planes: |x|, |y|, |z|

Distance between two points

  • PQ = (x − x) + (y − y) + (z − z)
  • OP = √(x + y + z)
  • distance of P(x, y, z) from the x-axis: √(y + z)
  • sphere, centre (a, b, c), radius r: (x − a) + (y − b) + (z − c) = r

Section formula in 3D

  • internal m: n: ((mx + nx)/(m + n), (my + ny)/(m + n), (mz + nz)/(m + n))
  • external m: n: ((mx − nx)/(m − n), (my − ny)/(m − n), (mz − nz)/(m − n))
  • mid-point: ((x + x)/2, (y + y)/2, (z + z)/2)
  • centroid: ((x + x + x)/3, (y + y + y)/3, (z + z + z)/3)
  • XY-plane unique intersection: signed ratio −z: z, z ≠ z

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