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CBSE Class 11 Mathematics★ 4-6 marks · CBSE Class 11 Mathematics Chapter 11

Introduction to Three-Dimensional Geometry

The world has THREE dimensions — and so does mathematics. This chapter extends coordinate geometry from the 2D plane to 3D SPACE: coordinates, distance formula, section formula, and octants. CBSE Class 11 Mathematics Chapter 11.

⏱ 18 min readEasy difficulty✓ Free · NCERT-aligned

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By the end of this chapter you'll be able to…

1Identify the three coordinate axes, three coordinate planes, and the eight octants in a 3D system
2Represent a point in 3D space as an ordered triplet (x, y, z) and state distances from coordinate planes
3Apply the 3D distance formula to find the distance between two points
4Use the section formula (internal division) and midpoint formula for points in 3D
5Classify the octant of a given point from the signs of its three coordinates

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Why this chapter matters

3D Geometry in Class 11 lays the coordinate foundation for the full 3D vectors, lines, and planes chapter in Class 12, which carries 8-10 marks. The distance and section formulas here are used directly in those higher-level problems.

Before you start — revise these

A 5-minute refresher here will save you 30 minutes of confusion below.

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Straight Lines

2D coordinate geometry (distance formula, midpoint) directly extended to 3D here

Introduction to Three-Dimensional Geometry

"The plane was not enough. The world required a third axis."

1. Chapter Overview

Coordinate geometry in Class 9-10 was 2D (x and y axes). This chapter adds the THIRD DIMENSION (z-axis). You'll work with POINTS IN 3D SPACE: (x, y, z). The chapter covers: coordinate axes and planes, distance formula in 3D, section formula in 3D, and dividing 3D space into OCTANTS.

2. Three-Dimensional Coordinate System

Axes

X-axis, Y-axis, Z-axis — all MUTUALLY PERPENDICULAR
Intersect at the ORIGIN O(0, 0, 0)
Three COORDINATE PLANES: XY-plane (z=0), YZ-plane (x=0), XZ-plane (y=0)
These three planes divide space into 8 regions called OCTANTS

Sign Convention (Octants)

Octant	x	y	z
I	+	+	+
II	—	+	+
III	—	—	+
IV	+	—	+
V	+	+	—
VI	—	+	—
VII	—	—	—
VIII	+	—	—

3. Coordinates of a Point in 3D

A point P in space is represented by the ordered triplet (x, y, z)
x = perpendicular distance from YZ-plane
y = perpendicular distance from XZ-plane
z = perpendicular distance from XY-plane

4. Distance Formula in 3D

The distance between two points P(x₁, y₁, z₁) and Q(x₂, y₂, z₂):

$P Q = (x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2} + (z_{2} - z_{1})^{2}$

This is a natural extension of the 2D distance formula — just add the z term!

Distance from origin (0, 0, 0): √(x² + y² + z²)

5. Section Formula in 3D

Internal Division

The point dividing P(x₁, y₁, z₁) and Q(x₂, y₂, z₂) INTERNALLY in the ratio m : n:

$(\frac{m x _{2} + n x _{1}}{m + n}, \frac{m y _{2} + n y _{1}}{m + n}, \frac{m z _{2} + n z _{1}}{m + n})$

Midpoint

Put m = n = 1 in the internal division formula:

$(\frac{x _{1} + x _{2}}{2}, \frac{y _{1} + y _{2}}{2}, \frac{z _{1} + z _{2}}{2})$

6. Exam Focus

Coordinates of a point — (x, y, z), distances from coordinate planes
Distance formula in 3D — √[(x₂-x₁)² + (y₂-y₁)² + (z₂-z₁)²]
Section formula — internal division and midpoint
Octants — signs of x, y, z in each

7. Conclusion

3D geometry is the NATURAL extension of everything you know from 2D:

Add a Z-AXIS. Now points have 3 coordinates instead of 2.
DISTANCE: Pythagoras in 3D — just add the third square term
SECTION: Divides exactly like 2D — just repeat the formula for z
This chapter is the FOUNDATION for 3D vector geometry (Class 12)

'We live in a 3D world. Mathematics in 2D was only ever preparation.'

Key formulas & results

Everything you need to memorise, in one card. Screenshot this for revision.

3D Distance Formula

PQ = √[(x₂−x₁)² + (y₂−y₁)² + (z₂−z₁)²]

Pythagoras theorem in three dimensions; the 2D formula with an additional (z₂−z₁)² term

Distance from Origin

OP = √(x² + y² + z²)

Special case of the distance formula with P₁ = origin (0,0,0)

Section Formula (Internal Division)

P = ((mx₂+nx₁)/(m+n), (my₂+ny₁)/(m+n), (mz₂+nz₁)/(m+n))

Point P divides P₁P₂ internally in ratio m:n; identical structure to 2D formula with z added

Midpoint Formula

M = ((x₁+x₂)/2, (y₁+y₂)/2, (z₁+z₂)/2)

Special case of section formula with m = n = 1

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Common mistakes & fixes

These are the exact errors that cost students marks in board exams. Read them once, save yourself the trouble.

WATCH OUT

✗ Forgetting to include the z-component in the distance formula

✓ The 3D distance formula has THREE squared terms: (x₂−x₁)², (y₂−y₁)², AND (z₂−z₁)². Omitting the z-term gives the 2D distance, which is incorrect for 3D points.

WATCH OUT

✗ Confusing which plane a point lies on when one coordinate is zero

✓ If z=0, the point lies on the XY-plane. If x=0, it lies on the YZ-plane. If y=0, it lies on the XZ-plane. Zero coordinate = lies IN that perpendicular plane.

WATCH OUT

✗ Getting the section formula ratio backwards (confusing m:n with n:m)

✓ If P divides P₁P₂ in ratio m:n, the formula uses m with P₂'s coordinates and n with P₁'s coordinates: x = (mx₂+nx₁)/(m+n). Always draw a diagram to verify the order.

WATCH OUT

✗ Stating that the octant sign pattern for z is the same as in 2D quadrants

✓ 3D has 8 octants. The first four octants I–IV have z>0; octants V–VIII have z<0. The signs of x and y follow the same quadrant pattern within each z-layer.

NCERT exercises (with solutions)

Every NCERT exercise from this chapter — what it covers and how many questions to expect.

Ex 11.1

Exercise 11.1

Identifying octants, distances from coordinate planes, plotting 3D points

Questions

Ex 11.2

Exercise 11.2

3D distance formula — between two points, equidistant conditions

Questions

Ex 11.3

Exercise 11.3

Section formula and midpoint formula in 3D

Questions

Practice problems

Try each one yourself before tapping "Show solution". Active recall > rereading.

Q1EASY· Distance Formula

Find the distance between the points A(1, 2, 3) and B(4, 6, 3).

▶ Show solution

AB = √[(4−1)²+(6−2)²+(3−3)²] = √[9+16+0] = √25 = 5.

Q2MEDIUM· Section Formula

Find the coordinates of the point that divides the line segment joining A(1, −2, 3) and B(3, 4, −5) internally in the ratio 2:3.

▶ Show solution

Using section formula with m=2, n=3: x = (2×3+3×1)/(2+3) = (6+3)/5 = 9/5. y = (2×4+3×(−2))/5 = (8−6)/5 = 2/5. z = (2×(−5)+3×3)/5 = (−10+9)/5 = −1/5. Point = (9/5, 2/5, −1/5).

Q3HARD· Equidistant Point

Show that the points A(1, 2, 3), B(−1, −2, −1), C(2, 3, 2) are vertices of a triangle and find which type by computing all three side lengths.

▶ Show solution

AB = √[(−1−1)²+(−2−2)²+(−1−3)²] = √[4+16+16] = √36 = 6. BC = √[(2+1)²+(3+2)²+(2+1)²] = √[9+25+9] = √43. AC = √[(2−1)²+(3−2)²+(2−3)²] = √[1+1+1] = √3. Since AB ≠ BC ≠ AC and AB² ≠ BC²+AC² (6²=36 ≠ 43+3=46), the triangle is scalene and not right-angled. All sides are different, so it is a scalene triangle.

5-minute revision

The whole chapter, distilled. Read this the night before the exam.

•Three coordinate planes: XY-plane (z=0), YZ-plane (x=0), XZ-plane (y=0); they divide space into 8 octants
•Octant I: (+,+,+); Octant II: (−,+,+); Octant III: (−,−,+); Octant IV: (+,−,+); Octants V–VIII have z<0
•Distance PQ = √[(x₂−x₁)²+(y₂−y₁)²+(z₂−z₁)²]; distance from origin = √(x²+y²+z²)
•Section formula (internal, m:n): ((mx₂+nx₁)/(m+n), (my₂+ny₁)/(m+n), (mz₂+nz₁)/(m+n))
•Midpoint = average of coordinates: ((x₁+x₂)/2, (y₁+y₂)/2, (z₁+z₂)/2)
•A point on the x-axis has coordinates (x,0,0); on the y-axis (0,y,0); on the z-axis (0,0,z)
•If a point lies in the XY-plane, its z-coordinate = 0; distance from XY-plane = |z|

CBSE marks blueprint

Where the marks come from in this chapter — so you can plan your prep.

Typical chapter weightage: 4-6 marks

Question type	Marks each	Typical count	What it tests
Short Answer	2	1	Distance between two 3D points, octant identification
Long Answer	4	1	Section formula, midpoint, showing geometric properties of triangles/quadrilaterals in 3D

Prep strategy

Treat every 3D formula as the 2D version with a z-term added — this mental model eliminates the need to memorise separate formulas
Practice identifying octants from coordinate signs using the table of 8 octants until it is automatic
For section formula problems, always write out the ratio m:n explicitly before substituting to avoid getting it backwards

Where this shows up in the real world

This chapter isn't just an exam topic — it lives in the world around you.

3D Modelling and Computer Graphics

Every vertex in a 3D video game or CAD model is represented as an (x,y,z) coordinate; the 3D distance formula is used to compute distances between objects for collision detection.

GPS and Air Traffic Control

Aircraft positions are specified as (longitude, latitude, altitude) — a 3D coordinate system. Distance between aircraft uses the 3D distance formula for collision avoidance calculations.

Crystallography

The positions of atoms in a crystal lattice are 3D coordinates; inter-atomic distances computed using the distance formula determine the crystal's physical and chemical properties.

Exam strategy

Battle-tested tips from teachers and toppers for this chapter.

For octant questions, determine the sign of each coordinate (positive or negative) — the octant number follows directly from the sign pattern
In distance questions, write out the formula, substitute neatly, simplify under the root — do not skip algebraic steps
For section formula, always state the ratio m:n and which point is P₁ and which is P₂ before computing
Check your answer by verifying: midpoint must be equidistant from both endpoints

Going beyond the textbook

For olympiad aspirants and curious learners — topics that build on this chapter.

Centroid of a tetrahedron with vertices A, B, C, D is the average of all four coordinate triplets: G = ((x_A+x_B+x_C+x_D)/4, ...)
Collinearity in 3D: three points are collinear if the direction ratios of two connecting vectors are proportional — no area formula like 2D

Where else this chapter is tested

CBSE board isn't the only one — other exams test this chapter too.

CBSE Class 11 BoardHigh

JEE MainHigh

JEE AdvancedMedium

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Questions students ask

The real ones — pulled from the Q&A community and tutor sessions.

A quadrant is a division of the 2D plane into 4 regions (by x and y axes). An octant is a division of 3D space into 8 regions (by x, y, and z axes). 2D has 2²=4 regions; 3D has 2³=8 regions.

Distance from XY-plane = |z|. Distance from YZ-plane = |x|. Distance from XZ-plane = |y|. The distance equals the absolute value of the perpendicular coordinate.

Yes, but for external division in ratio m:n, use (mx₂−nx₁)/(m−n) — subtraction instead of addition. The point lies outside the segment on the m-side.

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