Class 11 Maths · Original EZT practice
Introduction to Three Dimensional Geometry (Class 11 Chapter 11): summary and practice
Three Dimensional Geometry adds a third axis: the x, y and z axes, the eight octants with their sign patterns, coordinates of a point in space, and the distance formula sqrt((x2 - x1)^2 + (y2 - y1)^2 + (z2 - z1)^2). All questions are original EZT tutor practice following the chapter's own section order.
6 original worked examples, 6 fresh self-check questions and 4 mixed questions, with explained answers. These are not copied textbook exercises, an official chapter index or a full answer key. No affiliation with NCERT is claimed.
Official sources for topic and edition context, checked 21 October 2026:
1. Why a third axis
Two coordinates locate a point on a floor; locating a point in a room needs a third number - its height. The z-axis stands perpendicular to the familiar XY plane.
2. Axes and coordinate planes
Three mutually perpendicular axes meet at the origin. Each pair defines a coordinate plane: XY, YZ and ZX. The three planes carve space into eight octants.
3. Coordinates of a point in space
A point is (x, y, z): signed distances from the YZ, ZX and XY planes respectively. The octant is read off the three signs, from (+, +, +) in the first octant outward.
4. Distance between two points
d = sqrt((x2 - x1)^2 + (y2 - y1)^2 + (z2 - z1)^2) - the 2D formula with one more squared gap. From the origin it is sqrt(x^2 + y^2 + z^2).
Choose, solve, check, then try a fresh question.
Before calculating, read the method prompt and name the condition that makes it useful. Write your own attempt. Open the steps only when you need them, and use the check to catch a wrong sign, unit or interpretation. Finish with the fresh question before reading its answer.
Distance between two points in space
Choose the method: Square all three gaps, add, take the root.
EZT practice question
Find the distance between (1, 2, 3) and (4, 6, 3).
Show the worked solution
- Gaps: 4 - 1 = 3, 6 - 2 = 4, 3 - 3 = 0.
- Squares: 9, 16, 0.
- d = sqrt(9 + 16 + 0) = sqrt(25).
Answer: d = 5.
Check: The z-coordinates match, so this collapses to a 2D 3-4-5 triangle - the answer 5 fits.
Watch for this: Do not drop the z-term by habit - include (z2 - z1)^2 even when it is 0.
Now try it yourself
Distance from the origin to (3, 4, 12)?
Check your answer
13.
sqrt(9 + 16 + 144) = sqrt(169).
Read the octant from the signs
Choose the method: Write the signs of x, y, z in order - that triple names the octant.
EZT practice question
Which octant holds (2, -3, 5)?
Show the worked solution
- x = 2 is positive.
- y = -3 is negative.
- z = 5 is positive.
Answer: The (+, -, +) octant.
Check: Each sign triple is one of the 2 x 2 x 2 = 8 octants; (+, -, +) is a valid distinct one.
Watch for this: Order matters: (+, -, +) and (-, +, +) are different octants.
Now try it yourself
Signs of the octant containing (-1, -2, 3)?
Check your answer
(-, -, +).
x and y negative, z positive.
Distance from the origin
Choose the method: sqrt(x^2 + y^2 + z^2).
EZT practice question
Which is farther from the origin: (1, 2, 2) or (2, 0, 2)?
Show the worked solution
- (1, 2, 2): sqrt(1 + 4 + 4) = 3.
- (2, 0, 2): sqrt(4 + 0 + 4) = sqrt(8), about 2.83.
- Compare.
Answer: (1, 2, 2) is farther - 3 against about 2.83.
Check: Squared distances 9 and 8 preserve the order even before rooting.
Watch for this: Compare the SQUARES first if the roots look awkward - 9 vs 8 settles it.
Now try it yourself
Distance of (2, -3, 6) from the origin?
Check your answer
7.
sqrt(4 + 9 + 36) = sqrt(49).
Average coordinates in 3D
Choose the method: The midpoint averages each coordinate - same idea as 2D with one more average.
EZT practice question
Midpoint of (2, 1, 4) and (6, 5, -2)?
Show the worked solution
- x: (2 + 6)/2 = 4.
- y: (1 + 5)/2 = 3.
- z: (4 + (-2))/2 = 1.
Answer: (4, 3, 1).
Check: (4, 3, 1) is sqrt(9 + 4 + 9) = sqrt(22) from each endpoint - equal, as a midpoint must be.
Watch for this: Average the z-coordinates too: (4 + (-2))/2 is 1, not 3.
Now try it yourself
Midpoint of (0, 0, 0) and (4, 6, 2)?
Check your answer
(2, 3, 1).
Halve each coordinate.
Name the coordinate plane
Choose the method: The plane is named by the two axes it contains; the third coordinate is 0 on it.
EZT practice question
Where does (5, 0, -2) sit?
Show the worked solution
- y = 0.
- The point uses only x and z.
- x and z span the ZX plane.
Answer: On the ZX plane.
Check: Every point with y = 0 lies in the ZX plane - (5, 0, -2) qualifies.
Watch for this: The XY plane is z = 0, the YZ plane is x = 0 - match the ZERO coordinate to the missing letter.
Now try it yourself
Where does (0, 7, 3) sit?
Check your answer
On the YZ plane.
x = 0.
Find an unknown coordinate from a distance
Choose the method: Set up the distance equation and solve the square.
EZT practice question
The point (x, 0, 0) is 5 units from (1, 0, 0). Find x.
Show the worked solution
- (x - 1)^2 + 0 + 0 = 25.
- (x - 1)^2 = 25.
- x - 1 = 5 or -5.
Answer: x = 6 or x = -4.
Check: Both lie on the x-axis at distance 5 from (1, 0, 0): |6 - 1| = 5 and |-4 - 1| = 5.
Watch for this: A square has TWO roots - dropping x = -4 loses half the answer.
Now try it yourself
(0, y, 0) is 3 units from the origin. Find y.
Check your answer
y = 3 or y = -3.
y^2 = 9.
Try a mixed set
Choose a method or explanation before opening each answer.
1. 3D distance formula?
Show answer and reason
sqrt((x2 - x1)^2 + (y2 - y1)^2 + (z2 - z1)^2).
The 2D formula plus the z gap.
2. From the origin?
Show answer and reason
sqrt(x^2 + y^2 + z^2).
Same formula, one end at (0, 0, 0).
3. How many octants?
Show answer and reason
Eight.
Three signs, two choices each: 2 x 2 x 2.
4. The XY plane is which equation?
Show answer and reason
z = 0.
The plane holds the x and y axes.
Try it yourself
Work these on paper before checking any notes. Answers are deliberately not shown here.
- Distance between (0, 0, 0) and (3, 4, 12).
- Signs of the octant containing (-1, -2, 3).
- Distance between (1, 2, 3) and (4, 6, 3).
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Questions about this practice page
- What is an octant?
- One of the eight regions into which the three coordinate planes divide space. Each octant is named by the signs of x, y and z inside it, starting from (+, +, +) as the first.
- How does the 3D distance formula extend the 2D one?
- It is the same Pythagoras idea applied twice: the 2D distance across the floor, then a second right triangle standing on it using the height gap. The result simply adds the (z2 - z1)^2 term under the root.
- Is the section formula part of this chapter?
- Not in the Reprint 2026-27. The chapter runs 11.1 to 11.4 and ends with the distance formula and summary - the 3D section formula is not in this edition.
- How do I remember which plane is XY, YZ or ZX?
- A coordinate plane is named after the two axes it contains, and the remaining coordinate is 0 everywhere on it. On the XY plane z = 0; on the YZ plane x = 0; on the ZX plane y = 0.
- Are these the NCERT exercise questions for 3D Geometry (Class 11)?
- No. EZT pages are original practice following the chapter's section order - use them to prepare for the textbook exercises, not as a substitute for your book.