Worksheets · Algebra 2

Points and planes in 3D worksheet

In three dimensions the x-axis comes toward you, the y-axis runs right and the z-axis goes up. To plot a point, walk out along x, then across in the y direction, then up or down for z. A plane ax + by + cz = d crosses each axis where the other two letters are zero, and those three intercepts are enough to sketch it.

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Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Points and Planes in Three Dimensions

Date Period

Answer each.

  1. Write the coordinates of the point.xyz111222333444555−5−5−5
  2. Plot the point (−2,4,−5)xyz111222333444555−5−5−5
  3. Starting at the origin, how do you get to (−5,−4,−2)?
  4. Find the intercepts: −x−7y−2z=−11
  5. Sketch the plane: −5x−5y−4z=20xyz111222333444555−5−5−5
  6. Write the traces: 9x−y−7z=−19
  7. Write the coordinates of the point.xyz111222333444555−5−5−5
  8. Plot the point (5,1,1)xyz111222333444555−5−5−5
  9. Starting at the origin, how do you get to (−6,4,2)?
  10. Find the intercepts: −9x+5y−2z=7
  11. Sketch the plane: −4x+y−z=4xyz111222333444555−5−5−5
  12. Write the traces: 6x+6y+9z=−20
  13. Write the coordinates of the point.xyz111222333444555−5−5−5
  14. Plot the point (1,−1,−4)xyz111222333444555−5−5−5
  15. Starting at the origin, how do you get to (−1,5,8)?
  16. Find the intercepts: −7x−4y+2z=19
  17. Sketch the plane: 5x+3y+5z=15xyz111222333444555−5−5−5
  18. Write the traces: −7x+7y−z=12
  19. Write the coordinates of the point.xyz111222333444555−5−5−5
  20. Plot the point (−3,−2,2)xyz111222333444555−5−5−5

Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Points and Planes in Three DimensionsAnswer keyVersion 1

Date Period

Answer each.

  1. Write the coordinates of the point.xyz111222333444555−5−5−5(1,5,5)
  2. Plot the point (−2,4,−5)xyz111222333444555−5−5−5(−2,4,−5)
  3. Starting at the origin, how do you get to (−5,−4,−2)?5 units back,  4 units left,  2 units down
  4. Find the intercepts: −x−7y−2z=−11x-intercept 11, y-intercept 117, z-intercept 112
  5. Sketch the plane: −5x−5y−4z=20xyz111222333444555−5−5−5Intercepts (−4,0,0), (0,−4,0), (0,0,−5)
  6. Write the traces: 9x−y−7z=−19xy: 9x−y=−19 xz: 9x−7z=−19 yz: y+7z=19
  7. Write the coordinates of the point.xyz111222333444555−5−5−5(2,2,−5)
  8. Plot the point (5,1,1)xyz111222333444555−5−5−5(5,1,1)
  9. Starting at the origin, how do you get to (−6,4,2)?6 units back,  4 units right,  2 units up
  10. Find the intercepts: −9x+5y−2z=7x-intercept −79, y-intercept 75, z-intercept −72
  11. Sketch the plane: −4x+y−z=4xyz111222333444555−5−5−5Intercepts (−1,0,0), (0,4,0), (0,0,−4)
  12. Write the traces: 6x+6y+9z=−20xy: 3x+3y=−10 xz: 6x+9z=−20 yz: 6y+9z=−20
  13. Write the coordinates of the point.xyz111222333444555−5−5−5(5,−5,0)
  14. Plot the point (1,−1,−4)xyz111222333444555−5−5−5(1,−1,−4)
  15. Starting at the origin, how do you get to (−1,5,8)?1 unit back,  5 units right,  8 units up
  16. Find the intercepts: −7x−4y+2z=19x-intercept −197, y-intercept −194, z-intercept 192
  17. Sketch the plane: 5x+3y+5z=15xyz111222333444555−5−5−5Intercepts (3,0,0), (0,5,0), (0,0,3)
  18. Write the traces: −7x+7y−z=12xy: 7x−7y=−12 xz: 7x+z=−12 yz: 7y−z=12
  19. Write the coordinates of the point.xyz111222333444555−5−5−5(2,−4,−5)
  20. Plot the point (−3,−2,2)xyz111222333444555−5−5−5(−3,−2,2)

Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

How to do these

Find the intercepts: 4x+6y+3z=12

  1. Set y and z to 0: 4x = 12, so x = 3.
  2. Set x and z to 0: 6y = 12, so y = 2.
  3. Set x and y to 0: 3z = 12, so z = 4.

The answer is x-intercept 3y-intercept 2z-intercept 4.

Where students go wrong

Drawing the x-axis to the right out of habit. In a three-dimensional picture y goes to the right and x comes out toward you, so a point with a large x sits low and to the left.

Also called 3D coordinate system, plotting points in 3D, graphing planes or traces of a plane.

What you can put on this worksheet

Read a plotted point
Write the coordinates of the point. — a point plotted on three-dimensional axes → (3,1,−3)
Plot a point
Plot the point (3,1,−3) — three-dimensional axes → (3,1,−3)
Describe where a point is
Starting at the origin, how do you get to (3,1,−3)? → 3 units forward,  1 unit right,  3 units down
The intercepts of a plane
Find the intercepts: 4x+y−4z=4 → x-intercept 1, y-intercept 4, z-intercept −1
Sketch a plane from its intercepts
Sketch the plane: −3x−3y−5z=15 — three-dimensional axes → Intercepts (−5,0,0), (0,−5,0), (0,0,−3)
The traces of a plane
Write the traces: 3x+y−3z=−12 → xy: 3x+y=−12 xz: x−z=−4 yz: y−3z=−12

Questions about these worksheets

Yes. Every points and planes in 3d sheet is free to print and free to download as a PDF. There is no account to make, no email to hand over and no limit on how many you take.

Yes. The answer key prints on a second page, and you can choose whether it shows each question beside its answer or just the answers in a list.

Yes. Print straight from the page, or download the PDF and print that. The sheet is laid out for paper rather than squeezed off a screen, so there is room to work under each question, and the answer key comes out on its own page.

Algebra 2 is usually taken around 10th or 11th grade, though schools vary and plenty of students meet it earlier or later. Pick the difficulty rather than the grade: the easy level suits a first lesson on it, the hard level suits review before a test.

Yes, as many as you like. The questions are built fresh each time rather than picked from a fixed set of files, so pressing Generate gives a new sheet. That is what you want for a second class, a retake, or two students sitting next to each other.

That is the closest comparison, and Kuta Software is good software. It is also paid software you install on a computer. These worksheets run in a browser tab for free, with no account and nothing to install. Like Kuta, the points and planes in 3d questions are generated when you ask for them rather than pulled from a fixed set of files, so two classes never get the same sheet, and the answer key comes with it.

Yes. There are three levels, and you can tick more than one for a sheet that mixes them, in which case the questions come out easiest first. The harder levels are not just larger numbers: they ask for something the easy ones do not.

You pick from 6: read a plotted point, plot a point, describe where a point is, the intercepts of a plane, sketch a plane from its intercepts and the traces of a plane. Tick as many as you want and set how many of each, or let it spread them evenly.

Drawing the x-axis to the right out of habit. In a three-dimensional picture y goes to the right and x comes out toward you, so a point with a large x sits low and to the left.