Worksheets · Geometry

Centroid of a triangle worksheet

The three medians all cross at one point, and on the coordinate plane it is simply the average of the three vertices. Average the x values, average the y values, done. The centroid also cuts each median 2 : 1, with the longer piece next to the vertex.

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

Sigma Prep · sigmaprep.io/worksheets

Name

Centroids of Triangles

Date Period

Answer each question about the centroid.

  1. Use the centroid formula

    Find the centroid of the triangle with vertices (3,−6), (0,6), (6,−3).
  2. Find the centroid of the triangle with vertices (−3,8), (3,−1), (−3,2).
  3. Find the centroid of the triangle with vertices (4,−4), (6,−2), (−4,3).
  4. Find the centroid of the triangle with vertices (−2,−5), (−6,−4), (2,−3).
  5. Find the centroid of the triangle with vertices (−1,−1), (−2,5), (6,2).

  1. Use the 2:12 : 1 centroid ratio

    G is the centroid and the median CZ=18. Find CG.ABCZG?
  2. G is the centroid and the median CZ=60. Find GZ.ABCZG?
  3. G is the centroid and CG=26. Find GZ.ABCZG26?
  4. G is the centroid and GZ=7. Find CG.ABCZG?7
  5. G is the centroid and GZ=8. Find CG.ABCZG?8

  1. Use a median or the centroid

    G is the centroid of triangle ABC and AM‾ is a median. AG=3x−12 and AM=5x−24. Find AM.
  2. AM‾ is a median of triangle ABC. BM=6x−55 and MC=2x+1. Find BC.
  3. AM‾ is a median of triangle ABC. BM=5x−29 and MC=6x−39. Find BC.
  4. AM‾ is a median of triangle ABC. BM=2x+17 and MC=x+20. Find BC.
  5. G is the centroid of triangle ABC and AM‾ is a median. AG=5x−3 and AM=4x+13. Find AM.

  1. Find the centroid on a grid

    Give the point where the three medians of △ABC cross.−10−10−5−5551010ABC
  2. Find the balance point (centroid) of △LMN.−10−10551010LMN−5
  3. Find the centroid of △PQR.−10−10−5551010PQR−5
  4. The medians of △EFG meet at its centroid. Find that point.−10−10−5−5551010EFG
  5. What are the coordinates of the centroid of △ABC?−10−10−5−551010ABC

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

Sigma Prep · sigmaprep.io/worksheets

Name

Centroids of TrianglesAnswer keyVersion 1

Date Period

Answer each question about the centroid.

  1. Use the centroid formula

    Find the centroid of the triangle with vertices (3,−6), (0,6), (6,−3).(3,−1)
  2. Find the centroid of the triangle with vertices (−3,8), (3,−1), (−3,2).(−1,3)
  3. Find the centroid of the triangle with vertices (4,−4), (6,−2), (−4,3).(2,−1)
  4. Find the centroid of the triangle with vertices (−2,−5), (−6,−4), (2,−3).(−2,−4)
  5. Find the centroid of the triangle with vertices (−1,−1), (−2,5), (6,2).(1,2)

  1. Use the 2:12 : 1 centroid ratio

    G is the centroid and the median CZ=18. Find CG.ABCZG?12
  2. G is the centroid and the median CZ=60. Find GZ.ABCZG?20
  3. G is the centroid and CG=26. Find GZ.ABCZG26?13
  4. G is the centroid and GZ=7. Find CG.ABCZG?714
  5. G is the centroid and GZ=8. Find CG.ABCZG?816

  1. Use a median or the centroid

    G is the centroid of triangle ABC and AM‾ is a median. AG=3x−12 and AM=5x−24. Find AM.36
  2. AM‾ is a median of triangle ABC. BM=6x−55 and MC=2x+1. Find BC.58
  3. AM‾ is a median of triangle ABC. BM=5x−29 and MC=6x−39. Find BC.42
  4. AM‾ is a median of triangle ABC. BM=2x+17 and MC=x+20. Find BC.46
  5. G is the centroid of triangle ABC and AM‾ is a median. AG=5x−3 and AM=4x+13. Find AM.33

  1. Find the centroid on a grid

    Give the point where the three medians of △ABC cross.−10−10−5−5551010ABC(−2,−3)
  2. Find the balance point (centroid) of △LMN.−10−10551010LMN−5(−2,−2)
  3. Find the centroid of △PQR.−10−10−5551010PQR−5(0,−6)
  4. The medians of △EFG meet at its centroid. Find that point.−10−10−5−5551010EFG(−4,−1)
  5. What are the coordinates of the centroid of △ABC?−10−10−5−551010ABC(1,0)

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

How to do these

(0,0), (6,0), (3,9)

  1. Average the x values: 0, 6 and 3.
  2. Average the y values: 0, 0 and 9.
  3. That gives (3, 3).

The answer is (3,3).

Where students go wrong

Averaging only two of the vertices. That gives a midpoint, which is on a median but is not where the three of them meet.

Also called center of gravity of a triangle, medians of a triangle or coordinate geometry centroid.

What you can put on this worksheet

The centroid from three vertices
Find the centroid of the triangle with vertices (−2,7), (1,6), (−5,2). → (−2,5)
The 2 : 1 ratio on a median
G is the centroid and GZ=7. Find CG. — a triangle with its three medians meeting at the centroid, labeled A, B, C, Z, G, ?, 7 → 14
Medians and the centroid, in x
AM‾ is a median of triangle ABC. BM=x+15 and MC=3x−3. Find BC. → 48
Centroid read off a grid
Find the balance point (centroid) of △PQR. → (−3,0)
Decimals, and a fraction in front of x
G is the centroid of triangle ABC and AM‾ is a median. AG=12x+20 and GM=3x−56. Find x. → 24

Questions about these worksheets

Yes. Every centroid of a triangle 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.

Geometry is usually taken around 9th or 10th 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 centroid of a triangle 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 two 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 5: the centroid from three vertices, the 2 : 1 ratio on a median, medians and the centroid, in x, centroid read off a grid and decimals, and a fraction in front of x. Tick as many as you want and set how many of each, or let it spread them evenly.

Averaging only two of the vertices. That gives a midpoint, which is on a median but is not where the three of them meet.