Worksheets · Geometry

Circumcenter and incenter worksheet

Two circles belong to every triangle. One passes through all three corners and its center, the circumcenter, is where the perpendicular bisectors meet. The other sits inside touching all three sides, and its center, the incenter, is where the angle bisectors meet.

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

Sigma Prep · sigmaprep.io/worksheets

Name

Circumcenter and Incenter

Date Period

Find each measure.

  1. Find the circumcenter of the triangle with vertices (7,0),(4,−9),(−20,−9)
  2. Find the circumradius of the triangle with vertices (19,−1),(4,14),(−5,−13)
  3. Find the radius of its inscribed circle91215
  4. Find the circumcenter of the triangle with vertices (−3,−11),(4,−4),(−4,−10)
  5. Find the circumradius of the triangle with vertices (−5,3),(−4,2),(−12,−4)
  6. A right triangle has legs 24 and 32 and hypotenuse 40. Find the distance from the incenter to each side
  7. Find the circumcenter of the triangle with vertices (−5,19),(7,13),(10,4)
  8. Find the circumradius of the triangle with vertices (−2,2),(4,4),(−4,−12)
  9. A right triangle has legs 12 and 16 and hypotenuse 20. Find the distance from the incenter to each side
  10. Find the circumcenter of the triangle with vertices (−1,−4),(0,−11),(−8,−11)
  11. Find the circumradius of the triangle with vertices (−4,13),(12,−17),(21,−2)
  12. A right triangle has legs 36 and 48 and hypotenuse 60. Find the distance from the incenter to each side
  13. Find the circumcenter of the triangle with vertices (−10,−7),(22,−15),(−1,8)
  14. Find the circumradius of the triangle with vertices (8,−8),(−15,15),(8,22)
  15. A right triangle has legs 9 and 12 and hypotenuse 15. Find the distance from the incenter to each side
  16. Find the circumcenter of the triangle with vertices (−5,−11),(−5,−3),(1,−11)
  17. Find the circumradius of the triangle with vertices (13,−2),(13,−12),(−11,−12)
  18. A right triangle has legs 6 and 8 and hypotenuse 10. Find the distance from the incenter to each side
  19. Find the circumcenter of the triangle with vertices (−6,−7),(−12,11),(−14,9)
  20. Find the circumradius of the triangle with vertices (−4,−7),(−4,1),(−11,0)

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

Sigma Prep · sigmaprep.io/worksheets

Name

Circumcenter and IncenterAnswer keyVersion 1

Date Period

Find each measure.

  1. Find the circumcenter of the triangle with vertices (7,0),(4,−9),(−20,−9)(−8,0)
  2. Find the circumradius of the triangle with vertices (19,−1),(4,14),(−5,−13)15
  3. Find the radius of its inscribed circle912153
  4. Find the circumcenter of the triangle with vertices (−3,−11),(4,−4),(−4,−10)(0,−7)
  5. Find the circumradius of the triangle with vertices (−5,3),(−4,2),(−12,−4)5
  6. A right triangle has legs 24 and 32 and hypotenuse 40. Find the distance from the incenter to each side8
  7. Find the circumcenter of the triangle with vertices (−5,19),(7,13),(10,4)(−5,4)
  8. Find the circumradius of the triangle with vertices (−2,2),(4,4),(−4,−12)10
  9. A right triangle has legs 12 and 16 and hypotenuse 20. Find the distance from the incenter to each side4
  10. Find the circumcenter of the triangle with vertices (−1,−4),(0,−11),(−8,−11)(−4,−8)
  11. Find the circumradius of the triangle with vertices (−4,13),(12,−17),(21,−2)17
  12. A right triangle has legs 36 and 48 and hypotenuse 60. Find the distance from the incenter to each side12
  13. Find the circumcenter of the triangle with vertices (−10,−7),(22,−15),(−1,8)(7,−7)
  14. Find the circumradius of the triangle with vertices (8,−8),(−15,15),(8,22)17
  15. A right triangle has legs 9 and 12 and hypotenuse 15. Find the distance from the incenter to each side3
  16. Find the circumcenter of the triangle with vertices (−5,−11),(−5,−3),(1,−11)(−2,−7)
  17. Find the circumradius of the triangle with vertices (13,−2),(13,−12),(−11,−12)13
  18. A right triangle has legs 6 and 8 and hypotenuse 10. Find the distance from the incenter to each side2
  19. Find the circumcenter of the triangle with vertices (−6,−7),(−12,11),(−14,9)(−6,3)
  20. Find the circumradius of the triangle with vertices (−4,−7),(−4,1),(−11,0)5

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

How to do these

A right triangle has legs 3 and 4 and hypotenuse 5. Find the radius of its inscribed circle

  1. For a right triangle the inradius is the two legs minus the hypotenuse, halved.
  2. 3 plus 4 minus 5 is 2.
  3. Half of 2 is 1.

The answer is 1.

Where students go wrong

Mixing the two up. The circumcenter is equally far from the three corners and can sit outside the triangle; the incenter is equally far from the three sides and is always inside.

Also called circumcenter of a triangle, incenter, points of concurrency or circumscribed circle.

What you can put on this worksheet

The circumcenter from three vertices
Find the circumcenter of the triangle with vertices (−9,3),(1,−7),(7,11) → (1,3)
The circumradius
Find the circumradius of the triangle with vertices (−9,3),(1,−7),(7,11) → 10
The inscribed circle of a right triangle
A right triangle has legs 18 and 24 and hypotenuse 30. Find the distance from the incenter to each side → 6

Questions about these worksheets

Yes. Every circumcenter and incenter 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 circumcenter and incenter 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 3: the circumcenter from three vertices, the circumradius and the inscribed circle of a right triangle. Tick as many as you want and set how many of each, or let it spread them evenly.

Mixing the two up. The circumcenter is equally far from the three corners and can sit outside the triangle; the incenter is equally far from the three sides and is always inside.