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.

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Name

Circumcenter and Incenter

Date Period

Answer each question.

  1. Find the circumcenter

    Find the circumcenter of the triangle with vertices (5,0), (3,−6), (−13,−6).
  2. Find the circumcenter of the triangle with vertices (12,3), (−13,−2), (4,−9).
  3. Find the circumcenter of the triangle with vertices (2,8), (−4,8), (−4,0).
  4. Find the circumcenter of the triangle with vertices (3,8), (−11,−6), (−5,12).

  1. Find the circumradius

    Find the circumradius of the triangle with vertices (19,−1), (4,14), (−5,−13).
  2. Find the circumradius of the triangle with vertices (−7,−4), (−7,2), (1,2).
  3. Find the circumradius of the triangle with vertices (−22,−15), (1,−22), (−24,−7).
  4. Find the circumradius of the triangle with vertices (−3,4), (1,12), (15,−2).

  1. Find the inradius of a right triangle

    How far is the incenter of the triangle from each side?91215
  2. The incenter of a right triangle with legs 30 and 16 and hypotenuse 34 is how far from each side?
  3. A right triangle has legs 21 and 72 and hypotenuse 75. Find the distance from the incenter to each side.

  1. Use the perpendicular and angle bisector theorems

    Line m is the perpendicular bisector of AB‾. P is on m. PA =4x+6 and PB =3x+8. Find PA.
  2. Line m is the perpendicular bisector of AB‾. P is on m. PA =6x−7 and PB =5x+5. Find PA.
  3. Line m is the perpendicular bisector of AB‾. P is on m. PA =5x−4 and PB =x+20. Find PA.

  1. Find the orthocenter

    Find the orthocenter of the triangle with vertices (−3,−5), (5,−5) and (3,7).
  2. Find the orthocenter of the triangle with vertices (−2,0), (4,0) and (−1,5).
  3. Find the orthocenter of the triangle with vertices (0,2), (9,2) and (1,4).

  1. Name the point of concurrency

    Where do the altitudes of a triangle meet?
  2. Which point is the balance point of a triangle cut from cardboard?
  3. Which point is the center of the circle inside the triangle that touches all three sides?

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

Sigma Prep · sigmaprep.io/worksheets

Name

Circumcenter and IncenterAnswer keyVersion 1

Date Period

Answer each question.

  1. Find the circumcenter

    Find the circumcenter of the triangle with vertices (5,0), (3,−6), (−13,−6).(−5,0)
  2. Find the circumcenter of the triangle with vertices (12,3), (−13,−2), (4,−9).(−1,3)
  3. Find the circumcenter of the triangle with vertices (2,8), (−4,8), (−4,0).(−1,4)
  4. Find the circumcenter of the triangle with vertices (3,8), (−11,−6), (−5,12).(−5,2)

  1. Find the circumradius

    Find the circumradius of the triangle with vertices (19,−1), (4,14), (−5,−13).15
  2. Find the circumradius of the triangle with vertices (−7,−4), (−7,2), (1,2).5
  3. Find the circumradius of the triangle with vertices (−22,−15), (1,−22), (−24,−7).17
  4. Find the circumradius of the triangle with vertices (−3,4), (1,12), (15,−2).10

  1. Find the inradius of a right triangle

    How far is the incenter of the triangle from each side?912153
  2. The incenter of a right triangle with legs 30 and 16 and hypotenuse 34 is how far from each side?6
  3. A right triangle has legs 21 and 72 and hypotenuse 75. Find the distance from the incenter to each side.9

  1. Use the perpendicular and angle bisector theorems

    Line m is the perpendicular bisector of AB‾. P is on m. PA =4x+6 and PB =3x+8. Find PA.14
  2. Line m is the perpendicular bisector of AB‾. P is on m. PA =6x−7 and PB =5x+5. Find PA.65
  3. Line m is the perpendicular bisector of AB‾. P is on m. PA =5x−4 and PB =x+20. Find PA.26

  1. Find the orthocenter

    Find the orthocenter of the triangle with vertices (−3,−5), (5,−5) and (3,7).(3,−4)
  2. Find the orthocenter of the triangle with vertices (−2,0), (4,0) and (−1,5).(−1,1)
  3. Find the orthocenter of the triangle with vertices (0,2), (9,2) and (1,4).(1,6)

  1. Name the point of concurrency

    Where do the altitudes of a triangle meet?The orthocenter
  2. Which point is the balance point of a triangle cut from cardboard?The centroid
  3. Which point is the center of the circle inside the triangle that touches all three sides?The incenter

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 15 and 36 and hypotenuse 39. Find the distance from the incenter to each side. → 6
Perpendicular and angle bisector theorems
Line m is the perpendicular bisector of AB‾. P is on m. PA =5x and PB =x+32. Find PA. → 40
The orthocenter from three vertices
Find the orthocenter of the triangle with vertices (0,0), (6,0) and (7,7). → (7,−1)
Name the point
Which point is the center of the circle that passes through all three vertices? → The circumcenter
The incenter is the same distance from every side
P is the incenter of the triangle. How far is P from each side? — a triangle with the point where its angle bisectors meet, and the distances to each side, labeled 6x − 32, 5x − 24, P → 16

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 7: the circumcenter from three vertices, the circumradius, the inscribed circle of a right triangle, perpendicular and angle bisector theorems, the orthocenter from three vertices, name the point and the incenter is the same distance from every side. 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.