Worksheets · Geometry

Tangents to circles worksheet

A tangent touches the circle once and meets the radius there at a right angle, so the radius, the tangent and the segment to the outside point make a right triangle. That is Pythagoras, and it also tests whether a segment is tangent at all. Two tangents drawn from the same outside point are equal.

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

Sigma Prep · sigmaprep.io/worksheets

Name

Tangents to Circles

Date Period

Answer each question.

  1. Use a tangent perpendicular to a radius

    PT‾ is tangent at T. Find x.OTP5x13
  2. Segment PT touches the circle only at T. Find the distance from P to the circle.OTP815
  3. PT‾ is tangent to circle O at T. Find x.OTPx725
  4. Segment PT touches the circle only at T. Find x.OTP125x

  1. Decide if the segment is tangent

    Is PT‾ tangent to the circle at T?OTP345
  2. Circle O has a diameter of 16. Is PT‾ perpendicular to the radius OT‾, so tangent to the circle?OTP1518
  3. OP‾ crosses the circle at the dot. Is PT‾ tangent at T?OTP20219
  4. Is PT‾ tangent to circle O? Use the side lengths to decide.OTP6812

  1. Use two tangents from one point

    Both segments from P are tangent. Find x.P7x − 244x + 24
  2. Both segments from P are tangent. Find x.P5x − 172x + 13
  3. Both segments from P are tangent. Find x.P2x + 43x + 1
  4. Both segments from P are tangent. Find x.P3x + 257x − 43

  1. Find the perimeter of a circumscribed polygon

    A triangle is drawn around a circle, touching it on all three sides. The tangent segments from its corners are 6, 5 and 9. Find the perimeter.
  2. Triangle ABC is drawn around a circle, touching it on all three sides. AB=13, BC=11, AC=6. Find the length of the tangent segment from A.
  3. Quadrilateral ABCD is drawn around a circle, touching it on all four sides. AB=11, BC=15, CD=11. Find AD.
  4. A triangle is drawn around a circle, touching it on all three sides. The tangent segments from its corners are 8, 5 and 12. Find the perimeter.

  1. Find a distance using a tangent

    PT‾ is tangent to circle O at T. The radius is 20 and PT=15. Find the shortest distance from P to the circle.
  2. PT‾ is tangent to circle O at T. The radius is 16 and PT=30. Find the shortest distance from P to the circle.
  3. PT‾ is tangent to circle O at T. PT=36, and the shortest distance from P to the circle is 24. Find the radius.
  4. PT‾ is tangent to circle O at T. PT=6, and the shortest distance from P to the circle is 2. Find the radius.

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

Sigma Prep · sigmaprep.io/worksheets

Name

Tangents to CirclesAnswer keyVersion 1

Date Period

Answer each question.

  1. Use a tangent perpendicular to a radius

    PT‾ is tangent at T. Find x.OTP5x1312
  2. Segment PT touches the circle only at T. Find the distance from P to the circle.OTP8159
  3. PT‾ is tangent to circle O at T. Find x.OTPx72524
  4. Segment PT touches the circle only at T. Find x.OTP125x13

  1. Decide if the segment is tangent

    Is PT‾ tangent to the circle at T?OTP345Yes
  2. Circle O has a diameter of 16. Is PT‾ perpendicular to the radius OT‾, so tangent to the circle?OTP1518No
  3. OP‾ crosses the circle at the dot. Is PT‾ tangent at T?OTP20219Yes
  4. Is PT‾ tangent to circle O? Use the side lengths to decide.OTP6812No

  1. Use two tangents from one point

    Both segments from P are tangent. Find x.P7x − 244x + 2416
  2. Both segments from P are tangent. Find x.P5x − 172x + 1310
  3. Both segments from P are tangent. Find x.P2x + 43x + 13
  4. Both segments from P are tangent. Find x.P3x + 257x − 4317

  1. Find the perimeter of a circumscribed polygon

    A triangle is drawn around a circle, touching it on all three sides. The tangent segments from its corners are 6, 5 and 9. Find the perimeter.40
  2. Triangle ABC is drawn around a circle, touching it on all three sides. AB=13, BC=11, AC=6. Find the length of the tangent segment from A.4
  3. Quadrilateral ABCD is drawn around a circle, touching it on all four sides. AB=11, BC=15, CD=11. Find AD.7
  4. A triangle is drawn around a circle, touching it on all three sides. The tangent segments from its corners are 8, 5 and 12. Find the perimeter.50

  1. Find a distance using a tangent

    PT‾ is tangent to circle O at T. The radius is 20 and PT=15. Find the shortest distance from P to the circle.5
  2. PT‾ is tangent to circle O at T. The radius is 16 and PT=30. Find the shortest distance from P to the circle.18
  3. PT‾ is tangent to circle O at T. PT=36, and the shortest distance from P to the circle is 24. Find the radius.15
  4. PT‾ is tangent to circle O at T. PT=6, and the shortest distance from P to the circle is 2. Find the radius.8

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

How to do these

PT‾ is tangent at T. Find x

  1. The angle at T is 90°, so OT, TP and OP are a right triangle.
  2. With OT = 5 and TP = 12, OP is the hypotenuse.
  3. 5² + 12² = 169.

The answer is 13.

Where students go wrong

Putting the right angle at the outside point. It is at the point of tangency, so the segment to the center is the hypotenuse.

Also called tangent and radius, two tangent segments or tangent segment theorem.

What you can put on this worksheet

A tangent meets a radius at a right angle
PT‾ is tangent to circle O at T. Find x. — a circle with a tangent segment and the segment from the center to the outside point, labeled O, T, P, 12, x, 13 → 5
Is the segment tangent
Does PT‾ meet the radius OT‾ at a right angle, making it a tangent? — a circle with a tangent segment and the segment from the center to the outside point, labeled O, T, P, 8, 15, 19 → No
Two tangents from one point
Both segments from P are tangent. Find x. — a circle with two tangent segments drawn from the same outside point, labeled P, 3x − 5, 2x + 1 → 6
Polygons drawn around a circle
A triangle is drawn around a circle, touching it on all three sides. The tangent segments from its corners are 8, 10 and 2. Find the perimeter. → 40
The radius from a tangent
PT‾ is tangent to circle O at T. The radius is 10 and PT=24. Find the shortest distance from P to the circle. → 16
Tangent or not, with the radius and the outside piece
Is PT‾ tangent to the circle at T? — a circle with a tangent segment and the segment from the center to the outside point, labeled O, T, P, 12, 16, 8 → Yes
Radius plus the outside piece: the missing length
PT‾ is tangent at T. Find the length of the tangent, x. — a circle with a tangent segment and the segment from the center to the outside point, labeled O, T, P, 12, x, 25 → 35
Angles with tangents and radii
PA‾ and PB‾ are tangent. Find x. — a circle with two tangents from one point and the radii to where they touch, labeled O, A, P, B, 135°, x° → 45
A triangle drawn around a circle
The circle is inscribed in the triangle. Find its perimeter. — a triangle drawn around a circle, touching it once on each side, labeled 17, 9 → 52

Questions about these worksheets

Yes. Every tangents to circles 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 tangents to circles 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 9: a tangent meets a radius at a right angle, is the segment tangent, two tangents from one point, polygons drawn around a circle, the radius from a tangent, tangent or not, with the radius and the outside piece, radius plus the outside piece: the missing length, angles with tangents and radii and a triangle drawn around a circle. Tick as many as you want and set how many of each, or let it spread them evenly.

Putting the right angle at the outside point. It is at the point of tangency, so the segment to the center is the hypotenuse.