Worksheets · Geometry

Segment lengths in circles worksheet

All three rules here are the same rule. From any point, the two pieces along one line multiply to the two pieces along the other. Inside the circle that is the two bits of each chord; outside it is the near piece times the whole secant, and a tangent counts as both, so it gets squared.

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Pick more than one for a sheet that mixes them.

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

Sigma Prep · sigmaprep.io/worksheets

Name

Segment Lengths in Circles

Date Period

Solve for x, or find the length asked for.

  1. Find lengths where two chords cross

    Find x.x1053
  2. Find x.812x6
  3. Find x.9x106
  4. Find x.10168x
  5. Find x.105x14

  1. Find lengths with a tangent and a secant

    From a point P, a tangent of length x and a secant are drawn to a circle. The secant's outside piece is 26 and its inside piece is 78. Find x.
  1. Find x.x41123
  1. From a point P, a tangent of length 80 and a secant are drawn to a circle. The secant's outside piece is 32 and its inside piece is x. Find x.
  1. Find x.12x30
  1. From a point P, a tangent of length 40 and a secant are drawn to a circle. The secant's outside piece is x and its inside piece is 39. Find x.
  1. Find x.7049x
  1. From a point P, a tangent of length 74 and a secant are drawn to a circle. The secant's outside piece is 37 and its inside piece is x. Find x.
  1. Find x.50112x
  1. From a point P, a tangent of length x and a secant are drawn to a circle. The secant's outside piece is 18 and its inside piece is 110. Find x.
  1. Find x.16x36

  1. Find lengths with two secants

    Find x.20x1624
  2. Find x.194722x
  3. Find x.8x97
  4. Find x.576x
  5. Find x.21x1831

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

Sigma Prep · sigmaprep.io/worksheets

Name

Segment Lengths in CirclesAnswer keyVersion 1

Date Period

Solve for x, or find the length asked for.

  1. Find lengths where two chords cross

    Find x.x10536
  2. Find x.812x69
  3. Find x.9x10615
  4. Find x.10168x5
  5. Find x.105x147

  1. Find lengths with a tangent and a secant

    From a point P, a tangent of length x and a secant are drawn to a circle. The secant's outside piece is 26 and its inside piece is 78. Find x.52
  1. Find x.x4112382
  1. From a point P, a tangent of length 80 and a secant are drawn to a circle. The secant's outside piece is 32 and its inside piece is x. Find x.168
  1. Find x.12x3063
  1. From a point P, a tangent of length 40 and a secant are drawn to a circle. The secant's outside piece is x and its inside piece is 39. Find x.25
  1. Find x.7049x51
  1. From a point P, a tangent of length 74 and a secant are drawn to a circle. The secant's outside piece is 37 and its inside piece is x. Find x.111
  1. Find x.50112x90
  1. From a point P, a tangent of length x and a secant are drawn to a circle. The secant's outside piece is 18 and its inside piece is 110. Find x.48
  1. Find x.16x3665

  1. Find lengths with two secants

    Find x.20x162412
  2. Find x.194722x35
  3. Find x.8x9710
  4. Find x.576x4
  5. Find x.21x183121

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

How to do these

Two chords cross inside a circle. One is cut into 4 and 9; the other into 6 and x. Find x

  1. The pieces of one chord multiply to the pieces of the other.
  2. 4 times 9 is 36.
  3. 36 divided by 6 is 6.

The answer is 6.

Where students go wrong

Using the outside piece of a secant twice, or the whole secant twice. It is near piece times whole secant, and the whole one includes the near piece. Only a tangent is multiplied by itself.

Also called power of a point, intersecting chords theorem or tangent secant theorem.

What you can put on this worksheet

Two chords crossing inside
Find x. — a circle with two chords crossing inside it, labeled 10, x, 12, 8 → 15
A tangent and a secant
Segment ED‾ touches a circle at D, and secant EG‾ crosses the circle at F and G, with F between E and G. EF=3 and FG=24. Find ED. → 9
Two secants
Two secants leave the same point outside a circle. For one, the part outside the circle is 3 and the part inside the circle is 3. For the other, the part outside the circle is 2. Find the total length for the other. → 9
A chord, its distance from the center, and the radius
A chord sits 20 from the center of a circle of radius 52. Find the length of the chord. → 96
A tangent and a secant, a piece written as x
From a point P, a tangent of length 21 and a secant are drawn to a circle. The secant's outside piece is 7 and its inside piece is x. Find x. → 56
Two secants, from a drawing
Find x. — two lines from one outside point, cutting or touching a circle, labeled 21, x, 17, 25 → 13
A tangent and a secant, from a drawing
Find x. — two lines from one outside point, cutting or touching a circle, labeled 48, 60, x → 72
Chords in x that make a quadratic, then a whole chord
Chords AB‾ and CD‾ cross at E. AE=2x+5, EB=x+1, CE=7 and ED=17. Find x. → 6

Questions about these worksheets

Yes. Every segment lengths in 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 segment lengths in 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 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 8: two chords crossing inside, a tangent and a secant, two secants, a chord, its distance from the center, and the radius, a tangent and a secant, a piece written as x, two secants, from a drawing, a tangent and a secant, from a drawing and chords in x that make a quadratic, then a whole chord. Tick as many as you want and set how many of each, or let it spread them evenly.

Using the outside piece of a secant twice, or the whole secant twice. It is near piece times whole secant, and the whole one includes the near piece. Only a tangent is multiplied by itself.