In circle P, AB is a diameter, m∠ BPC = 80° and m∠ APD = 100°. Classify each arc and find its measure: arc BAC, arc BD, arc ADB.

arc BAC: major, 280°; arc BD: minor, 80°; arc ADB: semicircle, 180°

In circle P, AB is a diameter, m∠ BPC = 55° and m∠ APD = 45°. Classify each arc and find its measure: arc BAC, arc BD, arc BCA.

arc BAC: major, 305°; arc BD: minor, 135°; arc BCA: semicircle, 180°

In circle P, AB is a diameter, m∠ BPC = 135° and m∠ APD = 95°. Classify each arc and find its measure: arc BAD, arc BD, arc ACB.

arc BAD: major, 275°; arc BD: minor, 85°; arc ACB: semicircle, 180°

Arcs and chords worksheet

An arc has the same measure as its central angle, and the whole way round is 360. For chords, the perpendicular from the center cuts the chord in half, which makes a right triangle you already know how to solve.

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Sigma Prep · sigmaprep.io/worksheets Name Arcs and Chords Date Period Answer each question. Find arc measures Find the major arc. Two adjacent arcs measure $99^{\circ}$ and $163^{\circ}$ . Find the remaining arc. Two adjacent arcs measure $90^{\circ}$ and $106^{\circ}$ . Find the remaining arc. Relate chord, radius and center distance In a circle of radius $87$ , a chord lies $60$ from the center. How long is the chord? Find $r$ . How far is the chord from the center? Use congruent chords to find $x$ In a circle, $\overline{A B} ≅ \overline{C D}$ , $m \overset{⌢}{AB} = {(4 x - 4)}^{\circ}$ and $m \overset{⌢}{CD} = {(x + 35)}^{\circ}$ . Find $x$ . In circle $O$ , chords $\overline{A B}$ and $\overline{C D}$ are congruent. $A B = 5 x - 7$ and $C D = 8 x - 25$ . Find $x$ . Chords $\overline{A B}$ and $\overline{C D}$ are both $13$ long. Their distances from the center are $3 x - 66$ and $6 x - 138$ . Find $x$ . In circle $O$ , chords $\overline{A B}$ and $\overline{C D}$ are congruent. $A B = 3 x + 10$ and $C D = 7 x + 2$ . Find $x$ . Classify and measure arcs In circle $P$ , $\overline{A B}$ is a diameter, $m ∠ B P C = 80^{\circ}$ and $m ∠ A P D = 100^{\circ}$ . Classify each arc and find its measure: $\overset{⌢}{BAC}$ , $\overset{⌢}{BD}$ , $\overset{⌢}{ADB}$ . In circle $P$ , $\overline{A B}$ is a diameter, $m ∠ B P C = 55^{\circ}$ and $m ∠ A P D = 45^{\circ}$ . Classify each arc and find its measure: $\overset{⌢}{BAC}$ , $\overset{⌢}{BD}$ , $\overset{⌢}{BCA}$ . Find $x$ or an arc on a diameter $\overline{A B}$ is a diameter of circle $O$ . $m ∠ A O C = {(5 x + 7)}^{\circ}$ and $m ∠ C O B = {(2 x + 12)}^{\circ}$ . Find the measure of major arc $\overset{⌢}{ABC}$ . $\overline{A B}$ is a diameter of circle $O$ . $m ∠ A O C = {(x - 10)}^{\circ}$ and $m ∠ C O B = {(5 x + 10)}^{\circ}$ . Find the measure of major arc $\overset{⌢}{ABC}$ . Use a radius perpendicular to a chord In circle $O$ , radius $\overline{O C}$ is perpendicular to chord $\overline{A B}$ at $M$ . $A B = 38$ . Find $M B$ . In circle $O$ , radius $\overline{O C}$ is perpendicular to chord $\overline{A B}$ at $M$ . $A M = 22$ . Find $A B$ . Find $x$ from central angles The central angles $∠ A O B$ , $∠ B O C$ , $∠ C O A$ fill the circle around $O$ . $m ∠ A O B = {(2 x + 54)}^{\circ}$ , $m ∠ B O C = {(x + 137)}^{\circ}$ , $m ∠ C O A = {(x + 129)}^{\circ}$ . Find $x$ . The central angles $∠ A O B$ , $∠ B O C$ , $∠ C O D$ , $∠ D O A$ fill the circle around $O$ . $m ∠ A O B = {(3 x + 33)}^{\circ}$ , $m ∠ B O C = {(6 x - 13)}^{\circ}$ , $m ∠ C O D = {(x + 136)}^{\circ}$ , $m ∠ D O A = {(6 x + 12)}^{\circ}$ . Find $x$ . Solve chord problems to the nearest tenth The perpendicular from the center of a circle of diameter $34$ to a chord is $10$ long. How long is the chord? Round to the nearest tenth. A circle has a chord $4$ long, and the perpendicular from the center to it is $12$ . Find the diameter. Round to the nearest tenth. Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Arcs and Chords

Date Period

Answer each question.

Find arc measures
Find the major arc.
Two adjacent arcs measure $99^{\circ}$ and $163^{\circ}$ . Find the remaining arc.
Two adjacent arcs measure $90^{\circ}$ and $106^{\circ}$ . Find the remaining arc.

Relate chord, radius and center distance
In a circle of radius $87$ , a chord lies $60$ from the center. How long is the chord?
Find $r$ .
How far is the chord from the center?

Use congruent chords to find $x$
In a circle, $\overline{A B} ≅ \overline{C D}$ , $m \overset{⌢}{AB} = {(4 x - 4)}^{\circ}$ and $m \overset{⌢}{CD} = {(x + 35)}^{\circ}$ . Find $x$ .
In circle $O$ , chords $\overline{A B}$ and $\overline{C D}$ are congruent. $A B = 5 x - 7$ and $C D = 8 x - 25$ . Find $x$ .
Chords $\overline{A B}$ and $\overline{C D}$ are both $13$ long. Their distances from the center are $3 x - 66$ and $6 x - 138$ . Find $x$ .
In circle $O$ , chords $\overline{A B}$ and $\overline{C D}$ are congruent. $A B = 3 x + 10$ and $C D = 7 x + 2$ . Find $x$ .

Classify and measure arcs
In circle $P$ , $\overline{A B}$ is a diameter, $m ∠ B P C = 80^{\circ}$ and $m ∠ A P D = 100^{\circ}$ . Classify each arc and find its measure: $\overset{⌢}{BAC}$ , $\overset{⌢}{BD}$ , $\overset{⌢}{ADB}$ .
In circle $P$ , $\overline{A B}$ is a diameter, $m ∠ B P C = 55^{\circ}$ and $m ∠ A P D = 45^{\circ}$ . Classify each arc and find its measure: $\overset{⌢}{BAC}$ , $\overset{⌢}{BD}$ , $\overset{⌢}{BCA}$ .

Find $x$ or an arc on a diameter
$\overline{A B}$ is a diameter of circle $O$ . $m ∠ A O C = {(5 x + 7)}^{\circ}$ and $m ∠ C O B = {(2 x + 12)}^{\circ}$ . Find the measure of major arc $\overset{⌢}{ABC}$ .
$\overline{A B}$ is a diameter of circle $O$ . $m ∠ A O C = {(x - 10)}^{\circ}$ and $m ∠ C O B = {(5 x + 10)}^{\circ}$ . Find the measure of major arc $\overset{⌢}{ABC}$ .

Use a radius perpendicular to a chord
In circle $O$ , radius $\overline{O C}$ is perpendicular to chord $\overline{A B}$ at $M$ . $A B = 38$ . Find $M B$ .
In circle $O$ , radius $\overline{O C}$ is perpendicular to chord $\overline{A B}$ at $M$ . $A M = 22$ . Find $A B$ .

Find $x$ from central angles
The central angles $∠ A O B$ , $∠ B O C$ , $∠ C O A$ fill the circle around $O$ . $m ∠ A O B = {(2 x + 54)}^{\circ}$ , $m ∠ B O C = {(x + 137)}^{\circ}$ , $m ∠ C O A = {(x + 129)}^{\circ}$ . Find $x$ .
The central angles $∠ A O B$ , $∠ B O C$ , $∠ C O D$ , $∠ D O A$ fill the circle around $O$ . $m ∠ A O B = {(3 x + 33)}^{\circ}$ , $m ∠ B O C = {(6 x - 13)}^{\circ}$ , $m ∠ C O D = {(x + 136)}^{\circ}$ , $m ∠ D O A = {(6 x + 12)}^{\circ}$ . Find $x$ .

Solve chord problems to the nearest tenth
The perpendicular from the center of a circle of diameter $34$ to a chord is $10$ long. How long is the chord? Round to the nearest tenth.
A circle has a chord $4$ long, and the perpendicular from the center to it is $12$ . Find the diameter. Round to the nearest tenth.



Sigma Prep · sigmaprep.io/worksheets Name Arcs and ChordsAnswer keyVersion 1 Date Period Answer each question. Find arc measures Find the major arc. $265^{\circ}$ Two adjacent arcs measure $99^{\circ}$ and $163^{\circ}$ . Find the remaining arc. $98^{\circ}$ Two adjacent arcs measure $90^{\circ}$ and $106^{\circ}$ . Find the remaining arc. $164^{\circ}$ Relate chord, radius and center distance In a circle of radius $87$ , a chord lies $60$ from the center. How long is the chord? $126$ Find $r$ . $34$ How far is the chord from the center? $7$ Use congruent chords to find $x$ In a circle, $\overline{A B} ≅ \overline{C D}$ , $m \overset{⌢}{AB} = {(4 x - 4)}^{\circ}$ and $m \overset{⌢}{CD} = {(x + 35)}^{\circ}$ . Find $x$ . $13$ In circle $O$ , chords $\overline{A B}$ and $\overline{C D}$ are congruent. $A B = 5 x - 7$ and $C D = 8 x - 25$ . Find $x$ . $6$ Chords $\overline{A B}$ and $\overline{C D}$ are both $13$ long. Their distances from the center are $3 x - 66$ and $6 x - 138$ . Find $x$ . $24$ In circle $O$ , chords $\overline{A B}$ and $\overline{C D}$ are congruent. $A B = 3 x + 10$ and $C D = 7 x + 2$ . Find $x$ . $2$ Classify and measure arcs In circle $P$ , $\overline{A B}$ is a diameter, $m ∠ B P C = 80^{\circ}$ and $m ∠ A P D = 100^{\circ}$ . Classify each arc and find its measure: $\overset{⌢}{BAC}$ , $\overset{⌢}{BD}$ , $\overset{⌢}{ADB}$ . $\overset{⌢}{BAC}$ : major, $280^{\circ}$ ; $\overset{⌢}{BD}$ : minor, $80^{\circ}$ ; $\overset{⌢}{ADB}$ : semicircle, $180^{\circ}$ In circle $P$ , $\overline{A B}$ is a diameter, $m ∠ B P C = 55^{\circ}$ and $m ∠ A P D = 45^{\circ}$ . Classify each arc and find its measure: $\overset{⌢}{BAC}$ , $\overset{⌢}{BD}$ , $\overset{⌢}{BCA}$ . $\overset{⌢}{BAC}$ : major, $305^{\circ}$ ; $\overset{⌢}{BD}$ : minor, $135^{\circ}$ ; $\overset{⌢}{BCA}$ : semicircle, $180^{\circ}$ Find $x$ or an arc on a diameter $\overline{A B}$ is a diameter of circle $O$ . $m ∠ A O C = {(5 x + 7)}^{\circ}$ and $m ∠ C O B = {(2 x + 12)}^{\circ}$ . Find the measure of major arc $\overset{⌢}{ABC}$ . $238^{\circ}$ $\overline{A B}$ is a diameter of circle $O$ . $m ∠ A O C = {(x - 10)}^{\circ}$ and $m ∠ C O B = {(5 x + 10)}^{\circ}$ . Find the measure of major arc $\overset{⌢}{ABC}$ . $340^{\circ}$ Use a radius perpendicular to a chord In circle $O$ , radius $\overline{O C}$ is perpendicular to chord $\overline{A B}$ at $M$ . $A B = 38$ . Find $M B$ . $19$ In circle $O$ , radius $\overline{O C}$ is perpendicular to chord $\overline{A B}$ at $M$ . $A M = 22$ . Find $A B$ . $44$ Find $x$ from central angles The central angles $∠ A O B$ , $∠ B O C$ , $∠ C O A$ fill the circle around $O$ . $m ∠ A O B = {(2 x + 54)}^{\circ}$ , $m ∠ B O C = {(x + 137)}^{\circ}$ , $m ∠ C O A = {(x + 129)}^{\circ}$ . Find $x$ . $10$ The central angles $∠ A O B$ , $∠ B O C$ , $∠ C O D$ , $∠ D O A$ fill the circle around $O$ . $m ∠ A O B = {(3 x + 33)}^{\circ}$ , $m ∠ B O C = {(6 x - 13)}^{\circ}$ , $m ∠ C O D = {(x + 136)}^{\circ}$ , $m ∠ D O A = {(6 x + 12)}^{\circ}$ . Find $x$ . $12$ Solve chord problems to the nearest tenth The perpendicular from the center of a circle of diameter $34$ to a chord is $10$ long. How long is the chord? Round to the nearest tenth. $27.5$ A circle has a chord $4$ long, and the perpendicular from the center to it is $12$ . Find the diameter. Round to the nearest tenth. $24.3$ Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Arcs and ChordsAnswer keyVersion 1

Date Period

Answer each question.

Find arc measures
Find the major arc. $265^{\circ}$
Two adjacent arcs measure $99^{\circ}$ and $163^{\circ}$ . Find the remaining arc. $98^{\circ}$
Two adjacent arcs measure $90^{\circ}$ and $106^{\circ}$ . Find the remaining arc. $164^{\circ}$

Relate chord, radius and center distance
In a circle of radius $87$ , a chord lies $60$ from the center. How long is the chord? $126$
Find $r$ . $34$
How far is the chord from the center? $7$

Use congruent chords to find $x$
In a circle, $\overline{A B} ≅ \overline{C D}$ , $m \overset{⌢}{AB} = {(4 x - 4)}^{\circ}$ and $m \overset{⌢}{CD} = {(x + 35)}^{\circ}$ . Find $x$ . $13$
In circle $O$ , chords $\overline{A B}$ and $\overline{C D}$ are congruent. $A B = 5 x - 7$ and $C D = 8 x - 25$ . Find $x$ . $6$
Chords $\overline{A B}$ and $\overline{C D}$ are both $13$ long. Their distances from the center are $3 x - 66$ and $6 x - 138$ . Find $x$ . $24$
In circle $O$ , chords $\overline{A B}$ and $\overline{C D}$ are congruent. $A B = 3 x + 10$ and $C D = 7 x + 2$ . Find $x$ . $2$

Classify and measure arcs
In circle $P$ , $\overline{A B}$ is a diameter, $m ∠ B P C = 80^{\circ}$ and $m ∠ A P D = 100^{\circ}$ . Classify each arc and find its measure: $\overset{⌢}{BAC}$ , $\overset{⌢}{BD}$ , $\overset{⌢}{ADB}$ . $\overset{⌢}{BAC}$ : major, $280^{\circ}$ ; $\overset{⌢}{BD}$ : minor, $80^{\circ}$ ; $\overset{⌢}{ADB}$ : semicircle, $180^{\circ}$
In circle $P$ , $\overline{A B}$ is a diameter, $m ∠ B P C = 55^{\circ}$ and $m ∠ A P D = 45^{\circ}$ . Classify each arc and find its measure: $\overset{⌢}{BAC}$ , $\overset{⌢}{BD}$ , $\overset{⌢}{BCA}$ . $\overset{⌢}{BAC}$ : major, $305^{\circ}$ ; $\overset{⌢}{BD}$ : minor, $135^{\circ}$ ; $\overset{⌢}{BCA}$ : semicircle, $180^{\circ}$

Find $x$ or an arc on a diameter
$\overline{A B}$ is a diameter of circle $O$ . $m ∠ A O C = {(5 x + 7)}^{\circ}$ and $m ∠ C O B = {(2 x + 12)}^{\circ}$ . Find the measure of major arc $\overset{⌢}{ABC}$ . $238^{\circ}$
$\overline{A B}$ is a diameter of circle $O$ . $m ∠ A O C = {(x - 10)}^{\circ}$ and $m ∠ C O B = {(5 x + 10)}^{\circ}$ . Find the measure of major arc $\overset{⌢}{ABC}$ . $340^{\circ}$

Use a radius perpendicular to a chord
In circle $O$ , radius $\overline{O C}$ is perpendicular to chord $\overline{A B}$ at $M$ . $A B = 38$ . Find $M B$ . $19$
In circle $O$ , radius $\overline{O C}$ is perpendicular to chord $\overline{A B}$ at $M$ . $A M = 22$ . Find $A B$ . $44$

Find $x$ from central angles
The central angles $∠ A O B$ , $∠ B O C$ , $∠ C O A$ fill the circle around $O$ . $m ∠ A O B = {(2 x + 54)}^{\circ}$ , $m ∠ B O C = {(x + 137)}^{\circ}$ , $m ∠ C O A = {(x + 129)}^{\circ}$ . Find $x$ . $10$
The central angles $∠ A O B$ , $∠ B O C$ , $∠ C O D$ , $∠ D O A$ fill the circle around $O$ . $m ∠ A O B = {(3 x + 33)}^{\circ}$ , $m ∠ B O C = {(6 x - 13)}^{\circ}$ , $m ∠ C O D = {(x + 136)}^{\circ}$ , $m ∠ D O A = {(6 x + 12)}^{\circ}$ . Find $x$ . $12$

Solve chord problems to the nearest tenth
The perpendicular from the center of a circle of diameter $34$ to a chord is $10$ long. How long is the chord? Round to the nearest tenth. $27.5$
A circle has a chord $4$ long, and the perpendicular from the center to it is $12$ . Find the diameter. Round to the nearest tenth. $24.3$

How to do these

A chord of length $24$ is $5$ from the center. Find the radius

Half the chord is 12.
12, 5 and the radius make a right triangle.
12 squared plus 5 squared is 169.

The answer is $13$ .

Where students go wrong

Using the whole chord in the right triangle. The perpendicular from the center bisects it, so only half the chord is a leg. Using all 24 gives a radius far too large.

Also called central angles, major and minor arcs or chord length.

What you can put on this worksheet

Arc measures and central angles: Two adjacent arcs measure $84^{\circ}$ and $193^{\circ}$ . Find the remaining arc. → $83^{\circ}$
Chords, distance from the center, radius: Find $x$ . — a circle with a chord and the distance to the center, labeled 48, x, 26 → $10$
Congruent chords and arcs, written in x: In a circle, $\overline{A B} ≅ \overline{C D}$ , $m \overset{⌢}{AB} = {(2 x + 122)}^{\circ}$ and $m \overset{⌢}{CD} = {(3 x + 117)}^{\circ}$ . Find $x$ . → $5$
Congruent chords, lengths written in x: In circle $O$ , chords $\overline{A B}$ and $\overline{C D}$ are congruent. $A B = 2 x - 6$ and $C D = 5 x - 33$ . Find $x$ . → $9$
Minor arcs, major arcs and semicircles: In circle $P$ , $\overline{A B}$ is a diameter, $m ∠ B P C = 135^{\circ}$ and $m ∠ A P D = 95^{\circ}$ . Classify each arc and find its measure: $\overset{⌢}{BAD}$ , $\overset{⌢}{BD}$ , $\overset{⌢}{ACB}$ . → $\overset{⌢}{BAD}$ : major, $275^{\circ}$ ; $\overset{⌢}{BD}$ : minor, $85^{\circ}$ ; $\overset{⌢}{ACB}$ : semicircle, $180^{\circ}$
Arcs on a diameter, written in x: $\overline{A B}$ is a diameter of circle $O$ . $m ∠ A O C = {(4 x - 16)}^{\circ}$ and $m ∠ C O B = {(4 x + 4)}^{\circ}$ . Find $x$ . → $24$
A radius perpendicular to a chord cuts it in half: In circle $O$ , radius $\overline{O C}$ is perpendicular to chord $\overline{A B}$ at $M$ . $A B = 28$ . Find $M B$ . → $14$
Central angles round the whole circle, in x: The central angles $∠ A O B$ , $∠ B O C$ , $∠ C O A$ fill the circle around $O$ . $m ∠ A O B = {(6 x + 53)}^{\circ}$ , $m ∠ B O C = {(x + 119)}^{\circ}$ , $m ∠ C O A = {(x + 116)}^{\circ}$ . Find $x$ . → $9$
Chords that do not come out whole, rounded: How far from the center of a circle of radius $9$ does a chord of length $4$ lie? Round to the nearest tenth. → $8.8$
Name the arc, the major arc, or the central angle: Name the central angle that cuts off $\overset{⌢}{LM}$ . — a circle with points, chords and arcs, labeled J, K, L, M, O → $∠ L O M$
Two diameters crossing: Diameters $\overline{W Y}$ and $\overline{X Z}$ cross at the center $O$ . Find $m \overset{⌢}{YZ}$ . — a circle with points, chords and arcs, labeled 94°, W, X, Y, Z, O → $94^{\circ}$
Distance to a chord, with the radius drawn elsewhere: The perpendicular from $O$ to $\overline{A B}$ meets it at $D$ . The radius $O C = 10$ and $A B = 12$ . Find $O D$ . — a circle with points, chords and arcs, labeled A, B, C, D, O → $8$

Questions about these worksheets

Yes. Every arcs and chords 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 arcs and chords 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 12: arc measures and central angles, chords, distance from the center, radius, congruent chords and arcs, written in x, congruent chords, lengths written in x, minor arcs, major arcs and semicircles, arcs on a diameter, written in x, a radius perpendicular to a chord cuts it in half, central angles round the whole circle, in x, chords that do not come out whole, rounded, name the arc, the major arc, or the central angle, two diameters crossing and distance to a chord, with the radius drawn elsewhere. Tick as many as you want and set how many of each, or let it spread them evenly.

Using the whole chord in the right triangle. The perpendicular from the center bisects it, so only half the chord is a leg. Using all 24 gives a radius far too large.

Arcs and chords worksheet

Find arc measures

Relate chord, radius and center distance

Use congruent chords to find xxx

Classify and measure arcs

Find xxx or an arc on a diameter

Use a radius perpendicular to a chord

Find xxx from central angles

Solve chord problems to the nearest tenth

Find arc measures

Relate chord, radius and center distance

Use congruent chords to find xxx

Classify and measure arcs

Find xxx or an arc on a diameter

Use a radius perpendicular to a chord

Find xxx from central angles

Solve chord problems to the nearest tenth

How to do these

Where students go wrong

What you can put on this worksheet

Questions about these worksheets

Are these arcs and chords worksheets free?+

Do the arcs and chords worksheets come with an answer key?+

Are these arcs and chords worksheets printable?+

What grade are these arcs and chords worksheets for?+

Can I get a different version of the same arcs and chords worksheet?+

Is this a free alternative to Kuta Software for arcs and chords?+

Can I choose how hard the arcs and chords questions are?+

What kinds of question are on a arcs and chords worksheet?+

What do students get wrong with arcs and chords?+

Use congruent chords to find $x$

Find $x$ or an arc on a diameter

Find $x$ from central angles

Use congruent chords to find $x$

Find $x$ or an arc on a diameter

Find $x$ from central angles