Angles in circles worksheet

Every angle in this unit is half of something. An angle sitting on the circle is half the arc it cuts off. An angle where two chords cross inside is half the sum of the two arcs. An angle formed outside, by secants or tangents, is half their difference. Inside adds, outside subtracts.

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Sigma Prep · sigmaprep.io/worksheets Name Angles in Circles Date Period Find each measure. Use inscribed angles Find the angle marked $x$ . Find the arc marked $x$ . Find the angle marked $x$ . Find the arc marked $x$ . Find the arc marked $x$ . Find $x$ where two chords cross Find $x$ . Find $x$ . Find the value of $x$ . Find the value of $x$ . Find the value of $x$ . Find $x$ with secants and tangents from an outside point The arc marked is the minor arc between the points of tangency. Find the value of $x$ . The arc marked is the minor arc between the points of tangency. Find $x$ . Find $x$ . The arc marked is the minor arc between the points of tangency. Find the value of $x$ . Find the value of $x$ . Find $x$ with a tangent and a chord Find the value of $x$ . Find the value of $x$ . Find $x$ . Find $x$ . Find $x$ . Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Angles in Circles

Date Period

Find each measure.

Use inscribed angles
Find the angle marked $x$ .
Find the arc marked $x$ .
Find the angle marked $x$ .
Find the arc marked $x$ .
Find the arc marked $x$ .

Find $x$ where two chords cross
Find $x$ .
Find $x$ .
Find the value of $x$ .
Find the value of $x$ .
Find the value of $x$ .

Find $x$ with secants and tangents from an outside point
The arc marked is the minor arc between the points of tangency. Find the value of $x$ .
The arc marked is the minor arc between the points of tangency. Find $x$ .
Find $x$ .
The arc marked is the minor arc between the points of tangency. Find the value of $x$ .
Find the value of $x$ .

Find $x$ with a tangent and a chord
Find the value of $x$ .
Find the value of $x$ .
Find $x$ .
Find $x$ .
Find $x$ .



Sigma Prep · sigmaprep.io/worksheets Name Angles in CirclesAnswer keyVersion 1 Date Period Find each measure. Use inscribed angles Find the angle marked $x$ . $45^{\circ}$ Find the arc marked $x$ . $92^{\circ}$ Find the angle marked $x$ . $60^{\circ}$ Find the arc marked $x$ . $118^{\circ}$ Find the arc marked $x$ . $82^{\circ}$ Find $x$ where two chords cross Find $x$ . $134$ Find $x$ . $126$ Find the value of $x$ . $44$ Find the value of $x$ . $162$ Find the value of $x$ . $160$ Find $x$ with secants and tangents from an outside point The arc marked is the minor arc between the points of tangency. Find the value of $x$ . $100$ The arc marked is the minor arc between the points of tangency. Find $x$ . $82$ Find $x$ . $46$ The arc marked is the minor arc between the points of tangency. Find the value of $x$ . $70$ Find the value of $x$ . $40$ Find $x$ with a tangent and a chord Find the value of $x$ . $196$ Find the value of $x$ . $105$ Find $x$ . $238$ Find $x$ . $112$ Find $x$ . $184$ Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Angles in CirclesAnswer keyVersion 1

Date Period

Find each measure.

Use inscribed angles
Find the angle marked $x$ . $45^{\circ}$
Find the arc marked $x$ . $92^{\circ}$
Find the angle marked $x$ . $60^{\circ}$
Find the arc marked $x$ . $118^{\circ}$
Find the arc marked $x$ . $82^{\circ}$

Find $x$ where two chords cross
Find $x$ . $134$
Find $x$ . $126$
Find the value of $x$ . $44$
Find the value of $x$ . $162$
Find the value of $x$ . $160$

Find $x$ with secants and tangents from an outside point
The arc marked is the minor arc between the points of tangency. Find the value of $x$ . $100$
The arc marked is the minor arc between the points of tangency. Find $x$ . $82$
Find $x$ . $46$
The arc marked is the minor arc between the points of tangency. Find the value of $x$ . $70$
Find the value of $x$ . $40$

Find $x$ with a tangent and a chord
Find the value of $x$ . $196$
Find the value of $x$ . $105$
Find $x$ . $238$
Find $x$ . $112$
Find $x$ . $184$

How to do these

A central angle cuts off an arc of $80^{\circ}$ . Find an inscribed angle that cuts off the same arc

A central angle equals its arc, so the arc is 80°.
An inscribed angle is half its arc.

The answer is $40^{\circ}$ .

Where students go wrong

Doubling instead of halving. The arc is the big number and the inscribed angle is the small one, so if your answer came out larger than the arc you have used the rule upside down.

Also called tangent chord angle or angles formed by secants.

What you can put on this worksheet

Inscribed angles: Find the arc marked $x$ . — a circle with an inscribed angle, labeled 41°, x → $82^{\circ}$
The angle between a tangent and a chord: A tangent and a chord meet at the point of contact, and the chord cuts off a minor arc of $88^{\circ}$ . Find the acute angle between them. → $44^{\circ}$
Two chords crossing inside: Chords $\overline{B D}$ and $\overline{C E}$ intersect at $F$ . $m \overset{⌢}{BC} = 57^{\circ}$ and $m \overset{⌢}{DE} = 47^{\circ}$ . Find $m ∠ D F E$ . → $52^{\circ}$
Two secants meeting outside: A far arc of $121^{\circ}$ and a near arc of $59^{\circ}$ lie between two secants that meet outside a circle. What angle do the secants make? → $31^{\circ}$
A tangent and a secant meeting outside: A tangent and a secant drawn from one outside point form an angle of $31^{\circ}$ . The near intercepted arc is $59^{\circ}$ . Find the far one. → $121^{\circ}$
Two tangents meeting outside: Two tangents are drawn to a circle from one outside point. The far arc between them measures $256^{\circ}$ . Find the angle where they meet. → $76^{\circ}$
Quadrilaterals inscribed in a circle: Find $m ∠ C$ . — a quadrilateral abcd inscribed in a circle, labeled A, 92°, B, C, D, 86° → $88^{\circ}$
Two inscribed angles on one arc: Find the value of $x$ . — two inscribed angles on the same arc, (3x + 30)° and (x + 46)° → $8$
Two chords crossing, from a drawing: Find $x$ . — a circle with two chords crossing inside, labeled x°, 112°, 136° → $124$
Secants and tangents meeting outside, from a drawing: Find $x$ . — a circle with two lines meeting outside, labeled x°, 192°, 60° → $66$
A tangent and a chord, from a drawing: Find $x$ . — a circle with a tangent and a chord, labeled x°, 200° → $100$
The far arc from 360 first: Find $x$ . — a circle with points, chords and arcs → $29^{\circ}$
Two tangents: the near arc from the angle: Tangents from $P$ touch the circle at $A$ and $B$ . The minor arc $\overset{⌢}{AB}$ measures $96^{\circ}$ . Find $m ∠ A P B$ . → $84^{\circ}$

Questions about these worksheets

Yes. Every angles 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 angles 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 13: inscribed angles, the angle between a tangent and a chord, two chords crossing inside, two secants meeting outside, a tangent and a secant meeting outside, two tangents meeting outside, quadrilaterals inscribed in a circle, two inscribed angles on one arc, two chords crossing, from a drawing, secants and tangents meeting outside, from a drawing, a tangent and a chord, from a drawing, the far arc from 360 first and two tangents: the near arc from the angle. Tick as many as you want and set how many of each, or let it spread them evenly.

Doubling instead of halving. The arc is the big number and the inscribed angle is the small one, so if your answer came out larger than the arc you have used the rule upside down.

Angles in circles worksheet

Select your difficulty

Use inscribed angles

Find xxx where two chords cross

Find xxx with secants and tangents from an outside point

Find xxx with a tangent and a chord

Use inscribed angles

Find xxx where two chords cross

Find xxx with secants and tangents from an outside point

Find xxx with a tangent and a chord

How to do these

Where students go wrong

What you can put on this worksheet

Questions about these worksheets

Are these angles in circles worksheets free?+

Do the angles in circles worksheets come with an answer key?+

Are these angles in circles worksheets printable?+

What grade are these angles in circles worksheets for?+

Can I get a different version of the same angles in circles worksheet?+

Is this a free alternative to Kuta Software for angles in circles?+

Can I choose how hard the angles in circles questions are?+

What kinds of question are on a angles in circles worksheet?+

What do students get wrong with angles in circles?+

Find $x$ where two chords cross

Find $x$ with secants and tangents from an outside point

Find $x$ with a tangent and a chord

Find $x$ where two chords cross

Find $x$ with secants and tangents from an outside point

Find $x$ with a tangent and a chord