Polygons and angles worksheet

The interior angles of an n-sided polygon add to n minus two, times 180. The exterior angles always add to 360 however many sides there are, which surprises people every time.

The preview can look cramped on a phone. The PDF and printed sheet come out normal.



Sigma Prep · sigmaprep.io/worksheets Name Polygons and Angles Date Period Find each measure, or answer as asked. Find interior and exterior angles Find the sum of the interior angles. Find each exterior angle of a regular octagon. Find each interior angle. Find each exterior angle. Find the number of sides Each interior angle of a regular polygon is $162^{\circ}$ . How many sides does it have? Each exterior angle of a regular polygon is $90^{\circ}$ . How many sides does it have? The interior angles of a polygon add to $900^{\circ}$ . How many sides does it have? Each exterior angle of a regular polygon is $60^{\circ}$ . How many sides does it have? Find $x$ from polygon angles A hexagon has angles ${(x + 143)}^{\circ}$ , ${(2 x + 77)}^{\circ}$ , ${(3 x + 63)}^{\circ}$ , $82^{\circ}$ , $106^{\circ}$ and $147^{\circ}$ . Find $x$ . A pentagon has angles $73^{\circ}$ , $120^{\circ}$ , ${(3 x + 110)}^{\circ}$ , ${(3 x + 59)}^{\circ}$ and ${(3 x + 88)}^{\circ}$ . Find $x$ . A quadrilateral has angles ${(2 x + 73)}^{\circ}$ , $91^{\circ}$ , $106^{\circ}$ and ${(x + 63)}^{\circ}$ . Find $x$ . A pentagon has angles $102^{\circ}$ , ${(x + 130)}^{\circ}$ , ${(3 x + 59)}^{\circ}$ , $101^{\circ}$ and ${(3 x + 15)}^{\circ}$ . Find $x$ . Find the missing exterior angle A convex hexagon has one exterior angle at each vertex: $70^{\circ}$ , $85^{\circ}$ , $45^{\circ}$ , $35^{\circ}$ , $75^{\circ}$ and $x^{\circ}$ . Find $x$ . A convex hexagon has one exterior angle at each vertex: $3 x^{\circ}$ , $113^{\circ}$ , $20^{\circ}$ , $75^{\circ}$ , ${(x + 11)}^{\circ}$ , ${(3 x + 15)}^{\circ}$ . Find $x$ . A convex heptagon has one exterior angle at each vertex: $25^{\circ}$ , ${(2 x - 4)}^{\circ}$ , $75^{\circ}$ , $4 x^{\circ}$ , $68^{\circ}$ , $20^{\circ}$ , $80^{\circ}$ . Find $x$ . A convex heptagon has one exterior angle at each vertex: $95^{\circ}$ , $80^{\circ}$ , $25^{\circ}$ , $45^{\circ}$ , $40^{\circ}$ , $30^{\circ}$ and $x^{\circ}$ . Find $x$ . Classify polygons and check angle sums Is the polygon regular or irregular? Is the polygon convex or concave? Name the polygon by its number of sides. Is there a polygon whose interior angles add to $1620^{\circ}$ ? Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Polygons and Angles

Date Period

Find each measure, or answer as asked.

Find interior and exterior angles
Find the sum of the interior angles.
Find each exterior angle of a regular octagon.
Find each interior angle.
Find each exterior angle.

Find the number of sides
Each interior angle of a regular polygon is $162^{\circ}$ . How many sides does it have?
Each exterior angle of a regular polygon is $90^{\circ}$ . How many sides does it have?
The interior angles of a polygon add to $900^{\circ}$ . How many sides does it have?
Each exterior angle of a regular polygon is $60^{\circ}$ . How many sides does it have?

Find $x$ from polygon angles
A hexagon has angles ${(x + 143)}^{\circ}$ , ${(2 x + 77)}^{\circ}$ , ${(3 x + 63)}^{\circ}$ , $82^{\circ}$ , $106^{\circ}$ and $147^{\circ}$ . Find $x$ .
A pentagon has angles $73^{\circ}$ , $120^{\circ}$ , ${(3 x + 110)}^{\circ}$ , ${(3 x + 59)}^{\circ}$ and ${(3 x + 88)}^{\circ}$ . Find $x$ .
A quadrilateral has angles ${(2 x + 73)}^{\circ}$ , $91^{\circ}$ , $106^{\circ}$ and ${(x + 63)}^{\circ}$ . Find $x$ .
A pentagon has angles $102^{\circ}$ , ${(x + 130)}^{\circ}$ , ${(3 x + 59)}^{\circ}$ , $101^{\circ}$ and ${(3 x + 15)}^{\circ}$ . Find $x$ .

Find the missing exterior angle
A convex hexagon has one exterior angle at each vertex: $70^{\circ}$ , $85^{\circ}$ , $45^{\circ}$ , $35^{\circ}$ , $75^{\circ}$ and $x^{\circ}$ . Find $x$ .
A convex hexagon has one exterior angle at each vertex: $3 x^{\circ}$ , $113^{\circ}$ , $20^{\circ}$ , $75^{\circ}$ , ${(x + 11)}^{\circ}$ , ${(3 x + 15)}^{\circ}$ . Find $x$ .
A convex heptagon has one exterior angle at each vertex: $25^{\circ}$ , ${(2 x - 4)}^{\circ}$ , $75^{\circ}$ , $4 x^{\circ}$ , $68^{\circ}$ , $20^{\circ}$ , $80^{\circ}$ . Find $x$ .
A convex heptagon has one exterior angle at each vertex: $95^{\circ}$ , $80^{\circ}$ , $25^{\circ}$ , $45^{\circ}$ , $40^{\circ}$ , $30^{\circ}$ and $x^{\circ}$ . Find $x$ .

Classify polygons and check angle sums
Is the polygon regular or irregular?
Is the polygon convex or concave?
Name the polygon by its number of sides.
Is there a polygon whose interior angles add to $1620^{\circ}$ ?



Sigma Prep · sigmaprep.io/worksheets Name Polygons and AnglesAnswer keyVersion 1 Date Period Find each measure, or answer as asked. Find interior and exterior angles Find the sum of the interior angles. $1260^{\circ}$ Find each exterior angle of a regular octagon. $45^{\circ}$ Find each interior angle. $120^{\circ}$ Find each exterior angle. $24^{\circ}$ Find the number of sides Each interior angle of a regular polygon is $162^{\circ}$ . How many sides does it have? $20$ Each exterior angle of a regular polygon is $90^{\circ}$ . How many sides does it have? $4$ The interior angles of a polygon add to $900^{\circ}$ . How many sides does it have? $7$ Each exterior angle of a regular polygon is $60^{\circ}$ . How many sides does it have? $6$ Find $x$ from polygon angles A hexagon has angles ${(x + 143)}^{\circ}$ , ${(2 x + 77)}^{\circ}$ , ${(3 x + 63)}^{\circ}$ , $82^{\circ}$ , $106^{\circ}$ and $147^{\circ}$ . Find $x$ . $17$ A pentagon has angles $73^{\circ}$ , $120^{\circ}$ , ${(3 x + 110)}^{\circ}$ , ${(3 x + 59)}^{\circ}$ and ${(3 x + 88)}^{\circ}$ . Find $x$ . $10$ A quadrilateral has angles ${(2 x + 73)}^{\circ}$ , $91^{\circ}$ , $106^{\circ}$ and ${(x + 63)}^{\circ}$ . Find $x$ . $9$ A pentagon has angles $102^{\circ}$ , ${(x + 130)}^{\circ}$ , ${(3 x + 59)}^{\circ}$ , $101^{\circ}$ and ${(3 x + 15)}^{\circ}$ . Find $x$ . $19$ Find the missing exterior angle A convex hexagon has one exterior angle at each vertex: $70^{\circ}$ , $85^{\circ}$ , $45^{\circ}$ , $35^{\circ}$ , $75^{\circ}$ and $x^{\circ}$ . Find $x$ . $50$ A convex hexagon has one exterior angle at each vertex: $3 x^{\circ}$ , $113^{\circ}$ , $20^{\circ}$ , $75^{\circ}$ , ${(x + 11)}^{\circ}$ , ${(3 x + 15)}^{\circ}$ . Find $x$ . $18$ A convex heptagon has one exterior angle at each vertex: $25^{\circ}$ , ${(2 x - 4)}^{\circ}$ , $75^{\circ}$ , $4 x^{\circ}$ , $68^{\circ}$ , $20^{\circ}$ , $80^{\circ}$ . Find $x$ . $16$ A convex heptagon has one exterior angle at each vertex: $95^{\circ}$ , $80^{\circ}$ , $25^{\circ}$ , $45^{\circ}$ , $40^{\circ}$ , $30^{\circ}$ and $x^{\circ}$ . Find $x$ . $45$ Classify polygons and check angle sums Is the polygon regular or irregular?Regular Is the polygon convex or concave?Concave Name the polygon by its number of sides.Decagon Is there a polygon whose interior angles add to $1620^{\circ}$ ?Yes Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Polygons and AnglesAnswer keyVersion 1

Date Period

Find each measure, or answer as asked.

Find interior and exterior angles
Find the sum of the interior angles. $1260^{\circ}$
Find each exterior angle of a regular octagon. $45^{\circ}$
Find each interior angle. $120^{\circ}$
Find each exterior angle. $24^{\circ}$

Find the number of sides
Each interior angle of a regular polygon is $162^{\circ}$ . How many sides does it have? $20$
Each exterior angle of a regular polygon is $90^{\circ}$ . How many sides does it have? $4$
The interior angles of a polygon add to $900^{\circ}$ . How many sides does it have? $7$
Each exterior angle of a regular polygon is $60^{\circ}$ . How many sides does it have? $6$

Find $x$ from polygon angles
A hexagon has angles ${(x + 143)}^{\circ}$ , ${(2 x + 77)}^{\circ}$ , ${(3 x + 63)}^{\circ}$ , $82^{\circ}$ , $106^{\circ}$ and $147^{\circ}$ . Find $x$ . $17$
A pentagon has angles $73^{\circ}$ , $120^{\circ}$ , ${(3 x + 110)}^{\circ}$ , ${(3 x + 59)}^{\circ}$ and ${(3 x + 88)}^{\circ}$ . Find $x$ . $10$
A quadrilateral has angles ${(2 x + 73)}^{\circ}$ , $91^{\circ}$ , $106^{\circ}$ and ${(x + 63)}^{\circ}$ . Find $x$ . $9$
A pentagon has angles $102^{\circ}$ , ${(x + 130)}^{\circ}$ , ${(3 x + 59)}^{\circ}$ , $101^{\circ}$ and ${(3 x + 15)}^{\circ}$ . Find $x$ . $19$

Find the missing exterior angle
A convex hexagon has one exterior angle at each vertex: $70^{\circ}$ , $85^{\circ}$ , $45^{\circ}$ , $35^{\circ}$ , $75^{\circ}$ and $x^{\circ}$ . Find $x$ . $50$
A convex hexagon has one exterior angle at each vertex: $3 x^{\circ}$ , $113^{\circ}$ , $20^{\circ}$ , $75^{\circ}$ , ${(x + 11)}^{\circ}$ , ${(3 x + 15)}^{\circ}$ . Find $x$ . $18$
A convex heptagon has one exterior angle at each vertex: $25^{\circ}$ , ${(2 x - 4)}^{\circ}$ , $75^{\circ}$ , $4 x^{\circ}$ , $68^{\circ}$ , $20^{\circ}$ , $80^{\circ}$ . Find $x$ . $16$
A convex heptagon has one exterior angle at each vertex: $95^{\circ}$ , $80^{\circ}$ , $25^{\circ}$ , $45^{\circ}$ , $40^{\circ}$ , $30^{\circ}$ and $x^{\circ}$ . Find $x$ . $45$

Classify polygons and check angle sums
Is the polygon regular or irregular?Regular
Is the polygon convex or concave?Concave
Name the polygon by its number of sides.Decagon
Is there a polygon whose interior angles add to $1620^{\circ}$ ?Yes

How to do these

Find the sum of the interior angles of a 7-sided polygon

Subtract 2 from the number of sides.
Multiply by 180.

The answer is $900^{\circ}$ .

Where students go wrong

Multiplying by the number of sides rather than two fewer. A pentagon is 540, not 900, and the quickest sanity check is a triangle: three sides has to give 180.

Also called interior angle sum, exterior angles or regular polygons.

What you can put on this worksheet

Interior and exterior angles: Find each exterior angle. — a regular 8-sided polygon → $45^{\circ}$
The number of sides from an angle: Each exterior angle of a regular polygon is $45^{\circ}$ . How many sides does it have? → $8$
Angles written in x: A quadrilateral has angles ${(3 x + 90)}^{\circ}$ , $74^{\circ}$ , ${(3 x + 75)}^{\circ}$ and $73^{\circ}$ . Find $x$ . → $8$
A missing exterior angle: A convex heptagon has one exterior angle at each vertex: $65^{\circ}$ , $80^{\circ}$ , $25^{\circ}$ , $60^{\circ}$ , $65^{\circ}$ , $40^{\circ}$ and $x^{\circ}$ . Find $x$ . → $25$
Name it, convex or concave, regular or not: Is the polygon convex or concave? — a polygon with 7 sides → Convex
Which combinations are possible: Can there be an equiangular concave pentagon? Answer possible or not possible. → Not possible

Questions about these worksheets

Yes. Every polygons and angles 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 polygons and angles 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 6: interior and exterior angles, the number of sides from an angle, angles written in x, a missing exterior angle, name it, convex or concave, regular or not and which combinations are possible. Tick as many as you want and set how many of each, or let it spread them evenly.

Multiplying by the number of sides rather than two fewer. A pentagon is 540, not 900, and the quickest sanity check is a triangle: three sides has to give 180.

Polygons and angles worksheet

Find interior and exterior angles

Find the number of sides

Find xxx from polygon angles

Find the missing exterior angle

Classify polygons and check angle sums

Find interior and exterior angles

Find the number of sides

Find xxx from polygon angles

Find the missing exterior angle

Classify polygons and check angle sums

How to do these

Where students go wrong

What you can put on this worksheet

Questions about these worksheets

Are these polygons and angles worksheets free?+

Do the polygons and angles worksheets come with an answer key?+

Are these polygons and angles worksheets printable?+

What grade are these polygons and angles worksheets for?+

Can I get a different version of the same polygons and angles worksheet?+

Is this a free alternative to Kuta Software for polygons and angles?+

Can I choose how hard the polygons and angles questions are?+

What kinds of question are on a polygons and angles worksheet?+

What do students get wrong with polygons and angles?+

Find $x$ from polygon angles

Find $x$ from polygon angles