Read the first worked example with the solution open so the structure is clear.
Try the practice problems before revealing each solution.
Use the related concepts and background knowledge badges if you feel stuck.
What to Focus On
Core idea:A polygon is any closed shape made of three or more straight segments joined end to end.
Common stuck point:The procedure for polygon is the easy part; the trap is calling a figure with a curved edge a polygon. Asking "Is the figure closed and made of three or more straight sides with no curves?" first is what keeps a correct-looking calculation from being attached to the wrong concept.
Sense of Study hint:Ask: Is the figure closed and made of three or more straight sides with no curves?
Worked Examples
Example 1
easy
What is the sum of the interior angles of a hexagon?
Answer
720°
First step
1
Step 1: Use the interior angle sum formula: (n−2)×180° where n is the number of sides.
Full solution
2
Step 2: A hexagon has n=6 sides.
3
Step 3: Sum =(6−2)×180°=4×180°=720°.
Any polygon can be divided into (n−2) triangles by drawing diagonals from one vertex. Since each triangle contributes 180°, the total interior angle sum is (n−2)×180°. For a hexagon, this gives 720°.
Example 2
medium
Each interior angle of a regular polygon is 150°. How many sides does it have?
Example 3
easy
How many sides does a polygon with interior-angle sum 1080∘ have?Octagon: interior angle sum = 1080°
Example 4
medium
A regular polygon has each exterior angle equal to 24∘. Find the number of sides and the interior angle.
Example 5
medium
The interior angles of a quadrilateral are in the ratio 1:2:3:4. Find each angle.
Example 6
medium
A regular polygon's interior angle is three times its exterior angle. Find the number of sides.Regular octagon: interior 135° = 3 × exterior 45°, so n = 8
Example 7
medium
The interior-angle sum of a polygon equals 2520∘. How many sides does it have?
Example 8
hard
A polygon has interior-angle sum 4140∘. Is it possible for this polygon to be regular with each interior angle a whole number of degrees? If so, what is each angle?
Example 9
hard
A regular polygon's interior angle exceeds its exterior angle by 132∘. Find the number of sides.
Example 10
hard
A polygon has interior-angle sum equal to four times its exterior-angle sum. Find the number of sides.
Example 11
challenge
In a convex polygon, all but one of the interior angles each measure 150∘, and the remaining interior angle measures 90∘. How many sides does the polygon have?
Practice Problems
Try these problems on your own first, then open the solution to compare your method.
Example 1
easy
What is the sum of the exterior angles of any convex polygon?
Example 2
hard
A polygon has an interior angle sum of 1980°. How many sides does it have?
Example 3
easy
What is the fewest number of sides a polygon can have?
Example 4
easy
Is a circle a polygon?
Example 5
easy
What is a 7-sided polygon called?
Example 6
easy
How many sides does a polygon with 9 vertices have?
Example 7
easy
Find the sum of the interior angles of a quadrilateral (4 sides).Quadrilateral interior angles sum to 360°
Example 8
easy
Is a shape with one curved side and two straight sides a polygon?
Example 9
easy
What makes a polygon 'regular'?
Example 10
easy
Find the perimeter of a regular pentagon with side length 7.Regular pentagon with side length 7
Example 11
medium
Find the sum of the interior angles of a hexagon (6 sides).Hexagon interior angle sum: 6 × 120° = 720°
Example 12
medium
Find each interior angle of a regular hexagon.Find each interior angle of a regular hexagon
Example 13
medium
The interior angles of a polygon sum to 540∘. How many sides does it have?Pentagon interior angle sum = 5 × 108° = 540°
Example 14
medium
What is the sum of the exterior angles of any polygon, one at each vertex?
Example 15
medium
A regular polygon has each exterior angle equal to 40∘. How many sides does it have?
Example 16
medium
How many diagonals does a pentagon have?
Example 17
medium
Why does a regular hexagon tile the plane with no gaps, but a regular pentagon does not?
Example 18
medium
A polygon's interior angle sum is 1080∘. How many sides, and what is the shape's name?Octagon (8 sides): interior angle sum = 8 × 135° = 1080°
Example 19
challenge
Each interior angle of a regular polygon is 150∘. How many sides does it have?
Example 20
challenge
A convex polygon has 35 diagonals. How many sides does it have?
Example 21
challenge
Explain why the interior angle sum formula (n−2)×180∘ works, using triangles.
Example 22
challenge
As the number of sides n of a regular polygon grows, what value does each interior angle approach, and why does the shape approach a circle?
Example 23
easy
Find the sum of the interior angles of a pentagon (5 sides).Pentagon: (5 − 2) × 180° = 540°
Example 24
easy
Each interior angle of a regular polygon is 108∘. How many sides does it have?Regular pentagon — each interior angle = 108°
Example 25
easy
A regular polygon has 12 sides. Find the measure of each interior angle.
Example 26
easy
Find each exterior angle of a regular hexagon.
Example 27
medium
How many diagonals does a decagon (10-sided polygon) have?
Example 28
medium
Find the perimeter of a regular octagon with side length 4.5 cm.Regular octagon: side = 4.5 cm, perimeter = 8 × 4.5 = 36 cm
Example 29
medium
A regular hexagon has perimeter 48 cm. Find each side length.Regular hexagon: 6 × 8 cm = 48 cm perimeter
Example 30
medium
A polygon has 44 diagonals. How many sides does it have?
Example 31
hard
For how many values of n≥3 is the interior angle of a regular n-gon a whole number of degrees?
Example 32
hard
A regular polygon has each interior angle measuring 172∘. How many sides does it have?