Polygons - Tenth Grade Geometry
Introduction to Polygons
Polygon: A closed plane figure formed by three or more line segments
Key Properties:
• Made of straight line segments (sides)
• Sides meet only at their endpoints (vertices)
• Closed figure (no gaps or openings)
• Two-dimensional (flat)
Minimum sides: 3 (triangle)
Key Properties:
• Made of straight line segments (sides)
• Sides meet only at their endpoints (vertices)
• Closed figure (no gaps or openings)
• Two-dimensional (flat)
Minimum sides: 3 (triangle)
1. Polygon Vocabulary
Basic Terms
Vertex (plural: Vertices): A point where two sides meet; corner point
Side: A line segment forming the boundary of the polygon
Diagonal: A line segment connecting two non-adjacent vertices
Perimeter: Total distance around the polygon (sum of all side lengths)
Interior: The region inside the polygon
Exterior: The region outside the polygon
Side: A line segment forming the boundary of the polygon
Diagonal: A line segment connecting two non-adjacent vertices
Perimeter: Total distance around the polygon (sum of all side lengths)
Interior: The region inside the polygon
Exterior: The region outside the polygon
Number of Diagonals Formula:
For a polygon with $n$ sides:
$$\text{Number of diagonals} = \frac{n(n-3)}{2}$$
Explanation:
• From each vertex, you can draw $(n-3)$ diagonals
• Total would be $n(n-3)$, but this counts each diagonal twice
• Divide by 2 to get actual number
For a polygon with $n$ sides:
$$\text{Number of diagonals} = \frac{n(n-3)}{2}$$
Explanation:
• From each vertex, you can draw $(n-3)$ diagonals
• Total would be $n(n-3)$, but this counts each diagonal twice
• Divide by 2 to get actual number
Naming Polygons by Number of Sides
Common Polygon Names:
• 3 sides: Triangle
• 4 sides: Quadrilateral
• 5 sides: Pentagon
• 6 sides: Hexagon
• 7 sides: Heptagon (or Septagon)
• 8 sides: Octagon
• 9 sides: Nonagon (or Enneagon)
• 10 sides: Decagon
• 12 sides: Dodecagon
• n sides: n-gon
• 3 sides: Triangle
• 4 sides: Quadrilateral
• 5 sides: Pentagon
• 6 sides: Hexagon
• 7 sides: Heptagon (or Septagon)
• 8 sides: Octagon
• 9 sides: Nonagon (or Enneagon)
• 10 sides: Decagon
• 12 sides: Dodecagon
• n sides: n-gon
Classification: Regular vs. Irregular
Regular Polygon:
• All sides are equal in length (equilateral)
• All interior angles are equal (equiangular)
• Has rotational and reflective symmetry
• Always convex
• Examples: Equilateral triangle, square, regular pentagon, regular hexagon
Irregular Polygon:
• At least one side has different length, OR
• At least one angle has different measure
• May be convex or concave
• Examples: Rectangle (not square), scalene triangle, most quadrilaterals
• All sides are equal in length (equilateral)
• All interior angles are equal (equiangular)
• Has rotational and reflective symmetry
• Always convex
• Examples: Equilateral triangle, square, regular pentagon, regular hexagon
Irregular Polygon:
• At least one side has different length, OR
• At least one angle has different measure
• May be convex or concave
• Examples: Rectangle (not square), scalene triangle, most quadrilaterals
Classification: Convex vs. Concave
Convex Polygon:
• All interior angles are less than 180°
• All vertices point "outward"
• Any line segment connecting two points inside stays completely inside
• All diagonals lie inside the polygon
• Can be regular or irregular
• Examples: All regular polygons, rectangles, most common shapes
Concave Polygon:
• At least one interior angle is greater than 180° (reflex angle)
• At least one vertex appears to "cave in"
• Some diagonals lie outside the polygon
• Always irregular
• Looks like it has a "dent" or "indentation"
• Examples: Star shapes, arrow shapes, L-shapes
• All interior angles are less than 180°
• All vertices point "outward"
• Any line segment connecting two points inside stays completely inside
• All diagonals lie inside the polygon
• Can be regular or irregular
• Examples: All regular polygons, rectangles, most common shapes
Concave Polygon:
• At least one interior angle is greater than 180° (reflex angle)
• At least one vertex appears to "cave in"
• Some diagonals lie outside the polygon
• Always irregular
• Looks like it has a "dent" or "indentation"
• Examples: Star shapes, arrow shapes, L-shapes
Example 1: Find number of diagonals
How many diagonals does a hexagon (6 sides) have?
$$\text{Diagonals} = \frac{n(n-3)}{2} = \frac{6(6-3)}{2} = \frac{6(3)}{2} = \frac{18}{2} = 9$$
Answer: A hexagon has 9 diagonals
How many diagonals does a hexagon (6 sides) have?
$$\text{Diagonals} = \frac{n(n-3)}{2} = \frac{6(6-3)}{2} = \frac{6(3)}{2} = \frac{18}{2} = 9$$
Answer: A hexagon has 9 diagonals
2. Interior Angles of Polygons
Interior Angle: An angle inside the polygon, formed by two adjacent sides
Number of Interior Angles: Always equals the number of sides
Key Concept: Sum depends on number of sides, not shape or size
Number of Interior Angles: Always equals the number of sides
Key Concept: Sum depends on number of sides, not shape or size
Sum of Interior Angles
Sum of Interior Angles Formula:
For a polygon with $n$ sides:
$$S = (n - 2) \times 180°$$
Where:
• $S$ = sum of all interior angles
• $n$ = number of sides
Why $(n-2)$?
A polygon can be divided into $(n-2)$ triangles from one vertex
Each triangle has angle sum of 180°
For a polygon with $n$ sides:
$$S = (n - 2) \times 180°$$
Where:
• $S$ = sum of all interior angles
• $n$ = number of sides
Why $(n-2)$?
A polygon can be divided into $(n-2)$ triangles from one vertex
Each triangle has angle sum of 180°
Example 1: Find sum of interior angles
Find the sum of interior angles of an octagon (8 sides)
$$S = (n - 2) \times 180°$$
$$S = (8 - 2) \times 180°$$
$$S = 6 \times 180°$$
$$S = 1080°$$
Answer: Sum of interior angles = 1080°
Find the sum of interior angles of an octagon (8 sides)
$$S = (n - 2) \times 180°$$
$$S = (8 - 2) \times 180°$$
$$S = 6 \times 180°$$
$$S = 1080°$$
Answer: Sum of interior angles = 1080°
Each Interior Angle of Regular Polygon
Each Interior Angle of Regular Polygon:
For a regular polygon with $n$ sides:
$$\text{Each angle} = \frac{(n-2) \times 180°}{n}$$
Or equivalently:
$$\text{Each angle} = \frac{S}{n}$$
Where:
• $S$ = sum of interior angles
• $n$ = number of sides
Note: This formula ONLY works for regular polygons!
For a regular polygon with $n$ sides:
$$\text{Each angle} = \frac{(n-2) \times 180°}{n}$$
Or equivalently:
$$\text{Each angle} = \frac{S}{n}$$
Where:
• $S$ = sum of interior angles
• $n$ = number of sides
Note: This formula ONLY works for regular polygons!
Example 2: Find each interior angle of regular polygon
Find each interior angle of a regular pentagon (5 sides)
Method 1: Using formula directly
$$\text{Each angle} = \frac{(n-2) \times 180°}{n}$$
$$= \frac{(5-2) \times 180°}{5}$$
$$= \frac{3 \times 180°}{5}$$
$$= \frac{540°}{5}$$
$$= 108°$$
Method 2: Find sum first, then divide
Sum = $(5-2) \times 180° = 540°$
Each angle = $540° \div 5 = 108°$
Answer: Each interior angle = 108°
Find each interior angle of a regular pentagon (5 sides)
Method 1: Using formula directly
$$\text{Each angle} = \frac{(n-2) \times 180°}{n}$$
$$= \frac{(5-2) \times 180°}{5}$$
$$= \frac{3 \times 180°}{5}$$
$$= \frac{540°}{5}$$
$$= 108°$$
Method 2: Find sum first, then divide
Sum = $(5-2) \times 180° = 540°$
Each angle = $540° \div 5 = 108°$
Answer: Each interior angle = 108°
Example 3: Find number of sides given interior angle
Each interior angle of a regular polygon is 120°. How many sides does it have?
$$\frac{(n-2) \times 180°}{n} = 120°$$
$(n-2) \times 180° = 120n$
$180n - 360° = 120n$
$180n - 120n = 360°$
$60n = 360°$
$n = 6$
Answer: The polygon is a hexagon (6 sides)
Each interior angle of a regular polygon is 120°. How many sides does it have?
$$\frac{(n-2) \times 180°}{n} = 120°$$
$(n-2) \times 180° = 120n$
$180n - 360° = 120n$
$180n - 120n = 360°$
$60n = 360°$
$n = 6$
Answer: The polygon is a hexagon (6 sides)
3. Exterior Angles of Polygons
Exterior Angle: Angle formed by one side and the extension of an adjacent side
At each vertex: Interior angle + Exterior angle = 180° (linear pair)
One per vertex: Consider only one exterior angle at each vertex
Amazing Property: Sum is ALWAYS 360° for ANY polygon!
At each vertex: Interior angle + Exterior angle = 180° (linear pair)
One per vertex: Consider only one exterior angle at each vertex
Amazing Property: Sum is ALWAYS 360° for ANY polygon!
Sum of Exterior Angles
Sum of Exterior Angles Theorem:
For ANY polygon (regular or irregular, convex):
$$\text{Sum of exterior angles} = 360°$$
Key Points:
• This is true for ALL polygons, regardless of number of sides
• Triangle: 360°
• Square: 360°
• Pentagon: 360°
• 100-gon: 360°
• Must take only ONE exterior angle at each vertex
For ANY polygon (regular or irregular, convex):
$$\text{Sum of exterior angles} = 360°$$
Key Points:
• This is true for ALL polygons, regardless of number of sides
• Triangle: 360°
• Square: 360°
• Pentagon: 360°
• 100-gon: 360°
• Must take only ONE exterior angle at each vertex
Each Exterior Angle of Regular Polygon
Each Exterior Angle of Regular Polygon:
For a regular polygon with $n$ sides:
$$\text{Each exterior angle} = \frac{360°}{n}$$
Relationship with Interior Angle:
$$\text{Exterior angle} = 180° - \text{Interior angle}$$
Or:
$$\text{Interior angle} = 180° - \text{Exterior angle}$$
For a regular polygon with $n$ sides:
$$\text{Each exterior angle} = \frac{360°}{n}$$
Relationship with Interior Angle:
$$\text{Exterior angle} = 180° - \text{Interior angle}$$
Or:
$$\text{Interior angle} = 180° - \text{Exterior angle}$$
Example 1: Find each exterior angle
Find each exterior angle of a regular octagon (8 sides)
$$\text{Each exterior angle} = \frac{360°}{n} = \frac{360°}{8} = 45°$$
Verify with interior angle:
Interior angle = $\frac{(8-2) \times 180°}{8} = \frac{1080°}{8} = 135°$
Exterior angle = $180° - 135° = 45°$ ✓
Answer: Each exterior angle = 45°
Find each exterior angle of a regular octagon (8 sides)
$$\text{Each exterior angle} = \frac{360°}{n} = \frac{360°}{8} = 45°$$
Verify with interior angle:
Interior angle = $\frac{(8-2) \times 180°}{8} = \frac{1080°}{8} = 135°$
Exterior angle = $180° - 135° = 45°$ ✓
Answer: Each exterior angle = 45°
Example 2: Find number of sides given exterior angle
Each exterior angle of a regular polygon is 40°. How many sides does it have?
$$\frac{360°}{n} = 40°$$
$360° = 40n$
$n = \frac{360°}{40°}$
$n = 9$
Answer: The polygon is a nonagon (9 sides)
Each exterior angle of a regular polygon is 40°. How many sides does it have?
$$\frac{360°}{n} = 40°$$
$360° = 40n$
$n = \frac{360°}{40°}$
$n = 9$
Answer: The polygon is a nonagon (9 sides)
Example 3: Irregular polygon
The exterior angles of a pentagon are 60°, 80°, 70°, 90°, and x°. Find x.
Sum of exterior angles = 360°
$60° + 80° + 70° + 90° + x = 360°$
$300° + x = 360°$
$x = 60°$
Answer: x = 60°
The exterior angles of a pentagon are 60°, 80°, 70°, 90°, and x°. Find x.
Sum of exterior angles = 360°
$60° + 80° + 70° + 90° + x = 360°$
$300° + x = 360°$
$x = 60°$
Answer: x = 60°
4. Review: Interior and Exterior Angles of Polygons
Relationship Between Interior and Exterior Angles
At Each Vertex:
$$\text{Interior angle} + \text{Exterior angle} = 180°$$
This is because:
• They form a linear pair (supplementary angles)
• One is inside, one is outside, they share a side
Therefore:
• If you know interior angle, exterior = $180°$ - interior
• If you know exterior angle, interior = $180°$ - exterior
$$\text{Interior angle} + \text{Exterior angle} = 180°$$
This is because:
• They form a linear pair (supplementary angles)
• One is inside, one is outside, they share a side
Therefore:
• If you know interior angle, exterior = $180°$ - interior
• If you know exterior angle, interior = $180°$ - exterior
Summary of All Formulas
Complete Formula Sheet:
For ANY polygon with n sides:
1. Sum of interior angles:
$$S = (n-2) \times 180°$$
2. Sum of exterior angles:
$$\text{Sum} = 360°$$ (always!)
3. Number of diagonals:
$$D = \frac{n(n-3)}{2}$$
For REGULAR polygon with n sides:
4. Each interior angle:
$$\text{Each} = \frac{(n-2) \times 180°}{n}$$
5. Each exterior angle:
$$\text{Each} = \frac{360°}{n}$$
6. Relationship at each vertex:
$$\text{Interior} + \text{Exterior} = 180°$$
For ANY polygon with n sides:
1. Sum of interior angles:
$$S = (n-2) \times 180°$$
2. Sum of exterior angles:
$$\text{Sum} = 360°$$ (always!)
3. Number of diagonals:
$$D = \frac{n(n-3)}{2}$$
For REGULAR polygon with n sides:
4. Each interior angle:
$$\text{Each} = \frac{(n-2) \times 180°}{n}$$
5. Each exterior angle:
$$\text{Each} = \frac{360°}{n}$$
6. Relationship at each vertex:
$$\text{Interior} + \text{Exterior} = 180°$$
Example 1: Complete table
For a regular hexagon, find all angle measures
$n = 6$ sides
Sum of interior angles:
$S = (6-2) \times 180° = 720°$
Each interior angle:
$\frac{720°}{6} = 120°$
Sum of exterior angles:
$360°$ (always)
Each exterior angle:
$\frac{360°}{6} = 60°$
Verify: $120° + 60° = 180°$ ✓
For a regular hexagon, find all angle measures
$n = 6$ sides
Sum of interior angles:
$S = (6-2) \times 180° = 720°$
Each interior angle:
$\frac{720°}{6} = 120°$
Sum of exterior angles:
$360°$ (always)
Each exterior angle:
$\frac{360°}{6} = 60°$
Verify: $120° + 60° = 180°$ ✓
Example 2: Problem-solving
The ratio of an interior angle to an exterior angle of a regular polygon is 5:1. Find the number of sides.
Let exterior angle = $x$
Then interior angle = $5x$
Since they are supplementary:
$5x + x = 180°$
$6x = 180°$
$x = 30°$ (exterior angle)
Using exterior angle formula:
$\frac{360°}{n} = 30°$
$n = \frac{360°}{30°} = 12$
Answer: The polygon is a dodecagon (12 sides)
The ratio of an interior angle to an exterior angle of a regular polygon is 5:1. Find the number of sides.
Let exterior angle = $x$
Then interior angle = $5x$
Since they are supplementary:
$5x + x = 180°$
$6x = 180°$
$x = 30°$ (exterior angle)
Using exterior angle formula:
$\frac{360°}{n} = 30°$
$n = \frac{360°}{30°} = 12$
Answer: The polygon is a dodecagon (12 sides)
5. Construct an Equilateral Triangle or Regular Hexagon
Construction: Drawing using only compass and straightedge
No measuring: Cannot use protractor or ruler measurements
Regular polygons: All sides and angles equal
No measuring: Cannot use protractor or ruler measurements
Regular polygons: All sides and angles equal
Construct Equilateral Triangle
Construction: Equilateral Triangle
Given: Line segment AB
Goal: Construct equilateral triangle ABC
Step 1: Draw line segment AB (this will be one side)
Step 2: Place compass point on A
Step 3: Set compass width to length AB
Step 4: Draw an arc above (or below) line AB
Step 5: Keep same compass width, place compass point on B
Step 6: Draw an arc that intersects the first arc
Step 7: Label intersection point as C
Step 8: Draw segments AC and BC
Result: △ABC is equilateral (all sides equal)
Properties:
• All sides: $AB = BC = CA$
• All angles: $60°$ each
• Sum of angles: $180°$
Given: Line segment AB
Goal: Construct equilateral triangle ABC
Step 1: Draw line segment AB (this will be one side)
Step 2: Place compass point on A
Step 3: Set compass width to length AB
Step 4: Draw an arc above (or below) line AB
Step 5: Keep same compass width, place compass point on B
Step 6: Draw an arc that intersects the first arc
Step 7: Label intersection point as C
Step 8: Draw segments AC and BC
Result: △ABC is equilateral (all sides equal)
Properties:
• All sides: $AB = BC = CA$
• All angles: $60°$ each
• Sum of angles: $180°$
Construct Regular Hexagon
Construction: Regular Hexagon (Method 1 - Using Circle)
Step 1: Draw a circle with center O
Step 2: Mark any point on the circle as A
Step 3: Keep compass width equal to radius
Step 4: Place compass on A, mark arc on circle as B
Step 5: Place compass on B, mark arc on circle as C
Step 6: Place compass on C, mark arc on circle as D
Step 7: Place compass on D, mark arc on circle as E
Step 8: Place compass on E, mark arc on circle as F
Step 9: Connect consecutive points: AB, BC, CD, DE, EF, FA
Result: Regular hexagon ABCDEF
Why this works:
• Radius = side length of hexagon
• Each central angle = $\frac{360°}{6} = 60°$
• Creates 6 equilateral triangles
Properties:
• All 6 sides equal
• All 6 interior angles = $120°$ each
• All 6 exterior angles = $60°$ each
Step 1: Draw a circle with center O
Step 2: Mark any point on the circle as A
Step 3: Keep compass width equal to radius
Step 4: Place compass on A, mark arc on circle as B
Step 5: Place compass on B, mark arc on circle as C
Step 6: Place compass on C, mark arc on circle as D
Step 7: Place compass on D, mark arc on circle as E
Step 8: Place compass on E, mark arc on circle as F
Step 9: Connect consecutive points: AB, BC, CD, DE, EF, FA
Result: Regular hexagon ABCDEF
Why this works:
• Radius = side length of hexagon
• Each central angle = $\frac{360°}{6} = 60°$
• Creates 6 equilateral triangles
Properties:
• All 6 sides equal
• All 6 interior angles = $120°$ each
• All 6 exterior angles = $60°$ each
Why Hexagon Construction Works:
When you use the radius as the compass width:
• You're dividing the circle into 6 equal arcs
• Each arc subtends a central angle of 60°
• The radius to consecutive points forms an equilateral triangle
• Six such triangles form a regular hexagon
• This is unique to hexagons: radius = side length!
When you use the radius as the compass width:
• You're dividing the circle into 6 equal arcs
• Each arc subtends a central angle of 60°
• The radius to consecutive points forms an equilateral triangle
• Six such triangles form a regular hexagon
• This is unique to hexagons: radius = side length!
6. Construct a Square
Square Properties:
• All four sides equal
• All four angles = 90°
• Diagonals are equal and perpendicular
• Regular quadrilateral
• All four sides equal
• All four angles = 90°
• Diagonals are equal and perpendicular
• Regular quadrilateral
Construction: Square (Method 1 - Using Line Segment)
Given: Line segment AB
Goal: Construct square ABCD
Step 1: Draw line segment AB
Step 2: Construct perpendicular at A
a) Place compass on A, draw arcs on both sides of A intersecting AB
b) From these intersection points, draw arcs above AB that intersect
c) Draw line through A and intersection point (perpendicular to AB)
Step 3: On perpendicular line, mark point D such that AD = AB
(Use compass width equal to AB)
Step 4: Construct perpendicular at B
Repeat Step 2 process at point B
Step 5: On this perpendicular, mark point C such that BC = AB
Step 6: Connect points to form square ABCD
(Or verify that D and C are correct by drawing DC)
Result: Square ABCD with all sides equal and all angles 90°
Given: Line segment AB
Goal: Construct square ABCD
Step 1: Draw line segment AB
Step 2: Construct perpendicular at A
a) Place compass on A, draw arcs on both sides of A intersecting AB
b) From these intersection points, draw arcs above AB that intersect
c) Draw line through A and intersection point (perpendicular to AB)
Step 3: On perpendicular line, mark point D such that AD = AB
(Use compass width equal to AB)
Step 4: Construct perpendicular at B
Repeat Step 2 process at point B
Step 5: On this perpendicular, mark point C such that BC = AB
Step 6: Connect points to form square ABCD
(Or verify that D and C are correct by drawing DC)
Result: Square ABCD with all sides equal and all angles 90°
Construction: Square (Method 2 - Using Circle)
Step 1: Draw a circle with center O
Step 2: Draw diameter AB through center
Step 3: Construct perpendicular diameter CD through center O
(Using perpendicular bisector construction on AB)
Step 4: Connect the four points: A, C, B, D in order
Result: Square ACBD inscribed in circle
Properties:
• All sides equal (chords of equal length)
• All angles 90° (inscribed in semicircle)
• Diagonals AB and CD are diameters
Step 1: Draw a circle with center O
Step 2: Draw diameter AB through center
Step 3: Construct perpendicular diameter CD through center O
(Using perpendicular bisector construction on AB)
Step 4: Connect the four points: A, C, B, D in order
Result: Square ACBD inscribed in circle
Properties:
• All sides equal (chords of equal length)
• All angles 90° (inscribed in semicircle)
• Diagonals AB and CD are diameters
Example: Verify square construction
After constructing square ABCD with side length 5 cm, verify it's correct:
Check 1: All sides equal
$AB = BC = CD = DA = 5$ cm ✓
Check 2: All angles 90°
$\angle A = \angle B = \angle C = \angle D = 90°$ ✓
Check 3: Diagonals equal
$AC = BD = 5\sqrt{2}$ cm (using Pythagorean theorem) ✓
Check 4: Angle sum
$90° + 90° + 90° + 90° = 360°$ ✓
After constructing square ABCD with side length 5 cm, verify it's correct:
Check 1: All sides equal
$AB = BC = CD = DA = 5$ cm ✓
Check 2: All angles 90°
$\angle A = \angle B = \angle C = \angle D = 90°$ ✓
Check 3: Diagonals equal
$AC = BD = 5\sqrt{2}$ cm (using Pythagorean theorem) ✓
Check 4: Angle sum
$90° + 90° + 90° + 90° = 360°$ ✓
Polygon Names and Properties
| Sides | Name | Sum of Interior Angles | Each Interior Angle (Regular) | Each Exterior Angle (Regular) |
|---|---|---|---|---|
| 3 | Triangle | 180° | 60° | 120° |
| 4 | Quadrilateral | 360° | 90° | 90° |
| 5 | Pentagon | 540° | 108° | 72° |
| 6 | Hexagon | 720° | 120° | 60° |
| 7 | Heptagon | 900° | ≈128.57° | ≈51.43° |
| 8 | Octagon | 1080° | 135° | 45° |
| 9 | Nonagon | 1260° | 140° | 40° |
| 10 | Decagon | 1440° | 144° | 36° |
| 12 | Dodecagon | 1800° | 150° | 30° |
Classification Summary
| Type | Definition | Examples |
|---|---|---|
| Regular Polygon | All sides equal AND all angles equal | Equilateral triangle, square, regular pentagon |
| Irregular Polygon | Sides or angles not all equal | Rectangle, scalene triangle, most shapes |
| Convex Polygon | All interior angles < 180° | All regular polygons, rectangles, triangles |
| Concave Polygon | At least one interior angle > 180° | Star shapes, arrows, shapes with "dents" |
Formula Quick Reference
| What to Find | Formula | Notes |
|---|---|---|
| Sum of Interior Angles | $(n-2) \times 180°$ | Works for ANY polygon |
| Each Interior Angle | $\frac{(n-2) \times 180°}{n}$ | ONLY for regular polygons |
| Sum of Exterior Angles | $360°$ | ALWAYS 360° for any polygon |
| Each Exterior Angle | $\frac{360°}{n}$ | ONLY for regular polygons |
| Number of Diagonals | $\frac{n(n-3)}{2}$ | For any n-sided polygon |
| Interior + Exterior | $180°$ | At each vertex (linear pair) |
Problem-Solving Strategies
| Given | Want to Find | Strategy |
|---|---|---|
| Number of sides (n) | Sum of interior angles | Use $(n-2) \times 180°$ |
| Number of sides (regular) | Each interior angle | Use $\frac{(n-2) \times 180°}{n}$ |
| Each interior angle (regular) | Number of sides | Set up equation and solve for n |
| Number of sides (regular) | Each exterior angle | Use $\frac{360°}{n}$ |
| Each exterior angle (regular) | Number of sides | $n = \frac{360°}{\text{exterior angle}}$ |
| Interior angle | Exterior angle | $180° -$ interior angle |
Success Tips for Polygons:
✓ Sum of interior angles: $(n-2) \times 180°$ works for ANY polygon
✓ Sum of exterior angles: ALWAYS 360° for any polygon
✓ Regular polygon: All sides AND all angles equal
✓ Convex polygon: All interior angles < 180°
✓ Each interior angle formula works ONLY for regular polygons
✓ Interior + Exterior = 180° at each vertex (linear pair)
✓ To find n given angle: Set up equation and solve
✓ Hexagon construction: radius = side length (unique property!)
✓ Square construction: Use perpendiculars at right angles
✓ Number of diagonals: $\frac{n(n-3)}{2}$
✓ Sum of interior angles: $(n-2) \times 180°$ works for ANY polygon
✓ Sum of exterior angles: ALWAYS 360° for any polygon
✓ Regular polygon: All sides AND all angles equal
✓ Convex polygon: All interior angles < 180°
✓ Each interior angle formula works ONLY for regular polygons
✓ Interior + Exterior = 180° at each vertex (linear pair)
✓ To find n given angle: Set up equation and solve
✓ Hexagon construction: radius = side length (unique property!)
✓ Square construction: Use perpendiculars at right angles
✓ Number of diagonals: $\frac{n(n-3)}{2}$