The Central Idea
The geometry of a polygon is strongly controlled by its number of sides. If a polygon has \(n\) sides, then it also has \(n\) vertices and \(n\) interior angles.
Most importantly, a convex \(n\)-gon can be divided into \(n-2\) triangles from one vertex. Because every triangle has an interior angle sum of \(180^\circ\), the interior angle sum of the polygon is
\[ \boxed{(n-2)180^\circ}. \]This single relationship drives much of polygon angle analysis.
What Is a Polygon?
A polygon is a closed plane figure formed by a finite sequence of straight line segments.
The segments are the sides, and consecutive sides meet at vertices.
A polygon must be closed. A chain of segments with an opening is not a polygon.
Polygon Names
Polygons are commonly classified by their number of sides.
| Sides | Common Name |
|---|---|
| 3 | Triangle |
| 4 | Quadrilateral |
| 5 | Pentagon |
| 6 | Hexagon |
| 7 | Heptagon |
| 8 | Octagon |
| 9 | Nonagon |
| 10 | Decagon |
| 12 | Dodecagon |
For an arbitrary number of sides, the neutral term \(n\)-gon is often the clearest notation.
Convex and Concave Polygons
A polygon is convex when every segment joining two points of the polygon lies inside or on the polygon.
A polygon is concave when it has an inward indentation. Equivalently, at least one interior angle is greater than \(180^\circ\).
The standard angle-sum formula works for ordinary simple polygons whether they are convex or concave, although the common triangle-partition picture is easiest to see in the convex case.
Simple and Self-Intersecting Polygonal Paths
In elementary Euclidean geometry, the word polygon usually means a simple polygon: nonadjacent sides do not cross.
A self-intersecting polygonal path, such as a star-shaped crossing figure, requires additional conventions about interior regions and angle sums. The formulas in this lesson assume simple polygons unless stated otherwise.
Regular and Irregular Polygons
A regular polygon has all sides congruent and all interior angles congruent.
A polygon that does not satisfy both conditions is irregular.
Equal side lengths alone do not guarantee regularity, and equal angles alone do not guarantee regularity.
Why Triangles Control Polygon Angle Sums
Choose one vertex of a convex polygon and draw diagonals from that vertex to every nonadjacent vertex.
A quadrilateral divides into \(2\) triangles, a pentagon into \(3\), a hexagon into \(4\), and in general an \(n\)-gon into
\[ n-2 \]triangles.
Therefore, its interior angle sum is
\[ \boxed{S=(n-2)180^\circ}. \]Interior Angle Sum Examples
For a pentagon, \(n=5\):
\[ S=(5-2)180^\circ=540^\circ. \]For an octagon, \(n=8\):
\[ S=(8-2)180^\circ=1080^\circ. \]For a dodecagon, \(n=12\):
\[ S=(12-2)180^\circ=1800^\circ. \]Finding a Missing Interior Angle
Suppose a quadrilateral has interior angles
\[ 92^\circ,\qquad 84^\circ,\qquad 101^\circ,\qquad x. \]A quadrilateral has interior angle sum
\[ (4-2)180^\circ=360^\circ. \]Therefore,
\[ 92+84+101+x=360, \] \[ x=83. \]Thus,
\[ \boxed{x=83^\circ}. \]Interior Angle of a Regular Polygon
In a regular \(n\)-gon, all \(n\) interior angles are equal. Divide the total interior angle sum by \(n\):
\[ \boxed{I=\frac{(n-2)180^\circ}{n}}. \]This can also be written
\[ I=180^\circ-\frac{360^\circ}{n}. \]Example: Interior Angle of a Regular Octagon
For \(n=8\),
\[ I=\frac{(8-2)180^\circ}{8}. \] \[ I=\frac{1080^\circ}{8}=135^\circ. \]Therefore, every interior angle of a regular octagon is
\[ \boxed{135^\circ}. \]Exterior Angles
An exterior angle of a convex polygon is formed by one side and the extension of an adjacent side.
At each vertex, the interior angle \(I\) and its adjacent exterior angle \(E\) form a linear pair:
\[ \boxed{I+E=180^\circ}. \]The Exterior Angle Sum
Take one exterior angle at each vertex of a convex polygon, always turning in the same direction.
After traveling once around the polygon, the total change in direction is one complete turn:
\[ \boxed{360^\circ}. \]Therefore, the sum of one exterior angle at every vertex of any convex polygon is \(360^\circ\), regardless of the number of sides.
Exterior Angle of a Regular Polygon
In a regular polygon, all exterior angles are equal. Since their sum is \(360^\circ\), each exterior angle is
\[ \boxed{E=\frac{360^\circ}{n}}. \]For a regular decagon,
\[ E=\frac{360^\circ}{10}=36^\circ. \]Interior and Exterior Angles of a Regular Polygon
Because a regular polygon has equal interior and equal exterior angles,
\[ I=180^\circ-E. \]Combining this with \(E=360^\circ/n\) gives
\[ \boxed{I=180^\circ-\frac{360^\circ}{n}}. \]The two regular-polygon formulas are therefore two views of the same local linear-pair relationship.
Finding the Number of Sides from an Exterior Angle
Suppose each exterior angle of a regular polygon is \(24^\circ\).
Use
\[ 24=\frac{360}{n}. \]Therefore,
\[ n=\frac{360}{24}=15. \]The polygon is a
\[ \boxed{15\text{-gon}}. \]Finding the Number of Sides from an Interior Angle
Suppose each interior angle of a regular polygon is \(150^\circ\).
Its corresponding exterior angle is
\[ 180^\circ-150^\circ=30^\circ. \]Then
\[ n=\frac{360^\circ}{30^\circ}=12. \]Therefore, the polygon is a regular dodecagon.
Diagonals
A diagonal is a segment joining two nonadjacent vertices of a polygon.
From one vertex of an \(n\)-gon, you cannot draw a diagonal to that same vertex or to either adjacent vertex. Therefore, the number of diagonals from one vertex is
\[ \boxed{n-3}. \]Total Number of Diagonals
If each of the \(n\) vertices connects by diagonals to \(n-3\) other vertices, the initial count is
\[ n(n-3). \]But each diagonal has been counted twice, once from each endpoint. Therefore, the total number of distinct diagonals is
\[ \boxed{D=\frac{n(n-3)}{2}}. \]Example: Diagonals of a Decagon
For a decagon, \(n=10\):
\[ D=\frac{10(10-3)}{2}. \] \[ D=\frac{70}{2}=35. \]Thus a decagon has
\[ \boxed{35\text{ diagonals}}. \]Angle Sums and the Number of Sides
The interior-angle formula can be solved backward.
If the interior angle sum is \(S\), then
\[ S=(n-2)180^\circ. \]Therefore,
\[ \boxed{n=\frac{S}{180^\circ}+2}. \]For example, if the interior angle sum is \(1440^\circ\), then
\[ n=\frac{1440}{180}+2=10. \]The polygon has \(10\) sides.
Algebraic Interior Angles
Polygon angle problems often combine geometry with equations.
Suppose a pentagon has angles
\[ x,\qquad x+10,\qquad x+20,\qquad x+30,\qquad x+40. \]A pentagon has angle sum \(540^\circ\), so
\[ x+(x+10)+(x+20)+(x+30)+(x+40)=540. \] \[ 5x+100=540. \] \[ 5x=440. \] \[ \boxed{x=88^\circ}. \]Exterior Angles as Turning Angles
Exterior angles are especially useful because they describe how much a traveler must turn at each vertex to continue along the polygon.
In a regular hexagon, each turn is
\[ \frac{360^\circ}{6}=60^\circ. \]Six equal turns of \(60^\circ\) produce one full \(360^\circ\) revolution.
Regular Polygons and Circles
A regular polygon can be inscribed in a circle so that every vertex lies on the circle.
Connecting the center to consecutive vertices divides the figure into congruent isosceles triangles. Each central angle is
\[ \boxed{\frac{360^\circ}{n}}. \]Notice that this has the same numerical measure as each exterior angle of the regular polygon.
Increasing the Number of Sides
For regular polygons, increasing \(n\) causes each exterior angle \(360^\circ/n\) to become smaller.
Consequently, each interior angle
\[ 180^\circ-\frac{360^\circ}{n} \]becomes closer to \(180^\circ\).
This explains why a regular polygon with many sides appears locally flatter along its boundary.
Common Error: Using \(n180^\circ\)
The interior angle sum of an \(n\)-gon is not
\[ n180^\circ. \]The polygon partitions into \(n-2\) triangles, not \(n\) triangles. Therefore, the correct expression is
\[ \boxed{(n-2)180^\circ}. \]Common Error: Confusing the Sum with One Angle
For a regular octagon, the total interior angle sum is
\[ 1080^\circ, \]but each individual interior angle is
\[ \frac{1080^\circ}{8}=135^\circ. \]Always distinguish the sum of all angles from the measure of one angle.
Common Error: Assuming Every Polygon Has Equal Angles
The formula
\[ \frac{(n-2)180^\circ}{n} \]gives each interior angle only when the polygon is regular or otherwise known to be equiangular.
An arbitrary pentagon still has total angle sum \(540^\circ\), but its five angles need not each be \(108^\circ\).
Common Error: Exterior Angle Sum Depends on \(n\)
The size of each exterior angle of a regular polygon depends on \(n\), but the sum of one exterior angle at each vertex does not.
\[ \boxed{E_1+E_2+\cdots+E_n=360^\circ}. \]Common Error: Counting Diagonals Twice
The count \(n(n-3)\) counts every diagonal once from each endpoint.
Divide by \(2\) to obtain distinct diagonals:
\[ \boxed{D=\frac{n(n-3)}{2}}. \]A Reliable Procedure for Interior-Angle Problems
- Identify the number of sides \(n\).
- Compute the total interior angle sum \((n-2)180^\circ\).
- If the polygon is regular, divide the sum by \(n\).
- If angles contain variables, set their sum equal to the total.
- Solve the resulting equation.
- Check that the angle measures fit the geometry of the figure.
A Reliable Procedure for Regular-Polygon Exterior Angles
- Use the fact that all exterior angles sum to \(360^\circ\).
- For a regular \(n\)-gon, compute \(E=360^\circ/n\).
- Use \(I+E=180^\circ\) when an interior angle is needed.
- If the exterior angle is known, solve \(n=360^\circ/E\).
- Verify that \(n\) is an integer at least \(3\).
Practice 1: Interior Angle Sum
Find the sum of the interior angles of a heptagon.
Practice 2: Interior Angle Sum
Find the sum of the interior angles of a \(14\)-gon.
Practice 3: One Regular Interior Angle
Find each interior angle of a regular nonagon.
Practice 4: One Regular Exterior Angle
Find each exterior angle of a regular dodecagon.
Practice 5: Number of Sides from an Exterior Angle
A regular polygon has exterior angles of \(20^\circ\). How many sides does it have?
Practice 6: Number of Sides from an Interior Angle
Each interior angle of a regular polygon measures \(165^\circ\). How many sides does the polygon have?
Practice 7: Missing Angle
A quadrilateral has interior angles \(75^\circ\), \(110^\circ\), \(98^\circ\), and \(x\). Find \(x\).
Practice 8: Algebraic Pentagon
A pentagon has interior angles
\[ x,\qquad x+5,\qquad x+10,\qquad x+15,\qquad x+20. \]Find \(x\).
Practice 9: Diagonals from One Vertex
How many diagonals can be drawn from one vertex of a \(15\)-gon?
Practice 10: Total Diagonals
How many diagonals does a nonagon have?
Practice 11: Recover the Number of Sides
A polygon has interior angle sum \(1980^\circ\). How many sides does it have?
Practice 12: Identify the Error
A student says that each interior angle of any hexagon must be \(120^\circ\) because
\[ \frac{(6-2)180^\circ}{6}=120^\circ. \]Explain the error.
Practice Answers
Practice 1
\[ (7-2)180^\circ=\boxed{900^\circ}. \]Practice 2
\[ (14-2)180^\circ=12(180^\circ)=\boxed{2160^\circ}. \]Practice 3
\[ I=\frac{(9-2)180^\circ}{9}=\frac{1260^\circ}{9}=\boxed{140^\circ}. \]Practice 4
\[ E=\frac{360^\circ}{12}=\boxed{30^\circ}. \]Practice 5
\[ n=\frac{360^\circ}{20^\circ}=\boxed{18}. \]Practice 6
The exterior angle is
\[ 180^\circ-165^\circ=15^\circ. \]Therefore,
\[ n=\frac{360^\circ}{15^\circ}=\boxed{24}. \]Practice 7
\[ 75+110+98+x=360. \] \[ x=360-283=\boxed{77^\circ}. \]Practice 8
A pentagon has total interior angle sum \(540^\circ\):
\[ x+(x+5)+(x+10)+(x+15)+(x+20)=540. \] \[ 5x+50=540. \] \[ 5x=490. \] \[ \boxed{x=98^\circ}. \]Practice 9
\[ n-3=15-3=\boxed{12}. \]Practice 10
\[ D=\frac{9(9-3)}{2}=\frac{54}{2}=\boxed{27}. \]Practice 11
\[ 1980=(n-2)180. \] \[ 11=n-2. \] \[ \boxed{n=13}. \]Practice 12
The calculation \(120^\circ\) gives each interior angle only for a regular or otherwise equiangular hexagon. Every simple hexagon has total interior angle sum \(720^\circ\), but an irregular hexagon can distribute that total among unequal angles.
What to Remember
For a simple \(n\)-gon:
- there are \(n\) sides, \(n\) vertices, and \(n\) interior angles;
- the interior angle sum is \((n-2)180^\circ\);
- one exterior angle at each vertex sums to \(360^\circ\);
- a regular polygon has equal sides and equal angles;
- each regular exterior angle is \(360^\circ/n\);
- each regular interior angle is \((n-2)180^\circ/n\);
- the total number of diagonals is \(n(n-3)/2\).
The most useful organizing habit is to determine first whether a problem is asking for a total angle sum, one regular angle, or an unknown angle inside a fixed total.
Where We Go Next
Polygons provide a broad family of Euclidean figures, but four-sided polygons have enough special structure to deserve closer study.
The next lesson, Quadrilaterals and Their Properties, organizes parallelograms, rectangles, rhombi, squares, trapezoids, kites, and their characteristic side, angle, and diagonal relationships.