Lernzettel: Regular Polygons

📋 Course Outline

1. Polygon Types and Terminology
2. Convexity and Concavity
3. Interior Angle Sums
4. Exterior Angle Sums
5. Angles of Regular Polygons
6. Symmetry of Regular Polygons
7. Circles and Central Angles
8. Perimeter and Area Formulas

📖 1. Polygon Types and Terminology

🔑 Key Concepts & Definitions

• Polygon : A simple closed plane figure formed by three or more line segments joined end to end, with no two successive segments collinear.

★ Must-know

• The sides of a polygon are its forming line segments, vertices are the common endpoints of two sides, and interior angles are the angles formed inside the polygon.

Further detail

• A polygon with n sides is commonly named an n-gon; the course lists triangle, quadrilateral, pentagon, hexagon, heptagon, octagon, nonagon, and decagon for 3 through 10 sides.

📖 2. Convexity and Concavity

🔑 Key Concepts & Definitions

• Convex Polygon : Has every interior angle less than 180°, and its vertices point outward.
• Concave Polygon : Has at least one interior angle greater than 180°, with vertices that can point inward and outward.

📝 Essential Points

📌 A polygon is convex if none of the extensions of its sides intersects the polygon; otherwise, it is concave.

💡 Memory Hook

Outward vertices and angles below 180° versus inward vertices and an angle above 180°

📖 3. Interior Angle Sums

★ Must-know

• Joining one vertex of an n-sided convex polygon to its nonadjacent vertices divides it into n−2 non-overlapping triangles.

📐 Formula — The sum of the interior angles of an n-sided polygon is $S=(n-2)\times180^\circ$.

Further detail

• A hexagon has an interior-angle sum of $S=(6-2)\times180^\circ=720^\circ$.

• If a polygon has an interior-angle sum of 1080°, then $(n-2)\times180^\circ=1080^\circ$ gives n=8, so the polygon is an octagon.

📖 4. Exterior Angle Sums

🔑 Key Concepts & Definitions

• Exterior Angle : Of a convex polygon is an angle outside the polygon formed by one side and the extension of an adjacent side.

★ Must-know

• At each vertex of a polygon, the interior angle and its adjacent exterior angle add to $180^\circ$.

• For every convex n-sided polygon, the sum of one exterior angle at each vertex is $360^\circ$.

Further detail

• A triangle with interior angles 60°, 40°, and 80° has corresponding exterior angles 120°, 140°, and 100°.

📖 5. Angles of Regular Polygons

🔑 Key Concepts & Definitions

• Regular Polygon : Both equiangular, with equal angles, and equilateral, with equal sides.

★ Must-know

📐 Formula — Each interior angle of a regular n-sided polygon measures $\frac{(n-2)\times180^\circ}{n}$.

📐 Formula — Each exterior angle of a regular n-sided polygon measures $\frac{360^\circ}{n}$.

Further detail

• Each interior angle of a regular pentagon is 108°, and each exterior angle is 72°.

📖 6. Symmetry of Regular Polygons

🔑 Key Concepts & Definitions

• Line of Symmetry : A line through which the two halves of a figure match exactly as mirror images.

★ Must-know

• An n-sided regular polygon has exactly n lines of symmetry.

Further detail

• For an odd-sided regular polygon, each symmetry line connects a vertex to the midpoint of the opposite side; for an even-sided regular polygon, symmetry lines connect opposite vertices or opposite side midpoints.

📖 7. Circles and Central Angles

🔑 Key Concepts & Definitions

• Apothem : The perpendicular segment from the center of a regular polygon to the midpoint of one of its sides, and it is the radius of the incircle.
• Circumcircle : The circle passing through all vertices of a regular polygon and centered at the polygon's center.

📝 Essential Points

• A polygon is inscribed in a circle when all its vertices lie on the circle, while a polygon is circumscribed about a circle when each side touches the circle.

📐 Formula — Each central angle of a regular n-sided polygon measures $\frac{360^\circ}{n}$.

📖 8. Perimeter and Area Formulas

★ Must-know

📐 Formula — For a regular n-gon inscribed in a circle of radius r, the side length is $s=2r\sin\left(\frac{180^\circ}{n}\right)$.

📐 Formula — For a regular n-gon with circumradius r, the apothem is $a=r\cos\left(\frac{180^\circ}{n}\right)$.

📐 Formula — The perimeter of a regular n-gon inscribed in a circle of radius r is $P=2nr\sin\left(\frac{180^\circ}{n}\right)$.

📐 Formula — The area of a regular polygon is $A=\frac{1}{2}aP$, where a is the apothem and P is the perimeter.

Further detail

• A regular hexagon inscribed in a circle of radius r has side length r, perimeter $P=6r$, apothem $a=\frac{\sqrt3}{2}r$, and area $A=\frac{3\sqrt3}{2}r^2$.

💡 Memory Hook

Radius → side and apothem → perimeter → area

📊 Synthesis Tables

Convex and Concave Polygons

Feature	Convex polygon	Concave polygon
Interior angles	All less than 180°	At least one greater than 180°
Vertices	Point outward	May point inward and outward
Side extensions	Do not intersect the polygon	At least one extension intersects the polygon

Regular Polygon Angle Measures

Angle	Measure	Key property
Interior angle	$(n-2)\times180^\circ/n$	All interior angles are equal
Exterior angle	$360^\circ/n$	One at each vertex
Central angle	$360^\circ/n$	Formed at the center by consecutive radii