Understanding Polygons, Quadrilaterals, and Polyhedrons

Posted on Jan 9, 2025 in Visual arts

Understanding Polygons

A polygon is a part of a plane delimited by a closed polygonal path (it is called a polygonal region). It can be:

Convex: Prolongation of all segments of the polygonal path divides the plane into two semi-planes, and the polygonal path is all in the same semi-plane.
Concave: Prolongation of at least one segment of the polygonal path divides the plane into two parts, each one in different semi-planes.
Regular: When all the sides and angles are equal.

Perimeter: The sum of the lengths of the sides of the polygon.
Diagonal: Line segments from one corner to another (not sides).
Interior Angles: Angles in the polygon region with a vertex on a vertex of the polygon.
Exterior Angles: Angle between any side of a shape and the prolongation of the line from the next side.

Incenter: The three angle bisectors of a triangle intersect at a point called the incenter (it’s the center of the inscribed circle). Any point on the bisector of an angle is equidistant to the sides of the angle.
Circumcenter: The three perpendicular bisectors of the three sides of a triangle intersect at a point called the circumcenter (center of the circumscribed circumference). Any point on the perpendicular bisector of a segment is equidistant to the extremes of the segment.
Orthocenter: It is a segment perpendicular to the base from any point on the opposite vertex. The three altitudes of a segment intersect at a point called the orthocenter.
Centroid: Median of a triangle (segment from a vertex to the middle point of the opposite side). The three medians intersect at a point called the centroid. The centroid divides the median into two parts: 1/3 and 2/3 of the total length.

A quadrilateral is a polygon with four sides and four vertices. They can be concave or convex, open or closed.

Parallelogram: Two pairs of parallel sides.
- Square: Equal sides and angles.
- Rectangle: Equal angles, parallel sides equal two by two.
- Rhombus: Equal sides, angles equal two by two.
- Rhomboid: Angles and sides equal two by two.
Trapezoid: One pair of parallel sides.
Kite: No parallel sides (symmetric, two angles are equal, sides equal two by two).

Circumference: Set of all points that are at the same distance from a point called the center.
Radius: A segment that joins the center and a point belonging to the circumference.
Diameter: Chord that passes through the center of the circumference; the longest chord possible; it is two times the radius.
Semicircle: Half of the circumference; arc that is between two opposite points.
Arc Sector: A part of a circle delimited by two radii and an arc.
Arc Segment: A part of the circle delimited by an arc and its associated chord.
Central Angles: Vertex is the center of the circumference.
Inscribed Angles in a Circumference: Angles with the vertex on a point of the circumference. Theorem: All inscribed angles measure half of the arc delimited by their sides. The measure of an inscribed angle is half of the measure of the associated central angle.

Isometry: A transformation of a figure that keeps the size and the shape of the figure.
Vector: Defined by direction, length, initial point, and sense. Must have an initial point.
Translation: Consists of moving all the points of the figure by a vector, which is called the translation vector. It keeps the same figure, dimensions of the figure, and the orientation of the figure. To compose consecutively two translations, sum the vectors of each translation.
Rotation: To rotate a figure about a point P, called the center of rotation, a concrete angle x consists of rotating all the points of the figure the angle x about P.

A polyhedron is a solid figure bounded by polygons.

Faces: Polygons that delimit the polyhedron.
Edges: Sides of the polygons of the faces; intersection between two faces.
Vertices: Vertices of the polygons; intersection between at least three polygons.
Diagonals: Segment joining two vertices that are not in the same face.

Euler’s Formula: F + V = E + 2 (where F is the number of faces, V is the number of vertices, and E is the number of edges).