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The goal of this Mind-Map is to summarize and outline the primary concepts surrounding geometric figures as they exist on the plane and in space for students grades K-5. The audience is student-teachers, to help with outlining and planning for integrating these concepts into their classrooms.
Three dimensional figures in space consist of surfaces on the exterior that enclose either a hollow or solid interior.
A simple closed surface is any surface without holes that contains a hollow interior.
A solid is a connection of all of the points on a figures surface and interior.
A cylinder joins two congruent simple closed curves that are parallel from different planes by connecting all points on each curve.
A cone has a simple closed curve for a base and an apex in space not on the plane of the curve. The cone is formed by connecting all points on the curve to the apex.
A simple closed surface that joins the vertices of two parallel congruent polygons from different planes.
A simple closed surface joining the vertices of a polygonal base with a single point in space.
A polyhedron (singular figure) is a simple closed surface formed from planar polygonal regions. Each distinct region is called a face.
Polyhedra (plural) are named according to their number of faces.
Regular Polyhedra
Three distinct features comprise a regular polyhedron:
1. It's surface is convex
2. It's faces are congruent regular polygons
3. The same number of faces meet at each vertex
Euler's Formula
In Euler's Formula there are three important denotations:
F - Number of faces on a polyhedron
V - Number of vertices
E - Number of edges
Euler's Formula shows that there are similarities between all regular polyhedra:
V + F = E + 2
Dodecahedron
Comprised of 12 regular pentagonal faces.
Icosahedron
Comprised of 20 equilateral triangular faces.
Octahedron
Comprised of eight equilateral triangular faces.
Tetrahedron
Comprised of four equilateral triangular faces.
Cube
Comprised of six square faces.
A sphere is a set of points in space at a constant distance from the center point of the figure.
A curve that extends off of the planar surface and into the space of a third dimension.
Space is a three dimensional area where there are an infinite amount of planes and lines.
Each plane creates three disjoint sets:
1. The plane itself
2. One side of the plane (a half space)
3. The other half space
Two planes can only be either parallel or intersecting.
Skew lines are non intersecting lines that are on different planes. They are not parallel to other lines.
The plane is a two dimensional flat surface that extends indefinitely in all directions. Think of the plane as an infinitely large, perfectly thin, sheet of paper.
Where two rays share the same endpoint (the angle's vertex) and extend outwardly from each other, an angle is formed.
Triangles are composed of three interior angles that sum to 180 degrees
When two parallel lines are intersected by a transversal, a series of angles is created. The angles that are the interior angles of the transversal line are equal by the alternate interior angles theory.
Where two straight lines cross, an intersection is created, leaving four angles. The angles that are opposite of each other are congruent angles (equal in measure). This is true by the vertical angles theory.
There are six primary types of angles. Angles can be formed from zero to 360 degrees forming a complete circle around a point.
Reflex Angle
A reflex angle is greater than 180 degrees but less than 360 degrees. A reflex angle depends on the direction from which you measure the angle.
Straight Angle
A straight angle is formed when two rays extend outwardly in exactly opposite directions from their vertex, it is always exactly 180 degrees
Obtuse Angle
An obtuse angle is greater than 90 degrees but less than 180 degrees
Right Angle
A right angle is exactly 90 degrees
Acute Angle
An acute angle is greater than zero but less than 90 degrees
Zero Angle
Two rays extending from a vertex outwardly through a similar point form a zero degree angle
A line passes through any two points on the plane in a perfectly straight fashion in exactly opposite directions for an infinite distance.
A ray is a line that extends from endpoint A through point B and indefinitely onward.
A line segment is the section of a given line between two designated points A and B. A and B are considered the end points of the line segment. The point located exactly in the middle of the line segment is called the midpoint. The absolute value (the length) of the line segment can be found by subtracting B from A.
Two lines that intersect at a 90 degree angle are said to be perpendicular
Non parallel lines share a common point which is referred to as the point of intersection
Two lines are said to be parallel if they share no common points between one another
A point is a single location in the plane that is defined by an exact set of coordinates.
Any three points or more that fall on the same line are said to be collinear
Regular Polygons are both equilateral and equiangular.
The interior angles of all regular polygons can be found with the following formula:
(n-2) * 180 / n
The exterior angles of regular polygons can be found with the following formula:
360 / n
A polygon is a simple closed polygonal curve
Polygons are given their names by the number of sides that they have
There are seven different classifications of quadrilaterals.
Square
A rectangle with four sides of equal length.
Rectangle
A parallelogram with four right angles.
Rhombus
A parallelogram with four sides of the exact same length.
Parallelogram
A parallelogram has two pairs of opposite sides that are parallel.
Isosceles Trapezoid
An isosceles trapezoid has two pairs of congruent angles along both parallels.
Trapezoid
A trapezoid has exactly one pair of parallel sides.
Kite
A kite has two distinct pairs of congruent adjacent sides.
There are six different classifications of triangles
Equilateral
An equilateral is a triangle with three sides of exactly the same length.
Isosceles
An isosceles triangle is any triangle that has two sides of the same length.
Scalene
A scalene triangle is any triangle with three sides of different length.
Obtuse
An obtuse triangle is any triangle with exactly one obtuse angle.
Right
A right triangle is any triangle with one 90 degree angle.
Acute
An acute triangle is any triangle with three acute interior angles.
A curve can be described as a set of points on a plane, that can be linked with one another without interrupting the link until all points are linked.
The total turn theorem states that every complete rotation that a curve makes around a central point constitutes one turn. The total turn of a curve is the multiple of 360 degrees that shows how many turns around that central point the curve makes.
A curve that has a finite number of sides and vertices.
A polygonal curve can be simple, not simple, closed, not closed, convex, concave, or a combination of these.
A curve is concave if any line segments that begin and end in the interior of the curve, pass through the exterior of the curve.
A curve is convex if all possible line segments are contained within the interior of the curve.
A simple closed curve has three separate regions:
1. The curve itself
2. It's interior region
3. It's exterior region
A closed curve has the same beginning point and end point.
A curve is not closed if the beginning and end points are not similar.
A curve that does not intersect with itself at any point.
It can be said to be not simple if it does intersect with itself.
