Our physical existence is made of 3 dimensions.

- Each dimension can be represented as a line.
- Two of those dimensions together becomes a plane.

## What You Probably Know

While geometrical figures may *appear* to be more useful, mainstream Western education tends to be more scant on teaching geometrical concepts compared to arithmetic and algebra.

As early as kindergarten education, we learn about geometrical figures in reality:

- Identifying precise depictions of 2-dimensional shapes like the triangle, circle, oval, square, and rectangle.
- Counting shapes (i.e., arithmetical geometry).
- Expanding 2-dimensional shapes into 3 dimensions like the cone, pyramid, and sphere.
- Quantifying those shapes using measurements.
- Combining 2-dimensional shapes into larger shapes (e.g., 2 squares together become a rectangle).

For a while, it isn’t explored much in lieu of advancing arithmetic, then gets revisited with arithmetic concepts included into it:

- Using a number line to demarcate distance, often while providing a visual for basic arithmetic.
- Indicating information on a graph.
- Using multiplication to indicate area.
- Adding fractions to a number line.
- Adding or multiplying outside line segments together to determine perimeter.
- Factoring surface areas by evenly grouping them into smaller shapes.

There’s a divergence around 4th grade that focuses heavily on geometric primitives:

- Depicting primitives like points, lines, segments, and rays.
- Indicating relationships between geometric primitives by measuring angles, the concepts of intersection, parallel lines, and how segments can create shapes.
- Adding to angle calculations by inferring specific angles when given other angles and relationships (e.g., if an angle is 62° and part of a right angle, what is the other angle?).

The emphasis on primitives and shapes synthesize into geometric analysis:

- Reviewing shapes that accounts for the number and length of sides, angle size, the presence of parallel or perpendicular lines, and symmetry.
- Using reasoning about shapes’ properties to solve specific problems (e.g., given the length of a rhombus side, what’s the length of another of its sides?).

From there, it’s revisiting and expanding on ideas again:

- Finding the volume of simple 3-dimensional objects like cubes.
- Representing area and side lengths with fractions.
- Graphing shapes on a Cartesian coordinate plane.
- Indicating the hierarchy of shapes (e.g., all parallelograms are trapezoids, and all trapezoids are quadrilaterals).
- Solving word problem rules that will form the basis for graphing algebraic notation on a coordinate plane.
- Combining fractions and decimals together to graph accurately on a number line.
- Using two parallel number lines to indicate comparable units of measurement.

Around middle school, geometry gets some new additions:

- Intuitively decomposing and rearranging shapes to make volume easier to measure.
- Types of triangles (acute, obtuse, right).
- Finding the area of specific shapes like parallelograms and triangles.
- Finding the surface area of 3-dimensional objects by decomposing them into 2-dimensional shapes.
- Learning the definition of a polyhedron using nets.
- Solving Fermi problems using inductive and deductive reasoning.
- Graphing correlants on a Cartesian coordinate plane.
- Using exponents to indicate area and volume of squares and cubes.

However, geometry starts becoming more complex once it incorporates algebra:

- Graphing proportional relationships (e.g., x = 2y).
- Indicating how a circle has a radius, diameter, and circumference.

## Geometry Tricks

Since area and volume are commutative, you can factor the shapes into smaller shapes and move them around to more easily calculate them:

- A parallelogram can be vertically split, then moved to create a square.
- Opposite vertices of a parallelogram can
*always*slice the shape into 2 triangles.

The area of a triangle is *always* 1/2 the area of a rectangle, so you can multiply the base by the height of that triangle, then divide it by 2 (i.e., a = bh/2)

A triangle’s 3 sides always add up to 180°, and each side adds another 180°, which creates some tricks:

- If you know 2 sides of a triangle, you can deduce the 3rd.