Geometry

Geometry is the study of shapes and how the relate one to another.

Points

Points are a point in space. They don't move over time.

Lines

Lines are perfectly straight, do not bend, and extend infinitely.

Lines run through an infinite number of points. They can be defined by two points that are not the same.

Any given point may either lie on the line or not.

Planes

Planes are flat surfaces, like pieces of paper, that never curve, and extend infinitely in all directions.

Planes, like lines, contain an infinite number of points, but also an infinite number of lines. They can be described by three different points which lie in the plane.

Parallel Lines

Two lines run parallel to each other if they lie in the same plane and never intersect.

Intersections

When two lines intersect, they cross each other at exactly one point. That point is the intersection of the two lines. There are four angles formed by the intersection, two pairs of equal angles.

• The sum of the angles along a line is 180°.
• Hence, The angles opposite each other in an intersection are always equal.
• The sum of the angles around an intersection is 360°.

Line Segments

Line segments are lines that run only between two points, and are naturally defined by those points.

Line segments have lengths, whereas lines have no length.

Triangles

Triangles are defined by three points, and is made up of the three line segments that connect those points.

Triangles are flat and lie in a plane.

The sum of the interior angles of a triangle is 180°.

A triangle is:

• The interior angle is on the inside.
• The exterior angle is one the outside.
• Scalene no sides have the same length.
  • No angles are equal to each other.
• Isosceles if two sides are equal length.
  • The base is the side that is not equal to the others.
  • The base angles are equal.
• Equilateral if all sides are equal length.
  • The interior angles are 60°.
• Acute if all three angles are less than 90°.
• Obtuse if one angle is more than 90°
• Oblique if no angle is 90&deg (IE, acute or obtuse.)
• Right if one angle is 90°.
  • The side opposite the right angle is the hypotenuse.
  • The sides next to the right angle are the legs.
  • The other two angles add up to 90°.
  • Right triangles form the basis of Trigonometry.

Quadrilaterals

• Quadrilaterals have 4 sides and 4 angles.
• Kites
• Trapezoids.
• Rectangles.
• Rhombus.
• Square.

Polygons

Polygons have more than 2 sides and include all triangles.

• The sum of the interior angles is <math> (\text{number of sides} - 2) * 180^\circ \,</math>
• If all the interior angles are equal, then all the sides are equal and the polygon is Regular. This means its points lie along a circle as well.
• If an angle bends inward (interior angle < 180°), it is convex.
• If an angle bends outward (interior angle > 180°), it is concave.
• If any side crosses any other side, then it is complex.
• If no side crosses any other side, then it is simple.
• If any angle is concave, the polygon is concave.
• If all angles are convex, and it is not complex, then it is convex.
• If all the points lie along a circle, it is cyclic.

Circles

• Tangent: Any line that touches the edge of the circle.
• chord: A line segment that connects two points along the circle.
• diameter: A chord that crosses the center, the longest chords in the circle.
• radius: 1/2 diameter, a line segment from the center to any point on the circle.
• circumference: The length of the edge of the circle, <math>\pi d = 2 \pi r \,</math>
• area: <math>\pi r^2 \,</math>

3D Shapes

From the above, you can create 3D or N-dimensional shapes.

• Surface area: the area of the sides of the 3D shape.
• Volume: The area inside the 3D shape.

• Spheres
• Ellipsoids
• Cones
• Conical sections: hyperbolas, circles, parabolas, etc...

Measurements

• Perimeter: total length of the edges.
• Area: Often difficult to calculate.
  • Rectangles: base × height
  • Squares: <math>\text{side}^2 \,</math>
  • Triangles: <math>\frac{1}{2}\text{base}*text{height} \,</math>, where base is a side, and height is the tallest the triangle is perpendicular to that side.
  • Circles: