3D Coordinates

Three-dimensional coordinates refer to points in 3D space plotted on x, y, and z axes.
The origin in 3D space is where the x, y, and z axes intersect, expressed as (0,0,0).
A point in space is represented as (x, y, z), with each value indicating the distance of the point from the origin along the corresponding axis.
The distance between two points (x₁, y₁, z₁) and (x₂, y₂, z₂) can be calculated using the formula √[(x₂ - x₁)² + (y₂ - y₁)² + (z₂ - z₁)²].
The midpoint of a line segment with endpoints (x₁, y₁, z₁) and (x₂, y₂, z₂) is given by [(x₁ + x₂)/2, (y₁ + y₂)/2, (z₁+ z₂)/2].

Volume

The volume of a 3D object is the amount of space it occupies.
It is important to remember the formulas for the volume of basic shapes like a cube, cuboid, cylinder, cone, sphere.
For these basic shapes, the volume can be calculated as follows:
- Cube: side³
- Cuboid: Length * width * height
- Cylinder: π * radius² * height
- Cone: 1/3 * π * radius² * height
- Sphere: 4/3 * π * radius³

Pythagoras’ Theorem states that in a right-angled triangle the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides.
Formula is: a² + b² = c² where a and b are the lengths of the two sides and c is the length of the hypotenuse.

Two figures are similar if the ratios of the lengths of their corresponding sides are equal.
The corresponding angles in similar shapes are equal.
Scale factor is the ratio of the lengths of two corresponding sides of similar figures.
If polygons are similar then the ratio of their areas is the square of the ratio of their sides, formula k², while for volume, it is the cube of their ratio, k³.

An arc is a segment of a circle, while a sector is a region enclosed by two radii and an arc.
The length of an arc (L) in a circle of radius r subtending an angle of θ radians at the center is given by the formula L = rθ.
The area A of a sector in a circle of radius r subtending an angle of θ radians at the center is given by the formula A = 1/2 * r² * θ.

Remember the basic properties of a circle:
- The distance around a circle is its circumference.
- A line from the centre of a circle to its edge is a radius.
- A line that goes from one point to another on the circle’s circumference without leaving the circle is a chord.
A tangent of a circle is a line that touches the circle at exactly one point.
Any line that touches a circle and goes through its center is a diameter. It is twice the radius.
A diameter that divides the circle into two equal parts each forming a semi-circle.

The most common geometry problems involve finding missing angles or lengths.
Be aware of alternate, corresponding, and co-interior angles when working with parallel lines.
Triangles have interior angles summing up to 180 degrees and quadrilaterals to 360 degrees.
Circles have 360 degrees.
Remember the properties of special quadrilaterals (rectangles, squares, rhombi, parallelograms, trapezia).
The exterior angle of a triangle is equal to the sum of the opposite interior angles.
The angles in a triangle or on a straight line form a linear pair and sum up to 180 degrees.
Two angles are complementary if their sum is 90 degrees and supplementary if their sum is 180 degrees.

Two shapes are similar if their corresponding sides are in proportion and corresponding angles are equal.
In similar triangles, corresponding sides and heights are in the same ratio, or scale factor.
The scale factor is usually expressed in the form 1 : n or as a fraction.