Essential Math Foundations: Shapes, Space, Numbers
Quadrilaterals: Shapes & Properties
What is a Quadrilateral?
A quadrilateral is a four-sided polygon.
The interior angles of a quadrilateral add up to 360°.
Types of Quadrilaterals
| Type | Properties |
|---|---|
| Square | All sides equal, all angles 90°, opposite sides parallel |
| Rectangle | Opposite sides equal and parallel, all angles 90° |
| Rhombus | All sides equal, opposite angles equal, diagonals bisect at right angles |
| Parallelogram | Opposite sides equal and parallel, opposite angles equal |
| Trapezium | One pair of opposite sides parallel (in Australia, called a trapezium) |
| Kite | Two pairs of adjacent sides equal, one pair of opposite angles equal |
Quadrilateral Angle Rules
Sum of interior angles = 360°
For parallel lines:
Alternate angles are equal
Corresponding angles are equal
Co-interior angles add up to 180°
Key Geometric Terms
Diagonal: A line joining two opposite corners.
Parallel: Lines that never meet and stay the same distance apart.
Congruent: Same size and shape.
Understanding 3D Shapes & Objects
What is a 3D Object?
A 3D object has length, width, and height (3 dimensions).
It has faces, edges, and vertices (corners).
Common 3D Shapes and Their Properties
| Shape | Faces | Edges | Vertices | Example |
|---|---|---|---|---|
| Cube | 6 | 12 | 8 | Dice |
| Rectangular Prism | 6 | 12 | 8 | Cereal box |
| Triangular Prism | 5 | 9 | 6 | Toblerone box |
| Pyramid (square base) | 5 | 8 | 5 | Egyptian pyramid |
| Cylinder | 3 | 2 | 0 | Can of soft drink |
| Cone | 2 | 1 | 1 | Party hat |
| Sphere | 1 (curved) | 0 | 0 | Soccer ball |
Key 3D Shape Formulas
Volume:
Cube: V=s3
Rectangular Prism: V=l × w × h
Triangular Prism: V=1⁄2 × b × h × l
Cylinder: V=πr2h
Surface Area:
Depends on the shape; usually add areas of all faces.
The 3D Coordinate System
What is the 3D Coordinate System?
The 3D coordinate system is used to locate points in three-dimensional space.
A point in 3D space is written as:
(x, y, z)
where:x = horizontal position (left-right)
y = vertical position (forward-back)
z = height (up-down)
Axes in 3D Space
There are three axes:
x-axis: Left to right
y-axis: Forward and backward (depth)
z-axis: Up and down
These axes meet at the origin, which is the point (0, 0, 0).
Plotting Points in 3D Space
To plot a point like (2, 3, 4):
Move 2 units along the x-axis.
Then 3 units along the y-axis.
Then go 4 units up on the z-axis.
Always follow the order: x, y, z
Visualizing 3D Coordinates
Imagine a corner of a room:
The floor is the x-y plane.
The height from floor to ceiling is the z-axis.
Any object in the room has a position described by its x, y, z coordinates.
Applications of 3D Coordinates
Used in:
3D graphics and games
Engineering and design (CAD)
Navigation and GPS
Mathematics and science for modelling
Understanding Fractions
Key Fraction Vocabulary
Numerator: Top number (how many parts)
Denominator: Bottom number (total parts)
Simplify: Reduce to smallest form
LCD (Lowest Common Denominator): Smallest shared denominator
1. Simplifying Fractions
Divide top and bottom by their HCF
Example:
18/24 = 3/4
2. Equivalent Fractions
Multiply or divide top & bottom by the same number
Example:
1/2 = 2/4 = 4/8
3. Adding & Subtracting Fractions
Same Denominator:
Add/subtract the numerators only
5/8 – 3/8 = 2/8 = 1/4
Different Denominators:
Find LCD
Convert both fractions
Add/subtract numerators
Example:
2/3 + 1/6
LCD = 6
4/6 + 1/6 = 5/6
4. Multiplying Fractions
Multiply tops and bottoms
3/4 × 2/5 = 6/20 = 3/10
5. Dividing Fractions
Keep, flip, multiply
Example:
3/5 ÷ 2/7 = 3/5 × 7/2 = 21/10 = 2 1/10
6. Mixed Numbers & Improper Fractions
Converting to Improper Fractions:
Multiply whole number by denominator + numerator
2 1/4 = 9/4
Converting to Mixed Numbers:
Divide numerator by denominator
11/3 = 3 2/3
Quick Fraction Tips
Always simplify your answers
Common Denominator is key for + and –
Flip the second fraction when dividing