[Leejohnston]

Geometry Essentials — Angles, Shapes, Area & Space

Geometry is the study of shapes, angles, distances, and the structure of space. 
From triangles and circles to vectors and 3D solids — geometry appears everywhere in science, technology, engineering, architecture, and physics.

This thread covers the core concepts and formulas every learner should know.

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1. Types of Angles

Right angle: 90° 
Acute angle: less than 90° 
Obtuse angle: between 90° and 180° 
Reflex angle: greater than 180°

Key facts: 
• angles on a straight line = 180° 
• angles around a point = 360° 
• vertically opposite angles are equal 
• alternate angles are equal 
• corresponding angles are equal 

These appear constantly in GCSE questions.

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2. Triangles

Angle sum in any triangle: 
180°

Types of triangles: 
• equilateral → all sides equal, all angles 60° 
• isosceles → two equal sides, base angles equal 
• right-angled → one angle = 90°

Pythagoras’ Theorem: 
Right-angled triangle only:

Code:

a² + b² = c²

Where c is the hypotenuse.

Example:

Code:

3² + 4² = 9 + 16 = 25 → c = 5

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3. Area Formulas (2D Shapes)

Rectangle: A = lw 
Triangle: A = ½bh 
Parallelogram: A = bh 
Trapezium (Trapezoid): A = ½(a + b)h 
Circle: A = πr² 
Circumference: C = 2πr

Example: 
Circle with radius 7: 
A = π × 49 ≈ 153.9

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4. Perimeter Basics

Perimeter = distance around a shape.

Examples:

Square of side 6 → P = 4 × 6 = 24 
Rectangle 3 × 8 → P = 2(3 + 8) = 22 

Circle: 
Circumference = 2πr

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5. 3D Shapes — Volume & Surface Area

Cube: V = a³ 
Cuboid: V = lwh 
Cylinder: V = πr²h 
Cone: V = ⅓πr²h 
Sphere: V = ⁴⁄₃πr³

Examples:

Cylinder radius 3, height 5: 
V = π × 9 × 5 = 45π

Sphere radius 4: 
V = ⁴⁄₃π × 64 = 256π/3

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6. Coordinates & Graphs

A point is written as (x, y).

Distance between two points:

Code:

distance = √[(x₂ - x₁)² + (y₂ - y₁)²]

This is Pythagoras in coordinate form.

Midpoint of a line:

Code:

((x₁ + x₂)/2 , (y₁ + y₂)/2)

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7. Transformations

Types:

Translation → sliding 
Reflection → flipping 
Rotation → turning 
Enlargement → scaling

Example translation: 
(3, 4) moved by (2, –1) → (5, 3)

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8. Vectors (Simple Introduction)

A vector has:
• magnitude (length) 
• direction 

Written as:

Code:

⟨a, b⟩

Example: 
Move 4 units right and 2 up:

Code:

⟨4, 2⟩

Vector addition:

Code:

⟨a, b⟩ + ⟨c, d⟩ = ⟨a + c, b + d⟩

Scalar multiplication:

Code:

k⟨a, b⟩ = ⟨ka, kb⟩

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9. Key Geometry Mistakes to Avoid

❌ Thinking the longest side is always the hypotenuse 
✔ Only true in right-angled triangles

❌ Forgetting angle facts (corresponding, alternate) 
✔ These appear in almost every exam

❌ Using diameter instead of radius 
✔ Area needs r, not d

❌ Mixing perimeter and area 
✔ perimeter = length around 
✔ area = space inside

❌ Confusing volume and surface area 
✔ volume = space inside 
✔ surface area = outer covering

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10. Practice Questions

1. Find x: angles on a straight line are 180°. 
Given angles 112° and x°.

2. Use Pythagoras: find the hypotenuse of sides 9 and 12.

3. Area of a circle, radius 5.

4. Volume of a cylinder (r = 4, h = 10).

5. Translate the point (–1, 3) by the vector ⟨5, –2⟩.

6. Find the distance between (2, 7) and (8, 4).

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Summary

This post covered:
• angles 
• triangles 
• Pythagoras 
• area & perimeter 
• 3D volume 
• coordinates 
• transformations 
• vectors 

Geometry is one of the most visual and powerful parts of mathematics — mastering these tools opens the door to physics, engineering, architecture, and more.