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Geometry Essentials — Angles, Shapes, Area & Space - Leejohnston - 11-13-2025
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 = c²Where c is the hypotenuse.
Example:
Code:
3² + 4² = 9 + 16 = 25 → c = 5-----------------------------------------------------------------------
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)-----------------------------------------------------------------------
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⟩-----------------------------------------------------------------------
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.