Circular Measure (Copy)
CIRCULAR MEASURE — O LEVEL & IGCSE ADDITIONAL MATHEMATICS FORMULA SHEET + COMMON MISTAKES
| Topic | Formula / Rule | Common Mistake |
|---|---|---|
| Radian Definition | Angle subtended when arc length = radius | Mixing with degrees |
| π Radians | 180° | Using 360° |
| Full Circle | 2π radians | Writing π |
| Degree to Radian Conversion | θ(rad) = θ(deg) × π/180 | Multiplying by 180 instead |
| Radian to Degree Conversion | θ(deg) = θ(rad) × 180/π | Dividing incorrectly |
| Arc Length Formula | s = rθ | Using degrees instead of radians |
| Sector Area Formula | A = 1/2 r²θ | Forgetting 1/2 |
| Circumference | 2πr | Writing πr |
| Circle Area | πr² | Forgetting square |
| Arc Length in Degrees | s = θ/360 × 2πr | Using radians formula directly |
| Sector Area in Degrees | A = θ/360 × πr² | Missing fraction |
| Small Angle Formula | s = rθ | Using for degrees |
| Angle at Centre | Measured in radians/degrees | Mixing units |
| Semicircle | π radians | Writing 2π |
| Quarter Circle | π/2 radians | Writing π/4 |
| Three-Quarter Circle | 3Ï€/2 radians | Mixing numerator |
| Arc Length Rearrangement | θ = s/r | Wrong rearrangement |
| Radius Rearrangement | r = s/θ | Rearranging incorrectly |
| Sector Area Rearrangement | θ = 2A/r² | Forgetting 2 |
| Segment Area | Sector area − triangle area | Adding instead |
| Chord Formula | c = 2r sin(θ/2) | Forgetting θ/2 |
| Triangle Area with Radians | A = 1/2 ab sinC | Using degree mode incorrectly |
| Angular Speed | ω = θ/t | Using degrees |
| Linear Speed | v = rω | Wrong formula order |
| Relationship Formula | s = vt | Confusing with s = rθ |
| Angular Acceleration | α = ω/t | Mixing symbols |
| Exact Circular Values | Use radians exactly | Decimalising too early |
| Calculator Mode | Use radian mode when required | Staying in degree mode |
| Sector Perimeter | 2r + arc length | Forgetting both radii |
| Major Sector | Larger region of circle | Using minor angle |
| Minor Sector | Smaller region | Using reflex angle |
| Reflex Angle | θ > π radians | Confusing with acute angle |
| Radian Approximation | π ≈ 3.142 | Using rough values too early |
Written and Compiled By Sir Hunain Zia (AYLOTI), World Record Holder With 154 Total A Grades, 11 World Records and 7 Distinctions, Educate A Change.
QUICK RADIAN CONVERSIONS TABLE
| Degrees | Radians |
|---|---|
| 30° | π/6 |
| 45° | π/4 |
| 60° | π/3 |
| 90° | π/2 |
| 120° | 2π/3 |
| 135° | 3π/4 |
| 150° | 5π/6 |
| 180° | π |
| 270° | 3π/2 |
| 360° | 2π |
QUICK CIRCULAR MEASURE FORMULAS
| Formula | Expression |
|---|---|
| Arc Length | s = rθ |
| Sector Area | A = 1/2 r²θ |
| Circumference | 2Ï€r |
| Circle Area | πr² |
| Angular Speed | ω = θ/t |
| Linear Speed | v = rω |
| Chord Length | c = 2r sin(θ/2) |
Written and Compiled By Sir Hunain Zia (AYLOTI), World Record Holder With 154 Total A Grades, 11 World Records and 7 Distinctions, Educate A Change.
QUICK DEGREE/RADIAN RULES
| Conversion | Formula |
|---|---|
| Degrees → Radians | × π/180 |
| Radians → Degrees | × 180/π |
| Full Circle | 2Ï€ radians |
| Half Circle | π radians |
| Quarter Circle | π/2 radians |
QUICK SECTOR RULES
| Quantity | Formula |
|---|---|
| Arc Length | s = rθ |
| Sector Area | A = 1/2 r²θ |
| Sector Perimeter | 2r + s |
| Segment Area | Sector area − triangle area |
EXACT VALUE TABLE
| Expression | Value |
|---|---|
| sin(Ï€/6) | 1/2 |
| cos(π/6) | √3/2 |
| tan(π/6) | 1/√3 |
| sin(π/4) | √2/2 |
| cos(π/4) | √2/2 |
| tan(Ï€/4) | 1 |
| sin(π/3) | √3/2 |
| cos(Ï€/3) | 1/2 |
| tan(π/3) | √3 |
Written and Compiled By Sir Hunain Zia (AYLOTI), World Record Holder With 154 Total A Grades, 11 World Records and 7 Distinctions, Educate A Change.
EXAM HACKS
| Situation | Fast Trick |
|---|---|
| Arc length question | Use s = rθ immediately |
| Sector area question | Use 1/2 r²θ |
| Degree question | Convert before solving |
| Calculator issue | Check RAD mode |
| Segment area | Subtract triangle area |
| Circular motion | Use v = rω |
| Sector perimeter | Add arc + 2 radii |
| Exact answers | Keep π form |
MOST COMMON LOSSES OF MARKS
| Mistake | Why Marks Are Lost |
|---|---|
| Using degrees in radian formulas | Entire answer incorrect |
| Forgetting calculator RAD mode | Wrong trig values |
| Missing 1/2 in sector area | Formula marks lost |
| Mixing arc and perimeter | Incorrect final answer |
| Wrong conversion direction | Unit errors |
| Forgetting radii in perimeter | Incomplete answer |
| Decimalising too early | Accuracy marks lost |
| Using wrong angle | Entire geometry incorrect |
| Confusing segment and sector | Wrong area |
| Incorrect rearrangement | Algebra marks lost |
Written and Compiled By Sir Hunain Zia (AYLOTI), World Record Holder With 154 Total A Grades, 11 World Records and 7 Distinctions, Educate A Change.