Arc length and sector area using radians

3.1.3 · Maths › Advanced Trigonometry

RADIAN kya hota hai? (Derive karne se pehle define karo)

YEH kyun zaroori hai: definition pehle hi bata deta hai ki "angle = arc ÷ radius". Neeche jo bhi hai woh sirf usi ek sentence ko re-arrange karna hai.

ARC LENGTH derive kaise karein

Proportion se shuru karte hain: $\frac{\text{arc length }s}{\text{whole circumference}}=\frac{\text{your angle }\theta }{\text{full angle }2\pi }$

Circumference $2\pi r$ substitute karo: $\frac{s}{2\pi r}=\frac{\theta }{2\pi }$

Dono sides ko $2\pi r$ se multiply karo. $2\pi$ cancel ho jaata hai:

Yeh itna chhota kyun hai: $2\pi$ cancel ho gaye kyunki radians angle ko "arc per radius" ki units mein measure karte hain. Re-arrange karo toh definition wapas milti hai: $\theta =s/r$.

SECTOR AREA derive kaise karein

$\frac{\text{sector area }A}{\text{whole area}}=\frac{\theta }{2\pi }$

Disc area $\pi r^2$ substitute karo: $\frac{A}{\pi r^2}=\frac{\theta }{2\pi }$

Dono sides ko $\pi r^2$ se multiply karo; $\pi$ cancel ho jaata hai aur $2\pi /2\pi \to$ ek $\frac{1}{2}$ bachta hai: $A=\pi r^2\cdot \frac{\theta }{2\pi }=\frac{1}{2}{r}^2\theta$

Forecast-then-Verify

Worked examples

Common mistakes (Unhe samjho)

Flashcards

Connections

• Radian measure and degree conversion
• Circumference and area of a circle
• Small-angle approximations ($s\approx r\theta$ use karta hai)
• Angular velocity ($v=r\omega$, $s=r\theta$ jaisa hi hai)
• Segment area (sector minus triangle)

Concept Map