Circles — centre, radius, diameter, chord, arc, sector, segment

1.2.9 · Maths › Basic Geometry

Overview

Ek circle un sabhi points ka set hai ek plane mein jo ek fixed point se equidistant hain. Iske parts ko samajhna geometric problems solve karne ke liye fundamental hai jo circular shapes involve karte hain, wheel design se lekar planetary orbits tak.

Core Definitions

First principles se Derivation:

• Diameter circle par do points ko connect karta hai, centre $O$ se guzarta hua
• In points ko $A$ aur $B$ kehte hain, $O$ unke beech mein hai
• Radius ki definition se: $AO=r$ aur $OB=r$
• Kyunki $A$, $O$, $B$ collinear hain (ek hi line par): $AB=AO+OB=r+r=2r$
• Yeh step kyun? Hum segment addition property use kar rahe hain: jab teen points ek straight line par hon, toh pehle se teesre tak ki distance pehle-se-beech aur beech-se-teesre ki distances ka sum hoti hai.

Key property: Centre se chord tak ka perpendicular chord ko bisect karta hai.

Bisection property ki Derivation:

1. Maano chord $AB$ ka midpoint $M$ hai, aur $OM\perp AB$
2. Radii $OA$ aur $OB$ draw karo (dono length $r$)
3. Triangles $OMA$ aur $OMB$ RHS se congruent hain (right angle $M$ par, hypotenuse $OA=OB=r$, common side $OM$)
4. Isliye $AM=MB$ (congruent triangles ke corresponding parts)
5. Yeh step kyun? Congruent triangles ke sabhi corresponding parts equal hote hain, toh chord ke dono halves equal hone chahiye.

Notation: Arc $AB$ ko $\stackrel{⌢}{\mathrm{AB}}$ likha jaata hai

Arc length formula: Central angle $\theta$ (radians mein) ke liye: $\text{Arc length}=r\theta$

Derivation:

1. Full circle circumference: $C=2\pi r$ (yeh $\pi$ ki definition se aata hai circumference aur diameter ka ratio ke roop mein)
2. Ek full circle angle $2\pi$ radians correspond karta hai
3. Angle $\theta$ ke liye, arc full circle ka $\frac{\theta }{2\pi }$ hissa hai
4. Arc length $=\frac{\theta }{2\pi }\times 2\pi r=r\theta$
5. Yeh step kyun? Hum proportional reasoning use kar rahe hain: arc length full circumference se waise hi related hai jaise angle full rotation se.

First principles se Derivation:

1. Full circle area: $A=\pi r^2$
   • Kyun? Sochо concentric circles jo ek filled disk approximate karti hain. Jab hum infinitesimally thin rings of radius $r$ aur thickness $dr$ add karte hain, har ring ki circumference $2\pi r$ aur area $2\pi rdr$ hoti hai. $0$ se $r$ tak integrate karne par: ${\int }_0^{r}2\pi rdr=\pi r^2$
2. Circle mein full angle: $2\pi$ radians
3. Angle $\theta$ wala sector circle ka $\frac{\theta }{2\pi }$ fraction hai
4. Sector area $=\frac{\theta }{2\pi }\times \pi r^2=\frac{1}{2}{r}^2\theta$
5. Yeh step kyun? Phir se proportional reasoning: sector area full circle area se waise hi related hai jaise angle full rotation se.

Special case: Agar $\theta$ degrees mein hai, pehle radians mein convert karo: ${\theta }_{\text{rad}}={\theta }_{\text{deg}}\times \frac{\pi }{180}$, toh: $A_{\text{sector}}=\frac{{\theta }_{\text{deg}}}{360}\times \pi r^2$

Derivation:

1. Segment tab banta hai jab ek chord sector ke across cut karta hai
2. Sector area mein segment aur ek triangle dono include hote hain jiske vertices centre aur chord endpoints par hain
3. Sirf segment paane ke liye, triangle subtract karo: $A_{\text{segment}}=\frac{1}{2}{r}^2\theta -\frac{1}{2}{r}^2\sin \theta$
4. Centre par angle $\theta$ ke liye, triangle ki do sides length $r$ (radii) hain jo angle $\theta$ par milti hain
5. Do sides $a$, $b$ aur included angle $\theta$ ke liye triangle area formula: $\frac{1}{2}ab\sin \theta$
6. Yahan: $A_{\text{triangle}}=\frac{1}{2}r\cdot r\cdot \sin \theta =\frac{1}{2}{r}^2\sin \theta$
7. Yeh step kyun? Triangle area formula height $=r\sin \theta$ se aata hai jab ek radius base ho.

Final formula: $A_{\text{segment}}=\frac{1}{2}{r}^2(\theta -\sin \theta )$

Worked Examples

Solution: (a) Diameter: $d=2r=2(5)=10$ cm

• Yeh step kyun? Diameter defined hai radius se do guna, centre se guzarta hua.

(b) Circumference: $C=2\pi r=2\pi (5)=10\pi \approx 31.42$ cm

• Yeh step kyun? Circumference formula $\pi$ ki definition se aata hai.

(c) Area: $A=\pi r^2=\pi (5{)}^2=25\pi \approx 78.54$ cm²

• Yeh step kyun? Area formula concentric rings ko integrate karke derive hota hai.

Solution: Pehle, angle ko radians mein convert karo: $\theta =60°\times \frac{\pi }{180}=\frac{\pi }{3}$ radians

• Yeh step kyun? Standard formulas radians chahte hain kyunki $s=r\theta$ tabhi kaam karta hai jab $\theta$ radians mein ho.

(a) Arc length: $s=r\theta =6\times \frac{\pi }{3}=2\pi \approx 6.28$ cm

• Yeh step kyun? Arc length angle ke proportional hai: full circumference ka $\frac{\theta }{2\pi }$ hissa.

(b) Sector area: $A=\frac{1}{2}{r}^2\theta =\frac{1}{2}(6{)}^2\times \frac{\pi }{3}=6\pi \approx 18.85$ cm²

• Yeh step kyun? Sector area bhi full circle area ka wohi fraction $\frac{\theta }{2\pi }$ hai.

Verification: Sector $\frac{60°}{360°}=\frac{1}{6}$ circle hona chahiye. Full area: $\pi (6{)}^2=36\pi$ cm² Ek-chhatwaan: $\frac{36\pi }{6}=6\pi$ cm² ✓

Solution: Angle radians mein: $\theta =90°\times \frac{\pi }{180}=\frac{\pi }{2}$ radians

• Yeh step kyun? Formula ke liye radians mein convert kar rahe hain.

Sector area: $A_{\text{sector}}=\frac{1}{2}{r}^2\theta =\frac{1}{2}(8{)}^2\times \frac{\pi }{2}=16\pi$ cm²

• Yeh step kyun? Pehle full "pizza slice" area dhundho.

Triangle area: $A_{\text{triangle}}=\frac{1}{2}{r}^2\sin \theta =\frac{1}{2}(8{)}^2\sin (90°)=\frac{1}{2}(64)(1)=32$ cm²

• Yeh step kyun? Do radii aur chord se bana triangle ka area $\frac{1}{2}{r}^2\sin \theta$ hai. $90°$ ke liye, $\sin (90°)=1$.

Segment area: $A_{\text{segment}}=A_{\text{sector}}-A_{\text{triangle}}=16\pi -32\approx 50.27-32=18.27$ cm²

• Yeh step kyun? Sector se triangle part remove karo taaki sirf curved segment bache.

Solution: Maano $M$ centre $O$ se chord $AB$ tak perpendicular ka foot hai.

• $OM=6$ cm (given)
• $OA=10$ cm (radius)
• Triangle $OMA$ $M$ par right-angled hai

Pythagorean theorem se: $A{M}^2=O{A}^2-O{M}^2=10^2-6^2=100-36=64$ $AM=8\text{ cm}$

• Yeh step kyun? Right triangle mein, radius hypotenuse hai, perpendicular distance aur half-chord baaki do sides hain.

Kyunki centre se perpendicular chord ko bisect karta hai: $AB=2\times AM=2\times 8=16\text{ cm}$

• Yeh step kyun? Perpendicular bisection property: $OM\perp AB$ implies $AM=MB$.

Common Mistakes

Memory Aids

Visual memory: Ek pizza imagine karo:

• Centre = jahan tum use pakad te ho
• Radius = centre se crust tak
• Diameter = centre se hokar poori width
• Chord = across cutting (centre se nahi)
• Arc = curved crust edge
• Sector = tumhara slice (triangular)
• Segment = tumhare slice ka sirf crust wala part

Now, agar tum circle par do points ko connect karne wali koi bhi straight line draw karo, woh ek chord hai. Diameter sabse bada possible chord hai. Do points ke beech circle ka curved part ek arc hai — jaise pizza slice ki crust.

Ek sector pizza slice jaisa hai jiska pointy end centre par ho — tumhare paas do radii hain (jaise knife cuts) aur curved crust. Ek segment woh hai jo bachta hai agar tum woh pointy triangle part cut kar do — sirf chord (straight line) aur circle ki edge ke beech curved crust section.

Hum care kyun karte hain? Circles har jagah hain! Clock faces (radius se har hour mark tak), wheels (sabhi spokes same length = radii), cakes ko fairly kaatna (equal sector angles = equal areas). In parts ko samajhna tumhe exactly calculate karne deta hai kitna pizza milega, ya circular track par kitna chale, ya round table design karna!

Active Recall Practice

#flashcards/maths

What is the centre of a circle? :: Woh fixed point jisse circle ke sabhi points equidistant hain. Symbol: $O$.

Why must angle be in radians for $s=r\theta$? :: Kyunki radians define hote hain arc length divided by radius ($\theta =\frac{s}{r}$) ke roop mein, jo formula ko dimensionally consistent banata hai. Degrees arbitrary units hain.

Connections

• Circle Equations — Coordinate geometry mein circles ke liye $(x-h{)}^2+(y-k{)}^2=r^2$ use karna
• Angle Properties in Circles — Inscribed angles, central angles, aur arcs se unka relationship
• Tangent and Secant Lines — Lines jo circles ko touch ya intersect karti hain
• Circle Theorems — Properties jaise "angle in semicircle is 90°"
• Circumference and Area of Circles — $C=2\pi r$ aur $A=\pi r^2$ ki detailed derivations
• Pythagorean Theorem — Chord-radius calculations mein extensively use hota hai
• Radian Measure — Radians circular measurements ke liye natural units kyun hain
• Sectors and Pizza Problem — Sector calculations ke real-world applications
• Integration — $A=\pi r^2$ concentric rings ko integrate karke kaise derive hota hai

Study tip: Har part ko ek actual circle par draw karo. Compass se physical sketching radius, chord, aur arc ke beech relationships ko sirf padhne se kahin zyada internalize karne mein help karta hai.

Concept Map