Area — triangle, parallelogram, trapezium, composite shapes

1.2.12 · Maths › Basic Geometry

Area Kyun Important Hai

Area humein batata hai:

Physical space: paint kitna chahiye, zameen ka size, carpet kitna chahiye
Optimization: ek fence ke andar garden space maximize karo
Foundation for calculus: integration hi "area under curves" hai

Key insight: saare area formulas rectangle se derive hote hain. Har shape ya toh rectangle hi hai disguise mein, ya phir use kaat ke rearrange kiya ja sakta hai ek mein.

Triangle Area

First Principles Se Derivation

YEH KYU KAAM KARTA HAI?

Ek parallelogram se shuru karo: Koi bhi triangle banao. Uski copy karo, 180° flip karo, aur ek side ke saath attach kar do.
Tumhe ek parallelogram milta hai: Same base $b$ , same height $h$ (perpendicular flip karne se change nahi hoti).
Parallelogram area: Hum yeh prove karenge ki yeh $bh$ hai next section mein, lekin intuitively: yeh ek "squashed rectangle" hai jiska same base aur height hai us original rectangle ke jitna jis se woh aaya.
Triangle half hai: Kyunki humne 2 identical triangles se 1 parallelogram banaya: $A_{△} = \frac{1}{2} A_{parallelogram} = \frac{1}{2} bh$

HEIGHT KYA HAI? Base se opposite vertex tak perpendicular distance. Slant side nahi! Agar tumhara triangle apni sabse lambi side pe baith raha hai, toh height baaki dono sides se chhoti ho sakti hai.

ISKO KAISE USE KAREIN:

Koi bhi side base ki tarah lo
Us base ki perpendicular height dhoondo (Pythagoras ki zaroorat pad sakti hai)
Multiply karo aur aadha karo

Figure — Area — triangle, parallelogram, trapezium, composite shapes

Parallelogram Area

Rectangle Se Derivation

YEH BASE × HEIGHT KYU HAI, BASE × SLANT SIDE KYU NAHI?

Ek parallelogram se shuru karo: Iske opposite sides parallel aur equal hoti hain.
Kaat ke rearrange karo: Ek top corner se base tak ek perpendicular girao, left side pe ek right triangle banega. Is triangle ko kaat lo.
Triangle ko slide karo: Ise shape ki right side pe le jao. Parallelogram ki slant edge triangle ki hypotenuse ban jaati hai, jo perfectly fit hoti hai.
Tumhe ek rectangle milta hai: Base $b$ , height $h$ (parallel sides ke beech perpendicular distance).
Rectangle area $bh$ hai: Isliye, parallelogram area bhi $bh$ hai.

TRICK KYA HAI? Perpendicular height $h$ slant side ki length nahi hai. Slant side lambi hoti hai, lekin area ke liye count nahi hoti — sirf vertical "squeeze" matter karta hai.

GALTIYON SE KAISE BACHEIN:

Hamesha parallel sides ke beech perpendicular distance dhoondo
Agar slant side $s$ aur angle $θ$ diya ho, toh compute karo $h = s sin θ$

Trapezium Area

Parallelogram Se Derivation

PARALLEL SIDES KA AVERAGE KYU LETE HAIN?

Duplicate aur flip karo: Trapezium ki copy karo, use 180° rotate karo, aur original ke saath ek non-parallel side ke along attach karo.
Tumhe ek parallelogram milta hai: Dono parallel sides $a$ aur $b$ ab parallelogram ke do bases banate hain, stacked. Total base length = $a + b$ . Height abhi bhi $h$ hai.
Parallelogram area: $(a + b) \times h$
Trapezium half hai: $A_{trapezium} = \frac{1}{2} (a + b) h$

Intuition: $\frac{a + b}{2}$ parallel sides ki average length hai — trapezium ko ek rectangle ki tarah socho jiska average width yahi ho.

KAISE MEASURE KAREIN:

Dono parallel sides identify karo (aksar top aur bottom)
Unke beech perpendicular distance napo (slant sides nahi)
Parallel sides ko add karo, height se multiply karo, 2 se divide karo

Composite Shapes

Method

KYA KARNA HAI:

Components identify karo: Un simple shapes ko outline karo jo composite shape banati hain
Har piece napo: Zaroori dimensions dhoondo (bases, heights, radii)
Individual areas calculate karo: Har simple shape ke formulas use karo
Combine karo: Areas add karo agar shapes joined hain; subtract karo agar ek doosre se kaata gaya hai

TRICKY CASES KAISE HANDLE KAREIN:

Overlaps: Double-counting se bachne ke liye overlap area subtract karo
Cutouts: Total area = outer shape − inner shape(s)
Irregular pieces: Lines extend karo simpler shapes banane ke liye, phir extra parts subtract karo

Strategy Summary

Shape	Formula	Key Point
Triangle	$\frac{1}{2} bh$	Height base ke perpendicular hoti hai
Parallelogram	$bh$	Base × slant side nahi!
Trapezium	$\frac{1}{2} (a + b) h$	Parallel sides ka average × height
Composite	Sum ya difference	Known shapes mein todo

Recall Feynman Explanation (Ek 12-Saal Ke Bachche Ko Samjhao)

Imagine karo tum floor pe square tiles laga rahe ho, har tile 1 meter × 1 meter ki hai. Area bas yeh count karna hai ki ek shape ke andar kitni tiles fit hoti hain.

Triangle: Agar tum do identical triangles lo aur unhe puzzle pieces ki tarah jod do, tumhe ek parallelogram milta hai. Toh ek triangle us parallelogram ka aadha hai. Isliye hum base × height multiply karte hain aur phir 2 se divide karte hain. Parallelogram: Yeh ek rectangle ki tarah hai jo sideway push ho gayi ho. Agar tum ek side se triangle kaat ke doosri side pe lagao, tumhe ek perfect rectangle milta hai. Isliye iska area ek rectangle ke equal hai jiska same base aur height ho.

Trapezium: Yeh ek rectangle ki tarah hai, lekin ek side chhoti hai. Agar tum do banao aur ek ko ulta karo, tum dono ko ek parallelogram mein fit kar sakte ho. Us parallelogram ka base dono parallel sides ko jodne ke barabar hoga. Toh ek trapezium us ka aadha hai.

Composite shapes: Jab shapes ek saath judi hoti hain (jaise L-shape ya roof wala ghar), toh bas har simple piece ka area alag se nikalo, phir unhe add kar do. Agar koi hole hai (jaise picture frame), toh bada area nikalo aur hole subtract karo.

Secret: sab kuch rectangles pe wapas aata hai, jo bas "kitni tiles across" times "kitni tiles upar" hoti hain.

Connections

1.2.1-Basic-shapes — in shapes ki definitions
1.2.11-Perimeter — perimeter vs area ka fark
1.3.5-Pythagorean-theorem — right triangles mein heights dhoodhna
2.1.4-Trigonometric-ratios — perpendicular heights dhoodhne ke liye sine use karna
3.4.2-Integrationas-area — calculus area ko curves tak generalize karta hai
1.2.13-Circle-area-and-circumference — circular regions ka area

#flashcards/maths

Triangle area ka formula kya hai? :: $A = \frac{1}{2} bh$ jahan $b$ base hai aur $h$ perpendicular height hai

Triangle area base × height ka aadha kyun hota hai?

Ek triangle ek parallelogram ka aadha hota hai jiska same base aur height ho (triangle ko duplicate aur flip karo yeh dekhne ke liye)

Parallelogram area ka formula kya hai?

A = bh

jahan

b

base hai aur

h

parallel sides ke beech perpendicular height hai

Parallelogram area base × slant side kyun nahi hota?

Slant side lambi hoti hai lekin angle pe hoti hai; sirf perpendicular height count hoti hai ki shape kitna vertical space occupy karti hai

Trapezium area ka formula kya hai?

A = \frac{1}{2} (a + b) h

jahan

a

aur

b

parallel sides hain aur

h

perpendicular height hai

Trapezium area mein

(a + b) /2

kya represent karta hai?

Dono parallel sides ki average length — trapezium ek rectangle ki tarah act karta hai is average width ke saath

Composite shape ka area kaise nikaalte hain?

Ise simple shapes (triangles, rectangles, etc.) mein todo, har ek ka area alag se nikalo, phir unhe jod do (ya subtract karo agar cutouts hain)

Ek parallelogram ka base 12 cm, slant side 8 cm, aur angle 30° hai, toh uska area kya hai? :: Height = $8 sin (30°) = 4$ cm, toh area = $12 \times 4 = 48$ cm²

Ek trapezium ke parallel sides 5 m aur 9 m hain, height 4 m hai. Uska area kya hai?

A = \frac{1}{2} (5 + 9) \times 4 = \frac{1}{2} (14) (4) = 28

m²

Parallelogram area calculate karte waqt sabse common galti kya hoti hai?

Perpendicular height ki jagah slant side length use karna

Jab ek shape mein cutout ho toh kya yaad rakhna chahiye?

Outer shape ke area se cutout ka area subtract karo

Area formulas mein "height" ka hamesha kya matlab hota hai?

Base se opposite side ya vertex tak perpendicular distance — kabhi bhi slant measurement nahi