✅ Pehle Yeh Aata Hai?
Yeh post shuru karne se pehle, in teen sawaalon ko 1 minute mein khud try kijiye:
- 5 cm side wale square ka area kitna hai?
- Square ka diagonal use kitne triangles mein baantta hai?
- Agar ek square ka area 16 cm² hai, to uski side kitni?
Answers dekho
- 5 × 5 = 25 cm²
- 2 barabar (congruent) triangles mein
- 4 cm, kyunki 4 × 4 = 16
Teeno sahi? Badhiya, aage chaliye! Koi galat hua? Koi baat nahi, area aur square root Book 1 ke Chapter 1 — A Square and a Cube mein dobara dekh lijiye.
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🎬 Kahani: Pandit Ji Ki Yagya Vedi
Gaon mein bada havan hona tha. Pandit ji ne Golu se kaha, “Beta, yeh chaukor vedi (square altar) 1 metre side ki hai. Agle havan ke liye mujhe bilkul double area wali chaukor vedi chahiye.”
Golu turant bola, “Aasaan hai! Side double kar do, 2 metre!” Pandit ji muskuraaye, “Zara ginti karke dekho, kitni int (bricks) lagengi?”
Golu ne hisaab lagaaya: 1 × 1 = 1 m² se 2 × 2 = 4 m². Yeh to 4 guna ho gaya, double nahi! 😮
Lagbhag 2800 saal pehle (c. 800 BCE) Bharat ke ganitagya Baudhayana ne apne Śulba-Sūtra mein, yagya vedi banane ke liye, theek yahi sawaal hal kiya tha. Aaj hum unka tareeka seekhenge! 🔥
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📖 Concept: Diagonal Pe Bana Square, Area Double
Step 1: Side double karna kaam kyun nahi karta?
Side 1 → 2 karo to area 1 × 1 = 1 se 2 × 2 = 4 ho jaata hai. Area side × side hai, isliye side 2 guna hone pe area 2 × 2 = 4 guna ho jaata hai.
WHY? Area do dishaon mein phailta hai, lambaai mein bhi aur chaudaai mein bhi. Dono taraf 2 guna, to kul 4 guna.
Step 2: Baudhayana ka jawab
Baudhayana Śulba-Sūtra (verse 1.9) kehta hai: kisi square ke diagonal pe banaya gaya square, us square ke area ka double hota hai.
WHY yeh idea? Hume side ko “2 guna” nahi, “thoda sa” badhaana tha. Diagonal side se lambaa hai par double nahi, bilkul sahi naap!
Step 3: Triangles gin ke saboot
Upar wale chhote square (orange) ko uska diagonal 2 barabar triangles mein kaatta hai. Ab diagonal pe square banao (neela, tircha). East-west aur north-south lines kheencho, to yeh neela square 4 waise hi triangles se bana dikhta hai.
Chhota square = 2 triangles, diagonal wala square = 4 triangles. 4 = 2 × 2, isliye area double! ✅
WHY saare triangles barabar? Har triangle ek hi size ke chhote square ka aadha hai (diagonal se kata hua), isliye sab congruent hain.
Step 4: Doubling ki seedhi
Yahi trick baar-baar lagao: 2 triangles wala square → 4 triangles wala → 8 triangles wala → 16… Har agla square pichhle ka double.
WHY kaam ka? Ek hi tareeke se area 2, 4, 8, 16 guna kar sakte ho, bina ek bhi number ke hisaab ke, sirf rassi aur khoonti (pegs) se. Vedi banane wale yahi karte the.
Step 5: Kaagaz se khud karke dekho ✂️
- Do ek jaise kaagaz ke square kaato (Square 1 aur Square 2).
- Square 2 ko dono diagonals se kaat ke 4 triangles banao.
- In 4 triangles ko Square 1 ki chaaron sides pe, bahar ki taraf chipkao, har triangle ki lambi side Square 1 ki ek side se mile.
- Ek bada, tircha square banega. Iska area = Square 1 + Square 2 = double! Iski side = Square 1 ka diagonal.
WHY kaam karta hai? Humne koi kaagaz na joda, na feka. Do squares ka poora kaagaz ek naye square mein aa gaya, isliye area exactly double.
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🔐 Deep Concept: Diagonal Ki Lambaai Kitni?
Agar chhote square ka area 1 hai, to diagonal wale square ka area 2 hai. To uski side (yaani diagonal) woh number hai jiska square 2 ho: √2.
√2 lagbhag 1.414 hai. Matlab diagonal, side se lagbhag 41% lambaa hota hai, double se kaafi kam. Isliye “side double karo” galat tha, aur “diagonal lo” sahi.
Aur ek raaz: √2 ko kabhi bhi poori tarah decimal ya fraction mein nahi likh sakte! Iski poori kahani Post 3 — Root 2 (√2) Kya Hai? mein. 🔐
sirf charumam.com readers ke liye 🔐
✏️ Solved Examples
Example 1 (Easy): 5 cm side wale square ka area double karna hai. Naye square ka area kitna hoga? Aur agar galti se side double kar di, to area kitna ho jaata?
Asli area = 5 × 5 = 25 cm². Double = 50 cm².
Side double (10 cm) karne pe area = 10 × 10 = 100 cm², jo 4 guna hai.
WHY sahi hai? 50 = 2 × 25 ✅, aur 100 = 4 × 25, isliye side double karna galat tareeka hai.
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Example 2 (Medium): 6 cm side ke square ke diagonal pe ek naya square banaya. Naye square ka area kitna hai?
Baudhayana ke niyam se, diagonal wala square = 2 × (asli area) = 2 × 36 = 72 cm².
WHY sahi hai? Asli square = 2 triangles, naya = 4 triangles. Har triangle 18 cm², to 4 × 18 = 72 ✅
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Example 3 (NCERT style): Doubling ki seedhi mein pehla square 2 triangles se bana hai aur har triangle ka area 3 cm² hai. Seedhi ke pehle 4 squares ke area batao.
Triangles: 2, 4, 8, 16. Area: 2 × 3 = 6, 4 × 3 = 12, 8 × 3 = 24, 16 × 3 = 48 cm².
Jawab: 6, 12, 24, 48 cm².
WHY sahi hai? Har agla area pichhle ka double: 6 → 12 → 24 → 48 ✅
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Golu ka formula: “Area double chahiye? Side mat badhao, diagonal pakdo!” 🎉
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🏠 Real Life Mein
- 🔥 Yagya vedi: Śulba-Sūtra ke niyam isi liye bane ki vedi ka area bilkul sahi naap mein badhe.
- 🧱 Floor tiles: Tiles ko 45° ghuma ke (diamond pattern) lagao, to beech mein bane square doubling wala pattern dikhate hain.
- 🪁 Patang aur origami: Kaagaz ke kone beech mein modne se aadhe area wala square banta hai (agle post mein).
- 🌾 Khet ka naksha: Chaukor khet ka area double karna ho to naya khet purane ke diagonal jitni side ka chahiye.
— 📌 charumam.com
🧭 Kab Use Karein, Kab Nahi
| Situation | Kya karein | Kyun |
|---|---|---|
| Area double karna hai | ✅ Diagonal pe square banao | Diagonal wala square = 2 × area |
| Area 4 guna karna hai | ✅ Side double karo | Area = side × side, 2 × 2 = 4 |
| Area 2 guna chahiye aur side double kar di | ❌ Galat | Area 4 guna ho jaayega |
| Area 8 guna karna hai | ✅ Diagonal trick 3 baar | 2 × 2 × 2 = 8 |
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❌→✅ Worked Mistake
Sawaal: 4 m side ka chaukor bageecha hai. Naya bageecha double area ka chahiye. Naye bageeche ki side?
❌ Galat: “Double area, to double side: 8 m.”
Galti kya hai? 8 × 8 = 64 m², jabki asli area 4 × 4 = 16 m² hai. 64 to 16 ka 4 guna hai! ❌
✅ Sahi:
- Naya area = 2 × 16 = 32 m².
- Naye square ki side = asli square ka diagonal.
- Lambaai: side × √2 ≈ 4 × 1.414 ≈ 5.66 m (check: 5.66 × 5.66 ≈ 32 ✅).
Yaad rakho: Side ko k guna karo, to area k × k guna hota hai.
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📝 10 MCQs
Q1. Square ki side double karne pe area kitne guna hota hai?
Solution dekho
Sahi jawab: (c)
- (a) galat: Area side × side hai, isliye 2 × 2 lagta hai.
- (b) galat: 3 ka koi rishta nahi.
- (c) sahi: 2 × 2 = 4 guna ✅
- (d) galat: 8 guna cube (volume) mein hota hai, side double karne pe.
Q2. Baudhayana ke anusaar, kisi square ke diagonal pe bane square ka area kitna hota hai?
Solution dekho
Sahi jawab: (b)
- (a) galat: Diagonal side se lambaa hai, to area zyada hoga.
- (b) sahi: Diagonal wala square = 4 triangles, asli = 2 triangles ✅
- (c) galat: Teen guna ke liye alag construction chahiye (Post 4/5).
- (d) galat: Aadha to ulta hai, woh agle post mein!
Q3. Square ka ek diagonal use kitne congruent triangles mein baantta hai?
Solution dekho
Sahi jawab: (a)
- (a) sahi: Ek diagonal = 2 barabar triangles ✅
- (b) galat: Teen triangles ek diagonal se nahi bante.
- (c) galat: 4 triangles dono diagonals se bante hain.
- (d) galat: 8 bahut zyada hai.
Q4. Baudhayana ne Śulba-Sūtra kab ke aas-paas likha?
Solution dekho
Sahi jawab: (a)
- (a) sahi: NCERT ke anusaar lagbhag 800 BCE ✅
- (b) galat: 300 BCE Euclid ka samay hai.
- (c) galat: 1994 mein Fermat ka theorem prove hua tha.
- (d) galat: Yeh bahut baad ka samay hai.
Q5. 5 cm side ke square ke diagonal pe bane square ka area?
Solution dekho
Sahi jawab: (b)
- (a) galat: Yeh asli square ka area hai.
- (b) sahi: 2 × 25 = 50 cm² ✅
- (c) galat: Yeh side double karne pe aata.
- (d) galat: 10 to perimeter ka bhi nahi, sirf 2 × 5 hai.
Q6. Doubling ki seedhi mein 4 triangles wale square ke baad agla square kitne triangles ka hoga?
Solution dekho
Sahi jawab: (c)
- (a) galat: Ek triangle jodne se double nahi hota.
- (b) galat: 6 = 1.5 guna hai.
- (c) sahi: Har baar double: 4 × 2 = 8 ✅
- (d) galat: 16 to uske baad wala hai.
Q7. Ek square ka area 18 cm² hai. Uske diagonal pe bane square ka area?
Solution dekho
Sahi jawab: (b)
- (a) galat: Aadha karna nahi tha.
- (b) sahi: 2 × 18 = 36 cm² ✅
- (c) galat: 72 = 4 × 18, galat.
- (d) galat: Area badalta hai.
Q8. Kaagaz wali activity mein Square 2 ke 4 triangle Square 1 ke chaaron taraf lagaye. Naye square ka area?
Solution dekho
Sahi jawab: (a)
- (a) sahi: Square 1 + Square 2 ka poora kaagaz = 2 × Square 1 ✅
- (b) galat: 4 guna ke liye 3 square aur chahiye hote.
- (c) galat: Kaagaz joda gaya hai, to area badhega.
- (d) galat: Area ghat nahi sakta.
Q9. Diagonal wale square ki side, asli square ki side se lagbhag kitni hoti hai?
Solution dekho
Sahi jawab: (b)
- (a) galat: Diagonal side se lamba hota hai.
- (b) sahi: √2 ≈ 1.414 guna ✅
- (c) galat: 2 guna se area 4 guna ho jaata.
- (d) galat: Bahut zyada.
Q10. Area 8 guna karna hai. Diagonal wali trick kitni baar lagaani hogi?
Solution dekho
Sahi jawab: (b)
- (a) galat: 2 baar = 4 guna.
- (b) sahi: 2 × 2 × 2 = 8, yaani 3 baar ✅
- (c) galat: 4 baar = 16 guna.
- (d) galat: Har baar double hota hai, 8 baar = 256 guna.
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❓ FAQs (Bachchon Ke Asli Sawaal)
1. Side double karne se area double kyun nahi hota?
Area = side × side. Side 2 guna to dono taraf 2 guna, kul 4 guna.
2. Baudhayana kaun the?
Prachin Bharat ke ganitagya jinhone lagbhag 800 BCE mein Baudhayana Śulba-Sūtra likha. Isme yagya vedi banane ke liye geometry ke niyam hain.
3. Śulba ka matlab kya hai?
Śulba (ya Śulva) ka matlab hai rassi. Purane samay mein naap aur construction rassi aur khoontiyon (pegs) se hote the.
4. Kya diagonal ki lambaai exact nikal sakte hain?
1 side ke square ka diagonal √2 hai, lagbhag 1.414. Iska decimal kabhi khatam nahi hota, isliye “exact” decimal nahi likh sakte. Post 3 mein detail.
5. Kya yeh tareeka rectangle pe bhi chalega?
Nahi, yeh khaas taur pe square ke liye hai, kyunki square ke diagonal se bane saare triangles barabar hote hain.
6. Area teen guna karna ho to?
Uske liye alag construction hai (Śulba-Sūtra verse 1.10), jo theorem seekhne ke baad Post 5 mein karenge.
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🏋️ Practice Zone (12 Sawaal)
Q1. Easy 4 cm side ke square ka area aur uske diagonal pe bane square ka area batao.
Solution dekho
16 cm² aur 2 × 16 = 32 cm².
Q2. Easy Side 3 guna karne pe area kitne guna hoga?
Solution dekho
3 × 3 = 9 guna.
Q3. Easy Ek square ka area 20 cm² hai. Diagonal wale square ka area?
Solution dekho
40 cm².
Q4. Easy Diagonal pe bane square ka area 50 cm² hai. Asli square ka area?
Solution dekho
50 ÷ 2 = 25 cm², yaani side 5 cm.
Q5. Medium Doubling seedhi: pehla square 2 triangles (har triangle 5 cm²). Teesre square ka area?
Solution dekho
Triangles 2, 4, 8. Teesra = 8 × 5 = 40 cm².
Q6. Medium 7 cm side ke square ka diagonal wala square banaya, phir uske diagonal pe ek aur square. Aakhri square ka area?
Solution dekho
49 → 98 → 196 cm² (4 guna, jo 14 cm side ke square ke barabar hai!).
Q7. Medium Golu ne 10 m side ki vedi ki side 20 m kar di. Pandit ji ko double chahiye tha. Kitna area faltu bana?
Solution dekho
Chahiye 200 m², bana 400 m². Faltu = 200 m².
Q8. Medium Kaagaz activity mein 6 cm ke do squares liye. Naya square kitne area ka bana?
Solution dekho
36 + 36 = 72 cm².
Q9. NCERT style Dikhao ki diagonal wale square mein 4 triangles hain aur asli mein 2, saare congruent kyun?
Solution dekho
Har triangle ek hi size ke square ka aadha hai (diagonal ek square ko 2 congruent triangles mein baantta hai). Isliye sab congruent.
Q10. NCERT style Bade square (side 2 units) ko 4 unit squares mein baanto. Beech ka tircha square (sides ke midpoints jodkar) kitna area ka hai?
Solution dekho
Har unit square ka aadha andar hai: 4 × ½ = 2 sq. units, yaani unit square ka double.
Q11. Medium Ek square ka area 1 hai. Doubling seedhi 5 baar lagaayi. Area?
Solution dekho
2⁵ = 32.
Q12. NCERT style Kya diagonal wala square banane mein koi number ka hisaab chahiye?
Solution dekho
Nahi! Sirf diagonal ki lambaai rassi se naap ke square banana hai. Isliye Baudhayana ka tareeka bina ginti ke chalta tha.
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🧠 HOTS (Sochne Wale Sawaal)
Q1. Ek square ka area 4 guna karna hai. Do tareeke batao.
Solution dekho
(1) Side double karo. (2) Diagonal trick do baar lagao (2 × 2 = 4). Dono ka jawab same square deta hai: side 2 guna.
Q2. Kya kisi square ke diagonal pe bane square ka diagonal, asli square ki side ka double hota hai?
Solution dekho
Haan! Diagonal wale square ka area 2A, uske diagonal wale ka 4A. 4A area ka square = side double. Matlab do baar diagonal = side × 2.
Q3. Golu kehta hai: “Diagonal lagbhag 1.5 × side hai.” Kya yeh sahi hai?
Solution dekho
Nahi. 1.5 × 1.5 = 2.25, jo 2 se zyada hai. Sahi value ≈ 1.414, kyunki 1.414 × 1.414 ≈ 2.
Q4. Ek chaukor kamre mein 100 tiles lagi hain. Double area ke chaukor kamre mein kitni tiles lagengi, aur kya uski side mein poori tiles aayengi?
Solution dekho
200 tiles. Side = 10√2 ≈ 14.14 tiles, jo poora number nahi, isliye kinare pe kuch tiles kaatni padengi!
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🏆 Olympiad Corner
Q1. Ek square ke chaaron sides ke midpoints jodne se andar ek square banta hai. Andar wale ka area, bahar wale ka kitna hissa hai?
Solution dekho
Aadha. Bahar ke square ke 4 kone ke triangles mil ke andar ke square ke barabar hain (agle post ka topic!).
Q2. Area 1 se shuru karke diagonal trick baar-baar lagao. Kitni baar ke baad area pehli baar 1000 se zyada hoga?
Solution dekho
2¹⁰ = 1024. Yaani 10 baar (2⁹ = 512 abhi kam hai).
Q3. Ek square ka diagonal 10 cm hai. Uska area kitna hai?
Solution dekho
Diagonal pe bana square = 10 × 10 = 100 = 2 × area. To area = 50 cm².
Q4. Square ABCD ki side 8 cm. Diagonal AC pe square banaya, phir us naye square ke diagonal pe ek aur. Teesre square ki side?
Solution dekho
Areas: 64 → 128 → 256. Side = √256 = 16 cm.
sirf charumam.com readers ke liye 🔐
💗 CharuMam Motivation
Golu ki pehli soch “side double” thi, aur woh galat nikli. Par usne ginti karke check kiya, aur galti pakad li. 🔍
Yahi asli ganit hai: pehla idea galat ho sakta hai, par jaanchne ki aadat aapko sahi raaste pe le jaati hai. 2800 saal pehle Baudhayana ne bhi yahi kiya. Aap bhi kar sakte ho! 🌟
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🟢 Master Recap
- Side k guna → area k × k guna. Side double = area 4 guna.
- Baudhayana (Śulba-Sūtra 1.9): diagonal pe bana square = asli ka double.
- Saboot: asli square = 2 triangles, diagonal wala = 4 triangles.
- Doubling seedhi: 2 → 4 → 8 → 16 triangles, har baar double.
- Kaagaz activity: 2 squares ka poora kaagaz = 1 double square.
- Diagonal ≈ 1.414 × side (√2 × side).
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👉 Aage Kya?
Double to ho gaya! Ab ulta karenge: square ka aadha area kaise banayein, kaagaz mod ke, aur kyun side aadhi karna phir galat hai! ✂️
Koi doubt? CharuMam AI Tutor se poochho, ya comment mein likho! 💬