✅ Pehle Yeh Aata Hai?
Yeh post shuru karne se pehle, in teen sawaalon ko 1 minute mein khud try kijiye:
- Regular hexagon ki kitni bhujaayein, aur kya sab barabar?
- Cube ke kitne kinaare (edges)?
- Ek cube kone pe khada karke upar se dekho, kya dikhta hai?
Answers dekho
- 6, sab barabar
- 12
- Regular hexagon
Teeno sahi? Shaabaash! Projection yaad karna ho to Post 8 dekh lijiye.
— 📌 charumam.com
🎬 Kahani: Golu Ka Tetris
Golu phone pe Tetris jaisa game khel raha tha: chaar-chaar squares ki 5 shapes (seedhi I, chaukor O, T, L aur S/Z) neeche girti hain.
Priya ne kaha, “Socho ye squares nahi, cubes hain! Inhe kaagaz pe 3D jaisa kaise banaoge?”
Golu ne seedhe kaagaz pe koshish ki, cube tedha-medha ban gaya. 😅 Priya ne ek khaas dotted kaagaz nikaala jisme dots tikone pattern mein the: isometric grid. “Isse cube ekdum sahi banega!” 📐
— 📌 charumam.com
📖 Concept: Barabar Naap Wala 3D Chitra
Step 1: Isometric ka matlab
Isometric = Greek mein “barabar naap”. Cube ko aise ghumaao (kone pe khada) ki projection mein uske saare kinaare barabar lambe dikhein. Projection ek regular hexagon banta hai (Post 8).
WHY yeh khaas? Barabar lambaai ka matlab: drawing pe seedha gin ke naap sakte ho. Engineers isi liye use karte hain.
Step 2: Isometric grid
Hexagons se poora kaagaz bharo, to dots ka ek tikona jaal banta hai: isometric grid. Is pe teen dishaayein hain:
- Oonchaai (height): seedhi khadi (|).
- Lambaai (length): ek tirchhi disha (daayein-neeche).
- Gehraai (depth): doosri tirchhi disha (baayein-neeche).
Step 3: Ek cube banao
- Ek dot se 1 unit upar (height) line.
- Neeche ke dot se 1 unit lambaai aur 1 unit gehraai ki disha mein lines.
- Inke siron se upar 1 unit, phir upar ke kinaare jodo.
- Teen faces (upar, daayein, baayein) dikhenge; chhupe kinaare halke/dotted ya mitaa do.
WHY kinaara-kinaara? Har kinaara teen dishaon mein se kisi ek mein hota hai. Units gino aur us disha mein chalo.
Step 4: 4 cubes ki shapes
Ek line mein 4 cubes: lambaai ki disha mein, gehraai ki disha mein, ya oonchaai mein (meenar). Har baar shape same, drawing alag! Tetris ke 5 shapes (I, O, T, L, S) bhi aise hi banao.
Step 5: Kyun kaam karta hai?
Isometric projection mein parallel kinaare parallel dikhte hain aur barabar lambaaiyaan barabar. Isliye grid pe gin ke banaaya chitra asli shape ka sahi roop hai.
— 📌 charumam.com
🔐 Deep Concept: Asambhav Triangle Ka Jaadu
Isometric drawing mein gehraai (doori) ka pata nahi chalta: paas aur door ki cheezein same size ki dikhti hain. Isi ka fayda utha ke kalaakaar “asambhav triangle” (Penrose triangle) banaate hain: teen seedhi pattiyaan jo 90° pe mudti hain aur ek triangle jaisi lagti hain!
Asli cubes se ise poora banaana mumkin nahi; ek kone pe pattiyaan asal mein door hoti hain, bas ek khaas disha se judi dikhti hain. Isliye ise optical illusion kehte hain. 🔺🌀🔐
sirf charumam.com readers ke liye 🔐
✏️ Solved Examples
Example 1 (Easy): Isometric grid pe ek cube ke kitne faces dikhte hain?
3 (upar, aur do side). Baaki 3 peeche/neeche chhupe.
WHY sahi hai? Kone se dekhne pe 3 faces ✅
📌 charumam.com
Example 2 (Medium): 2 × 2 × 1 (4 cubes ki chaukor patti) ke kitne kinaare isometric drawing mein dikhte hain (bahari)?
Shape ek cuboid 2 × 2 × 1 hai; bahar se 9 kinaare dikhte hain (3 chhupe), har lamba kinaara 2 unit.
WHY sahi hai? Cuboid jaisa hi: 12 mein se 9 dikhte ✅
📌 charumam.com
Example 3 (NCERT style): 4 cubes ko face se face chipkaane ke tetris ke 5 ke alaawa aur tareeke?
Haan! 3D mein 3 aur: do “screw” (ek doosre ki mirror) aur ek “tripod” (ek cube pe teen dishaon mein ek-ek). Kul 8 (mirror alag gino to).
WHY sahi hai? Ye sapaat (flat) nahi hain, isliye 2D tetris mein nahi ✅
📌 charumam.com
Golu ka formula: “Upar seedha, baaki do tirchhe. Gino aur kheencho!” 🎉
— 📌 charumam.com
🏠 Real Life Mein
- 🏭 Engineering drawing: machine parts isometric mein.
- 🎮 Video games: kai purane aur naye games isometric view mein (sheher, khet wale games).
- 🏠 Furniture assembly guide: almari ke hisse isometric chitron mein.
- 🎨 Kala: M.C. Escher ki asambhav seedhiyaan aur jharne.
— 📌 charumam.com
🧭 Kab Use Karein, Kab Nahi
| Situation | Kya karein | Kyun |
|---|---|---|
| Cube ke saare kinaare barabar dikhane | ✅ Isometric | Barabar naap |
| Upar-neeche | ✅ Seedhi (|) line | Height axis |
| Lambaai/gehraai | ✅ Tirchhi grid lines | Do slanted axes |
| Door ki cheez chhoti dikhaani | ❌ Isometric nahi | Isometric mein size nahi badalta |
— 📌 charumam.com
❌→✅ Worked Mistake
Sawaal: 3 cubes ki meenar isometric mein.
❌ Galat: Oonchaai ki line tirchhi disha mein kheench di.
Galti kya hai? Oonchaai hamesha seedhi khadi (|). Tirchhi dishaayein sirf lambaai aur gehraai ke liye ❌
✅ Sahi: Ek cube ka hexagon, uske upar seedha 1 unit, phir ek aur. Kul oonchaai 3 unit, seedhi.
Yaad rakho: Height = |, length/depth = tirchhe.
sirf charumam.com pe milenge!
📝 10 MCQs
Q1. Isometric ka matlab?
Solution dekho
Sahi jawab: (a)
- (a) sahi: Greek: iso + metric ✅
- (b) galat: Nahi.
- (c) galat: Nahi.
- (d) galat: Nahi.
Q2. Cube kone pe, projection?
Solution dekho
Sahi jawab: (b)
- (a) galat: Seedha.
- (b) sahi: Sahi ✅
- (c) galat: Nahi.
- (d) galat: Nahi.
Q3. Isometric grid pe height ki disha?
Solution dekho
Sahi jawab: (b)
- (a) galat: Length/depth.
- (b) sahi: Sahi ✅
- (c) galat: Nahi.
- (d) galat: Nahi.
Q4. Isometric cube ke dikhte faces?
Solution dekho
Sahi jawab: (c)
- (a) galat: Kam.
- (b) galat: Kam.
- (c) sahi: Sahi ✅
- (d) galat: Saare nahi.
Q5. Tetris ki sapaat shapes (4 squares)?
Solution dekho
Sahi jawab: (c)
- (a) galat: Kam.
- (b) galat: Kam.
- (c) sahi: I, O, T, L, S ✅
- (d) galat: 3D mein 8.
Q6. 4 cubes ki kul shapes (3D, mirror alag)?
Solution dekho
Sahi jawab: (c)
- (a) galat: Sapaat wali.
- (b) galat: Mirror same gino to.
- (c) sahi: Sahi ✅
- (d) galat: Zyada.
Q7. Parallel kinaare isometric mein?
Solution dekho
Sahi jawab: (a)
- (a) sahi: Sahi ✅
- (b) galat: Nahi.
- (c) galat: Nahi.
- (d) galat: Nahi.
Q8. Asambhav triangle kyun dikhta hai?
Solution dekho
Sahi jawab: (a)
- (a) sahi: Sahi ✅
- (b) galat: Ganit sahi hai.
- (c) galat: Nahi.
- (d) galat: Asli mein nahi ban sakta.
Q9. Isometric mein door ki cheez?
Solution dekho
Sahi jawab: (b)
- (a) galat: Perspective mein hota.
- (b) sahi: Sahi ✅
- (c) galat: Nahi.
- (d) galat: Nahi.
Q10. Chhupe kinaare kaise?
Solution dekho
Sahi jawab: (b)
- (a) galat: Nahi.
- (b) sahi: Sahi ✅
- (c) galat: Zaroori nahi.
- (d) galat: Dhyaan se.
— 📌 charumam.com
❓ FAQs (Bachchon Ke Asli Sawaal)
1. Isometric grid kahan milega?
Stationery shop pe “isometric paper”, ya internet se print. Khud bhi tikona dot pattern bana sakte ho.
2. Perspective drawing se farak?
Perspective mein door ki cheez chhoti, isometric mein sab same naap.
3. Teen dishaayein yaad kaise?
Ek khadi, do tirchhi (Y ulta jaisa ⅄).
4. Kya har 3D shape isometric mein ban sakti?
Haan, par gol cheezein (cylinder) ellipse ban ke dikhti hain.
5. Tripod tetracube kya hai?
Ek cube jisse teen dishaon mein ek-ek cube juda (kone jaisa).
6. Asambhav triangle kisne banaaya?
Oscar Reutersvärd (1934) aur baad mein Penrose ne mashhoor kiya.
— 📌 charumam.com
🏋️ Practice Zone (12 Sawaal)
Q1. Easy Ek cube draw karo (isometric).
Solution dekho
Hexagon + andar Y jaisi 3 lines.
Q2. Easy Height kis disha?
Solution dekho
Seedhi khadi (|).
Q3. Easy 4 cubes ek line, lambaai disha mein draw karo.
Solution dekho
4 unit lambi patti.
Q4. Easy Kitne faces dikhte (ek cube)?
Solution dekho
3.
Q5. Medium 2 × 2 × 2 cube (8 cubes) ke dikhte chhote faces?
Solution dekho
Har bade face pe 4, 3 face → 12.
Q6. Medium Tetris ke 5 shapes cubes bana ke draw karo.
Solution dekho
I, O, T, L, S, har cube ko isometric mein.
Q7. Medium 3 × 3 × 3 cube ke dikhte chhote faces?
Solution dekho
3 × 9 = 27.
Q8. NCERT style Kya 4 cubes 5 ke alaawa joda ja sakte hain? Dikhao.
Solution dekho
Haan: 2 screw + tripod (3D).
Q9. NCERT style Asambhav triangle ka front, top, side (cubes se) kya hoga?
Solution dekho
Har view mein L jaisi pattiyaan; asli mein teeno ek saath mel nahi khaate.
Q10. Medium Isometric pe 2 × 1 × 3 cuboid ke dikhte kinaare?
Solution dekho
9.
Q11. Medium Meenar (3 cubes) ka top view?
Solution dekho
Ek square.
Q12. NCERT style Isometric kaam kyun karta hai?
Solution dekho
Parallel → parallel, barabar → barabar.
— 📌 charumam.com
🧠 HOTS (Sochne Wale Sawaal)
Q1. Kya isometric drawing se pata chalta hai ki kaunsa cube aage hai?
Solution dekho
Hamesha nahi; isliye kai baar ek se zyada 3D shapes ek hi drawing deti hain.
Q2. n × n × n cube ke dikhte chhote faces?
Solution dekho
3n² (teen faces, har ek n²).
Q3. Kya cube ka isometric chitra 3 rhombus se bana hai?
Solution dekho
Haan! Teeno faces rhombus (60°–120°) dikhte hain.
Q4. Isometric pe circle kaisa dikhega?
Solution dekho
Ellipse (anda jaisa).
sirf charumam.com pe milenge!
🏆 Olympiad Corner
Q1. 3 × 3 × 3 cube ke saare dikhte chhote cubes (kam se kam ek face dikhe)?
Solution dekho
Teen faces: 27 − 8 (jo kone ke andar chhupe: 2 × 2 × 2 peeche) = 19.
Q2. Isometric chitra mein cube ke hexagon ka andar wala bindu kya hai?
Solution dekho
Cube ka saamne wala kona (ya peeche wala, dono ek pe aate).
Q3. Teen rhombus ke angles?
Solution dekho
60° aur 120°.
Q4. 4 cubes ki shapes jo plane mein na ho sakein?
Solution dekho
3 (2 screw + 1 tripod).
sirf charumam.com readers ke liye 🔐
💗 CharuMam Motivation
Golu ka pehla cube tedha bana, par sahi grid aur tareeke se dusra ekdum perfect! 📐
Sahi saadhan aur sahi tareeka seekhne mein sharm nahi. Poora chapter khatam! Aapne fractals se lekar 3D drawing tak sab seekh liya. Badhaai ho! 🎓
— 📌 charumam.com
🟢 Master Recap
- Isometric = barabar naap; cube kone pe → regular hexagon.
- Grid: height seedhi, length aur depth do tirchhi dishaayein.
- Kinaara-kinaara gin ke banao; chhupe kinaare halke.
- Parallel → parallel, barabar → barabar.
- 4 cubes: 5 sapaat (tetris) + 3 3D = 8.
- Asambhav triangle: gehraai ka pata na chalne se illusion.
— charumam.com 🎯
👉 Aage Kya?
🎉 Chapter 4 poora! Chapter hub se koi bhi post dobara dekho.
Koi doubt? CharuMam AI Tutor se poochho, ya comment mein likho! 💬