Cuboid Pe Sabse Chhota Raasta | Class 8 Maths Book 2 Ch 4

📘 Class 8 · Ganita Prakash Book 2Ch 4 · Exploring Some Geometric ThemesPost 7 / 9⏱ 25–30 min

Cuboid Pe Sabse Chhota Raasta 🐜

IS POST MEIN

Pehle Yeh Aata Hai?Kahani Concept Deep Concept Examples Real Life Galti 10 MCQs FAQs Practice HOTS Olympiad Recap

✅ Pehle Yeh Aata Hai?

Yeh post shuru karne se pehle, in teen sawaalon ko 1 minute mein khud try kijiye:

6² + 7² = ?
√85 kin do poore numbers ke beech?
Cuboid ke kitne faces?

Answers dekho

36 + 49 = 85
9 aur 10 (81 aur 100)
6

Teeno sahi? Badhiya! Theorem yaad karna ho to Ch 2 ka Theorem Se Missing Side Nikalna dekh lijiye. Nets ke liye Post 6.

— 📌 charumam.com

🎬 Kahani: Cheenti Aur Laddoo

Rasoi mein mithai ka ek band dibba rakha tha (6 cm lamba, 4 cm chauda, 3 cm ooncha). Neeche ke ek kone pe ek cheenti thi, aur bilkul ulte (upar ke door wale) kone pe chipka tha ek chhota sa laddoo ka tukda! 🐜🟡

Cheenti dibbe ke andar nahi ja sakti, use sataha pe chal ke jaana hai. Golu bola, “Seedha kone se kone, andar se!” Priya hasi, “Cheenti diwaar tod ke nahi jaayegi!” Sataha pe sabse chhota raasta kaun sa? 🤔

— 📌 charumam.com

📖 Concept: Dibba Kholo, Line Kheencho

Step 1: Sapaat pe sabse chhota raasta

Sapaat kaagaz pe do points ke beech sabse chhota raasta seedhi line hai. Par dibbe ki sataha mudi hui hai, wahan seedhi line kaise?

Step 2: Trick: dibbe ko net mein kholo

Dibbe ko kholo (net banao). Sataha pe jo bhi raasta hai, net pe bhi utna hi lamba raasta banta hai. To net pe A se G tak seedhi line kheencho: wahi sabse chhota!

WHY lambaai same? Kholne mein kaagaz na khinchta hai na sikudta, sirf mudta hai.

Saamne aur upar ka face khola: A se G seedhi line = √(6² + 7²) = √85.

Step 3: Teen alag tareeke se kholo

Kone se ulte kone tak jaane ke liye cheenti do faces paar karti hai. Kaun se do faces, uske 3 vikalp hain:

Kholne ka tareeka	Rectangle	Lambaai
Saamne + upar	6 × (3 + 4) = 6 × 7	√(36 + 49) = √85 ≈ 9.22 cm
Saamne + side	(6 + 4) × 3 = 10 × 3	√(100 + 9) = √109 ≈ 10.44 cm
Side + upar	(6 + 3) × 4 = 9 × 4	√(81 + 16) = √97 ≈ 9.85 cm

Sabse chhota: √85 ≈ 9.22 cm. (Andar se seedha √61 ≈ 7.8 cm hota, par woh mumkin nahi.)

WHY saare check? Alag kholne se alag lambaai aati hai. Sirf ek dekh ke jawab mat do!

Step 4: Line net ke bahar na jaaye

Kabhi kisi net pe kheenchi seedhi line net ke bahar nikal jaati hai (khaali jagah se guzarti hai). Woh asli raasta nahi hai, kyunki cheenti hawa mein nahi chal sakti! Aisa net chhodo aur doosra kholo.

Step 5: Jo raasta net pe seedha nahi, woh chhota nahi

Agar koi raasta net pe toota hua (zig-zag) dikhe, to usse chhota seedha raasta usi net pe mil jaata hai.

— 📌 charumam.com

🔐 Deep Concept: Makdi Aur Makkhi Ki Mashhoor Paheli

Ek kamra (ya dibba) 30 lamba, 12 chauda, 12 ooncha. Makdi ek chhoti deewar ke beech mein, chhat se 1 neeche. Makkhi saamne wali chhoti deewar ke beech mein, farsh se 1 upar.

Seedha socho: makdi upar jaaye (1), chhat paar (30), neeche aaye (11) = 42. Par ek chaalaak tareeke se kamre ko kholo (5 faces se guzarta raasta), to ek right triangle banta hai jiske legs 24 aur 32: d² = 24² + 32² = 576 + 1024 = 1600 → d = 40! 😲

Lambe raaste (zyada faces) se chhota jawab! Isliye har kholne ka tareeka check karna zaroori hai, aur Baudhayana ka theorem har jagah kaam aata hai. 🔐

sirf charumam.com readers ke liye 🔐

✏️ Solved Examples

Example 1 (Easy): Cube (side 4 cm) ke ek kone se ulte kone tak, sataha pe sabse chhota raasta?

Kholo: 4 × (4 + 4) = 4 × 8 rectangle. d = √(16 + 64) = √80 ≈ 8.94 cm.

WHY sahi hai? Cube mein teeno tareeke same aate hain ✅

📌 charumam.com

Example 2 (Medium): Dibba 5 × 3 × 2. Kone se ulte kone tak teeno lambaaiyan aur sabse chhoti?

√(5² + 5²) = √50 ≈ 7.07; √(8² + 2²) = √68 ≈ 8.25; √(7² + 3²) = √58 ≈ 7.62. Sabse chhoti √50 ≈ 7.07.

WHY sahi hai? Sabse lambi side ko akela rakho, baaki do jodo: (5, 3 + 2) ✅

📌 charumam.com

Example 3 (NCERT style): Cheenti dibbe (8 × 4 × 2) ke ek face ke beech mein; laddoo saamne wale face ke beech mein. Socho kaise khologe.

Dono faces ke beech wala face (ya faces) kholo taaki teeno ek line mein aa jaayein; phir seedhi line kheencho aur check karo ki net ke andar hai. Har sambhav kholne ki lambaai gino, sabse chhoti lo.

WHY sahi hai? Yahi NCERT ka tareeka: net + seedhi line + saare vikalp ✅

📌 charumam.com

Priya ka mantra: “Kholo, kheencho, check karo, sabse chhota chuno!” 🎉

— 📌 charumam.com

🏠 Real Life Mein

🛣️ Pahaad pe sadak: mudi sataha pe sabse chhota raasta.
✈️ Hawai jahaaz: gol prithvi pe sabse chhota raasta naksha pe seedha nahi dikhta.
📦 Rassi/tape: dibbe pe sabse kam tape lagaana.
🐜 Cheentiyaan sach mein khaane tak chhote raaste dhoondh leti hain!

— 📌 charumam.com

🧭 Kab Use Karein, Kab Nahi

Situation	Kya karein	Kyun
Sataha pe raasta	✅ Net pe seedhi line	Lambaai same rehti hai
Andar se seedha	❌ Mumkin nahi (cheenti ke liye)	Sataha pe hi chalna hai
Ek hi net check kiya	❌ Adhoora	Alag nets, alag lambaai
Line net ke bahar	❌ Raasta nahi	Khaali jagah se guzarti hai

— 📌 charumam.com

❌→✅ Worked Mistake

Sawaal: Dibba 6 × 4 × 3, kone se ulte kone (sataha pe).

❌ Galat: “Saamne + side kholo: √(10² + 3²) = √109 ≈ 10.44. Yahi jawab.”

Galti kya hai? Sirf ek tareeka dekha. Doosre tareeke se chhota mil sakta hai ❌

✅ Sahi: Teeno check: √85, √109, √97. Sabse chhota √85 ≈ 9.22 cm.

Yaad rakho: Har kholne ka tareeka, phir minimum.

sirf charumam.com pe milenge!

📝 10 MCQs

Q1. Sapaat pe sabse chhota raasta?

(a) Gol

(b) Seedhi line

(c) Zig-zag

(d) Koi bhi

Solution dekho

Sahi jawab: (b)

(a) galat: Nahi.
(b) sahi: Sahi ✅
(c) galat: Lamba.
(d) galat: Nahi.

Q2. Sataha ka raasta net pe?

(a) Chhota ho jaata

(b) Utna hi lamba

(c) Lamba ho jaata

(d) Gayab

Solution dekho

Sahi jawab: (b)

(a) galat: Nahi.
(b) sahi: Kaagaz sirf mudta hai ✅
(c) galat: Nahi.
(d) galat: Nahi.

Q3. 6 × 4 × 3 dibbe mein sabse chhota (kone se kone)?

(a) √85

(b) √109

(c) √97

(d) √61

Solution dekho

Sahi jawab: (a)

(a) sahi: Sahi ✅
(b) galat: Lamba.
(c) galat: Beech ka.
(d) galat: Andar se, mumkin nahi.

Q4. √61 kyun nahi?

(a) Andar se guzarta hai

(b) Bahut lamba

(c) Galat hisaab

(d) Net pe nahi

Solution dekho

Sahi jawab: (a)

(a) sahi: Cheenti andar nahi ja sakti ✅
(b) galat: Chhota hai, par asambhav.
(c) galat: Hisaab sahi.
(d) galat: Sataha pe hai hi nahi.

Q5. Cube (side 4) kone se kone sataha pe?

(a) √48

(b) √80

(c) 8

(d) 12

Solution dekho

Sahi jawab: (b)

(a) galat: Andar ka diagonal.
(b) sahi: 4 × 8 rectangle ✅
(c) galat: Nahi.
(d) galat: Nahi.

Q6. Net pe line bahar jaaye to?

(a) Wahi jawab

(b) Doosra net try karo

(c) Line chhoti karo

(d) Kuch nahi

Solution dekho

Sahi jawab: (b)

(a) galat: Asli raasta nahi.
(b) sahi: Sahi ✅
(c) galat: Galat.
(d) galat: Galat.

Q7. Makdi-makkhi paheli ka jawab?

(a) 42

(b) 40

(c) 24

(d) 32

Solution dekho

Sahi jawab: (b)

(a) galat: Seedha raasta, sabse chhota nahi.
(b) sahi: √(24² + 32²) ✅
(c) galat: Ek leg.
(d) galat: Doosra leg.

Q8. 24² + 32² = ?

(a) 1600

(b) 1024

(c) 576

(d) 56

Solution dekho

Sahi jawab: (a)

(a) sahi: Sahi ✅
(b) galat: Sirf 32².
(c) galat: Sirf 24².
(d) galat: Jod nahi.

Q9. 5 × 3 × 2 ka sabse chhota tareeka?

(a) (5, 3+2)

(b) (3, 5+2)

(c) (2, 5+3)

(d) Koi bhi same

Solution dekho

Sahi jawab: (a)

(a) sahi: √50 ✅
(b) galat: √58
(c) galat: √68
(d) galat: Alag hain.

Q10. Zig-zag raasta net pe?

(a) Sabse chhota

(b) Usse chhota seedha milta

(c) Hamesha sahi

(d) Mumkin nahi

Solution dekho

Sahi jawab: (b)

(a) galat: Nahi.
(b) sahi: Sahi ✅
(c) galat: Nahi.
(d) galat: Mumkin hai par lamba.

— 📌 charumam.com

❓ FAQs (Bachchon Ke Asli Sawaal)

1. Net kaise chunein?

Jin faces se raasta guzarta hai unhe ek saath kholo. Kai tareeke try karo.

2. Kya cube mein bhi alag tareeke se alag jawab?

Kone se kone ke liye cube mein teeno same. Par doosre points ke liye alag ho sakte hain.

3. Shortcut rule hai?

Kone se ulte kone ke liye: sabse lambi side akeli, baaki do jodo: √(l² + (b + h)²) jahan l sabse bada.

4. Andar wala diagonal kya hai?

√(l² + b² + h²): dibbe ke andar se. Cheenti ke kaam ka nahi.

5. Makdi wala raasta kitne faces se?

5 faces se, phir bhi 40 < 42!

6. Theorem kahan aaya?

Net pe right triangle banta hai; hypotenuse = raasta.

— 📌 charumam.com

🏋️ Practice Zone (12 Sawaal)

Q1. Easy Kaagaz 4 cm × 3 cm ka diagonal?

Solution dekho

5 cm.

Q2. Easy Cube side 1, kone se kone sataha pe?

Solution dekho

√(1 + 4) = √5 ≈ 2.24.

Q3. Easy 8 × 6 rectangle ka diagonal?

Solution dekho

10.

Q4. Easy Net pe raasta seedha kyun?

Solution dekho

Sapaat pe seedhi line sabse chhoti.

Q5. Medium Dibba 12 × 5 × 4, kone se kone: teeno tareeke aur sabse chhota?

Solution dekho

√(12² + 9²) = 15, √(5² + 16²) ≈ 16.76, √(4² + 17²) ≈ 17.46. Sabse chhota 15.

Q6. Medium Dibba 8 × 4 × 2, kone se kone sabse chhota?

Solution dekho

√(8² + 6²) = 10.

Q7. Medium Cube side 3, kone se kone?

Solution dekho

√(9 + 36) = √45 ≈ 6.71.

Q8. NCERT style 6 × 4 × 3 ke teeno raaste likho.

Solution dekho

√85, √109, √97.

Q9. NCERT style Makdi-makkhi mein 24 aur 32 legs; raasta?

Solution dekho

40.

Q10. Medium Andar ka diagonal 6 × 4 × 3?

Solution dekho

√61 ≈ 7.81.

Q11. Medium Kya andar ka diagonal hamesha sataha ke raaste se chhota?

Solution dekho

Haan, andar seedhi line sabse chhoti.

Q12. NCERT style Dibba 2 × 2 × 1, kone se kone sataha pe?

Solution dekho

√(2² + 3²) = √13 ≈ 3.61.

— 📌 charumam.com

🧠 HOTS (Sochne Wale Sawaal)

Q1. Kyun “sabse lambi side akeli, baaki do jodo” sabse chhota deta hai?

Solution dekho

(a + b)² + c² = a² + b² + c² + 2ab. 2ab sabse chhota tab, jab a, b sabse chhoti sides hon.

Q2. Laddoo upar ke face ke beech mein, cheenti neeche ke ek kone pe (dibba 6 × 4 × 3). Ek raasta nikaalo.

Solution dekho

Saamne + upar kholo: horizontal 3, vertical 3 + 2 = 5 → √(9 + 25) = √34 ≈ 5.83. (Doosre tareeke bhi check karo.)

Q3. Gol (cylinder) dibbe pe sabse chhota raasta?

Solution dekho

Cylinder ka net rectangle hai; usi pe seedhi line (jaise seedhi ke chaaron taraf lipti patti).

Q4. Do raaston ki lambaai barabar kab?

Solution dekho

Cube jaisi barabar sides mein.

sirf charumam.com pe milenge!

🏆 Olympiad Corner

Q1. Kamra 30 × 12 × 12 mein makdi-makkhi (1 upar/neeche): seedha raasta 42, chaalaak 40. Kitna bachaaya?

Solution dekho

2 units.

Q2. Dibba a × b × c (a ≥ b ≥ c), kone se kone sabse chhota?

Solution dekho

√(a² + (b + c)²).

Q3. Cube side 10, kone se kone sataha pe (approx)?

Solution dekho

√(100 + 400) = √500 ≈ 22.36.

Q4. Andar diagonal aur sataha raaste ka farak, cube side 1?

Solution dekho

√5 − √3 ≈ 2.236 − 1.732 = 0.50.

sirf charumam.com readers ke liye 🔐

💗 CharuMam Motivation

Cheenti diwaar tod nahi sakti, par sahi raasta chun sakti hai. 🐜

Hum bhi kai baar seedha raasta nahi le paate. Par ruk ke saare vikalp dekhna, aur sabse achha chunna, yahi samajhdaari hai. Jaldbaazi ka pehla raasta hamesha sabse achha nahi hota! 🧭

— 📌 charumam.com

🟢 Master Recap

Sataha pe raasta = net pe utni hi lambi line.
Net pe seedhi line = sabse chhota (agar net ke andar rahe).
Har kholne ka tareeka check karo; lambaaiyan alag.
6 × 4 × 3 kone se kone: √85, √97, √109 → √85 ≈ 9.22.
Makdi-makkhi: 24² + 32² = 40² (42 se chhota!).

— charumam.com 🎯

👉 Aage Kya?

Ab solids ki parchhaai! Agle post mein projections: front, top aur side view, aur shadows ka raaz. 🔦

← Pichla: Post 6 — Nets: Kaagaz Se Solid 📚 Chapter 4 ke saare posts Agla: Post 8 — Projections: Front, Top Aur Side View →

Koi doubt? CharuMam AI Tutor se poochho, ya comment mein likho! 💬

✅ Pehle Yeh Aata Hai?

🎬 Kahani: Cheenti Aur Laddoo

📖 Concept: Dibba Kholo, Line Kheencho

Step 1: Sapaat pe sabse chhota raasta

Step 2: Trick: dibbe ko net mein kholo

Step 3: Teen alag tareeke se kholo

Step 4: Line net ke bahar na jaaye

Step 5: Jo raasta net pe seedha nahi, woh chhota nahi

🔐 Deep Concept: Makdi Aur Makkhi Ki Mashhoor Paheli

✏️ Solved Examples

🏠 Real Life Mein

🧭 Kab Use Karein, Kab Nahi

❌→✅ Worked Mistake

📝 10 MCQs

❓ FAQs (Bachchon Ke Asli Sawaal)

🏋️ Practice Zone (12 Sawaal)

🧠 HOTS (Sochne Wale Sawaal)

🏆 Olympiad Corner

💗 CharuMam Motivation

🟢 Master Recap

👉 Aage Kya?

You may also like