Government exams jaise SSC CGL, CHSL, MTS, Banking (IBPS, SBI), aur Railways mein Mensuration ek mandatory section hai. Geometry ke baad, Mensuration hi woh topic hai jisme students sabse zyada confuse hote hain. Is topic ko do parts mein baata gaya hai: Mensuration 2D (Area & Perimeter) aur Mensuration 3D (Volume & Surface Area).
Is exhaustive guide mein hum dono 2D aur 3D shapes ko detail mein cover karenge taaki aapka ek bhi mark na kate.
Part 1: Mensuration 2D (Area & Perimeter)
1. Rectangle (Ayat)
Rectangle wo shape hai jiski aamne-saamne ki sides barabar aur parallel hoti hain, aur charo angles 90 degree ke hote hain.
- Length (l): Lambai
- Breadth (b): Chaudai
Perimeter = 2(l + b)
Diagonal = √(l² + b²)
2. Square (Varg)
Square ek special rectangle hai jiski charo sides barabar hoti hain.
- Side (a): Bhuja
Perimeter = 4a
Diagonal = a√2
3. Circle (Vrat)
Circle ek round shape hai jisme center se boundary tak ki doori (Radius) same hoti hai.
- Radius (r): Trijya
- Diameter (d): Vyas (d = 2r)
Circumference (Perimeter) = 2πr
4. Triangle (Tribhuj)
Triangle ke area ka basic formula hai:
Equilateral Triangle (Sambahu Tribhuj): Jiski teeno sides barabar hon.
Part 2: Mensuration 3D (Volume & Surface Area)
1. Cylinder (Belan)
Cylinder ek aisi 3D shape hai jiska base aur top dono circular (vratakar) hote hain aur unke beech ek fixed height hoti hai.
- Radius (r): Base circle ka radius.
- Height (h): Top aur bottom ke beech ki vertical doori.
TSA = 2πr(r + h)
Volume = πr²h
Exam Pro-Tip: Jab bhi kisi wire (taar) ki baat ho, toh yaad rakhen ki wire hamesha ek Cylinder hota hai. Wire ki length hi uski height (h) ban jati hai.
2. Cone (Shanku)
Cone ek aisi shape hai jiska base toh circular hota hai, lekin upar jate-jate woh ek point (Apex) par mil jata hai. Example: Ice-cream cone ya Birthday cap.
- Slant Height (l): l = √(r² + h²)
TSA = πr(l + r)
Volume = (1/3)πr²h
3. Sphere and Hemisphere (Gola aur Ardh-Gola)
Sphere ek perfect symmetrical shape hai. Example: Football.
Sphere Surface Area = 4πr²
Hemisphere (Ardh-Gola):
TSA = 3πr²
Volume = (2/3)πr³
Comparison Table for Quick Revision
| Shape | Curved/Lateral Area | Total Surface Area | Volume |
|---|---|---|---|
| Cylinder | 2πrh | 2πr(r+h) | πr²h |
| Cone | πrl | πr(l+r) | (1/3)πr²h |
| Sphere | 4πr² | 4πr² | (4/3)πr³ |
| Hemisphere | 2πr² | 3πr² | (2/3)πr³ |
Most Important Exam Tricks (Shortcuts) ⚡
A. The π Trick (Divisibility by 11)
Mensuration ke adhiktat formulas mein π (22/7) hota hai. Iska matlab hai ki aapka answer hamesha 11 ka multiple hoga. Options check karein aur jo 11 se divide ho wahi answer choose karein.
B. Melting and Recasting
Jab ek shape ko pighla kar dusri shape banayi jati hai, toh hamesha Volume Equal hota hai.
C. Ratio Method for Percentage Change
Agar radius 10% badh jaye, toh area ya volume par kya asar padega?
- For Area (2D): 10 + 10 + (10×10)/100 = 21% Increase
- For Volume (3D): 33.1% Increase (Successive Percentage)
Conclusion: Practice is Key
Mensuration 2D aur 3D dono hi scoring topics hain. Agar aapko formulas yaad hain aur calculation tricks aati hain, toh aap seconds mein answer nikal sakte hain. Ikkish Prep ka maqsad aapko inhi complex topics ko aasan bhasha mein samjhana hai.
Apni calculation speed badhane ke liye hamare Speed Math Tool ka use karein.