Surface area and volume of all above 3D shapes

1.2.14 · Maths › Basic Geometry

Core Principle: 2D Se Banao

Saare 3D formulas do ideas se aate hain:

Volume: Height ke saath cross-sectional areas ko integrate karo
Surface Area: Shape ko 2D pieces mein unfold karo aur unke areas ka sum karo

1. Cube

Derivation

Volume:

Cube square slices ka stack hai, har ek ka area $a^{2}$ hai
Inhe height $a$ tak stack karo: $V = base area \times height = a^{2} \cdot a = a^{3}$

Surface Area:

Cube ke 6 faces hote hain, har ek $a$ side ka square hai
Har face ka area $a^{2}$ hai
Total: $S A = 6 a^{2}$

2. Cuboid (Rectangular Prism)

Derivation

Volume:

Base ek rectangle hai: area = $l \times w$
Height $h$ tak stack karo: $V = l \times w \times h$

Surface Area:

6 faces 3 pairs mein aate hain: $(l \times w)$ top/bottom, $(l \times h)$ front/back, $(w \times h)$ left/right
Total area: $S A = 2 (l w + l h + w h)$

3. Cylinder

Derivation

Volume:

Base ek circle hai: area = $π r^{2}$
Height $h$ tak stack karo: $V = π r^{2} h$

Surface Area:

Do circular ends: $2 \times π r^{2}$
Curved surface: ek rectangle mein unroll karo jiska width = circle ki circumference = $2 π r$ , height = $h$
- Rectangle ka area: $2 π r \times h$
Total: $S A = 2 π r^{2} + 2 π r h = 2 π r (r + h)$

4. Cone

Derivation

Volume:

Cone same base aur height wale cylinder ka $\frac{1}{3}$ hota hai
$\frac{1}{3}$ kyun? Jaise hum cone mein upar jaate hain, circular slices zero ho jaati hain. Integration (ya Cavalieri's principle) yeh ratio dikhata hai. $V = \frac{1}{3} π r^{2} h$

Surface Area:

Base: $π r^{2}$
Curved surface: radius $l$ aur arc length $2 π r$ wale circle ke ek sector mein unfold hota hai
- Sector area: $\frac{arc length}{2 π l} \times π l^{2} = \frac{2 π r}{2 π l} \times π l^{2} = π r l$
Total: $S A = π r^{2} + π r l = π r (r + l)$

5. Sphere

Derivation

Volume:

Socho sphere ko horizontally slice kar rahe ho. Har slice ek circle hai.
Center se $y$ height par, slice ka radius $r^{2} - y^{2}$ hai (Pythagorean theorem)
Slice area: $π (r^{2} - y^{2})$
$- r$ se $r$ tak integrate karo: $V = \int_{- r}^{r} π (r^{2} - y^{2}) d y = π [r^{2} y - \frac{y ^{3}}{3}]_{- r}^{r} = π (2 r^{3} - \frac{2 r ^{3}}{3}) = \frac{4}{3} π r^{3}$

Surface Area:

Archimedes ki insight: Sphere ko ek cylinder ( $r$ radius, $2 r$ height) mein wrap karo.
Sphere ki surface area cylinder ki lateral area ke barabar hoti hai: $2 π r \times 2 r = 4 π r^{2}$
Ise calculus se rigorously prove kiya ja sakta hai ( $x^{2} + y^{2} = r^{2}$ ko revolve karke).

6. Hemisphere

Derivation

Volume:

Sphere ke volume ka aadha: $V = \frac{1}{2} \times \frac{4}{3} π r^{3} = \frac{2}{3} π r^{3}$

Surface Area:

Curved surface: sphere ka aadha = $2 π r^{2}$
Flat base: $π r^{2}$
Total: $S A = 2 π r^{2} + π r^{2} = 3 π r^{2}$

7. Pyramid (Square Base)

Derivation

Volume:

Koi bhi pyramid $\frac{1}{3}$ × base area × height hota hai $V = \frac{1}{3} a^{2} h$

Surface Area:

Base: $a^{2}$
Chaar triangular faces: har ek ka base $a$ aur height $l$ (slant height) hai, toh area $\frac{1}{2} a l$
Total: $S A = a^{2} + 4 \times \frac{1}{2} a l = a^{2} + 2 a l = a (a + 2 l)$

Comparison Table

Shape	Volume Formula	Surface Area Formula
Cube	$a^{3}$	$6 a^{2}$
Cuboid	$l w h$	$2 (l w + l h + w h)$
Cylinder	$π r^{2} h$	$2 π r (r + h)$
Cone	$\frac{1}{3} π r^{2} h$	$π r (r + l)$
Sphere	$\frac{4}{3} π r^{3}$	$4 π r^{2}$
Hemisphere	$\frac{2}{3} π r^{3}$	$3 π r^{2}$
Pyramid	$\frac{1}{3} a^{2} h$	$a (a + 2 l)$

Common mistake Common Errors

Galti 1: Radius aur diameter mein confusion

Kyun sahi lagta hai: Dono circle measurements hain, mix up karna aasaan hai.
Fix: Hamesha identify karo ki kya diya gaya hai. Yaad rakho $d = 2 r$ . Agar problem mein "diameter 10 cm" likha hai, toh formulas mein $r = 5$ use karo.

Galti 2: Cone/pyramid lateral area ke liye slant height ki jagah height use karna

Kyun sahi lagta hai: Height zyada obvious lagti hai, aur hum ise volume ke liye use karte hain.
Fix: Lateral surface slant ke saath "wrap around" karta hai. Hamesha pehle Pythagorean theorem se $l$ calculate karo.

Galti 3: Cones aur pyramids ke liye $\frac{1}{3}$ factor bhool jaana

Kyun sahi lagta hai: Poora cylinder/prism formula use karne ka mann karta hai.
Fix: Visualize karo: cones/pyramids ek point tak taper hote hain, bahut kam hold karte hain. Hamesha $\frac{1}{3}$ se multiply karo.

Galti 4: Curved aur flat surfaces ko galat tarike se add karna

Kyun sahi lagta hai: "Total" aur "lateral" surface area mein mix up ho jaata hai.
Fix: Ek "unfolded" net draw karo. Har piece alag count karo.

Recall Feynman Technique: Ek 12-Saal Ke Bacche Ko Samjhao

Socho tum jaanna chahte ho ek bottle mein kitna paani bhar sakta hai (yeh volume hai) aur ise kitna plastic wrap karta hai (yeh surface area hai).

Ek box (cuboid) ke liye: Layers ko pancakes ki tarah stack karo—length × width ek pancake ka area deta hai, saare pancakes ke liye height se multiply karo. Ise wrap karne ke liye, tumhe chhe rectangles chahiye (top, bottom, chaar sides).

Ek cylinder (jaise ek can) ke liye: Base ek circle hai. Circles ko height tak stack karo. Ise wrap karne ke liye, tumhe do circle lids plus ek label chahiye jo uske around jaata hai (label unroll karo—yeh ek rectangle hai!).

Ek cone (ice cream cone) ke liye: Yeh ek cylinder jaisa hai lekin ek point tak press kiya gaya hai, isliye sirf $\frac{1}{3}$ utna hi hold karta hai. Wrapper zyada tricky hai—jab unfold karte hain toh yeh ek pie-slice shape hai. Ek ball (sphere) ke liye: Yeh magic hai! Archimedes ne discover kiya tha ki agar tum ek ball ko ek cylinder ke andar rakhte ho (same radius aur height = diameter), toh ball ki surface exactly cylinder ke curved part se match karti hai. Volume ke liye, yeh $\frac{4}{3} π r^{3}$ hai—dekhna mushkil hai, lekin saari circular slices add karne se aata hai.

Key idea: Volume = layers stack karo. Surface area = unfold karo aur pieces add karo.

Connections

Pythagorean Theorem — slant heights nikalne ke liye use hota hai
Area of2D Shapes — har 3D surface 2D pieces se bani hoti hai
Integration and Calculus — sphere, cone volumes ki rigorous derivation
Dimensional Analysis — kyun volume $r^{3}$ ke saath scale hota hai, area $r^{2}$ ke saath
Similar Solids — saare dimensions ko $k$ se scale karne par volume $k^{3}$ se multiply hota hai
Density and Mass — $mass = density \times volume$
Real-World Applications — packaging, architecture, engineering design

#flashcards/maths

Cube ke volume ka formula kya hai? :: $V = a^{3}$ jahan $a$ side length hai.

l

w

h

dimensions wale cuboid ka total surface area kya hai?

S A = 2 (l w + l h + w h)

Cylinder ka curved surface area kya hota hai?

C S A = 2 π r h

jahan

r

radius hai,

h

height hai.

Cone ka volume

\frac{1}{3} π r^{2} h

kyun hota hai?

Cone same base aur height wale cylinder ke volume ka exactly ek-tihai hota hai (integration ya Cavalieri's principle se prove hota hai).

Sphere ka surface area kya hai?

S A = 4 π r^{2}

$r$ aur $h$ diye hone par cone ki slant height $l$ kaise nikaalte hain? :: Use Pythagorean theorem: $l = r^{2} + h^{2}$

Hemisphere ka volume kya hota hai? :: $V = \frac{2}{3} π r^{3}$ (sphere ke volume ka aadha)

Hemisphere ka total surface area kya hota hai?

T S A = 3 π r^{2}

(curved surface

2 π r^{2}

+ flat base

π r^{2}

)

Square pyramid ka lateral surface area kya hota hai?

L S A = 2 a l

jahan

a

base ki side length hai aur

l

slant height hai.

Cone lateral area ke liye vertical height ki jagah slant height kyun use karte hain?

Kyunki curved surface base se apex tak slant ke along "wrap" karta hai, vertically nahi.

Radius aur diameter mein kya relation hai?

d = 2 r

(diameter radius ka do guna hota hai)

Agar ek 3D shape ke saare dimensions double kar diye jayein, toh volume mein kya change aata hai?

Volume

2^{3} = 8

se multiply ho jaata hai (volume linear dimensions ke cube ke saath scale karta hai).