3D shapes — cube, cuboid, cylinder, cone, sphere, prism, pyramid

1.2.13 · Maths › Basic Geometry

Core Concept: 2D se 3D Tak

WHY hume 3D shapes ki parwah kyun hai? Kyunki real world 3D hai. Ek kagaz ka tukda 2D hai (negligible thickness), lekin ek box, ek ball, ek can — sab 3D hain.

WHAT ek 3D shape ko define karta hai?

• Faces: flat ya curved surfaces
• Edges: jahan do faces milte hain
• Vertices: jahan edges milti hain (corners)
• Volume ($V$): andar ki space (cubic units mein measure hota hai: cm³, m³)
• Surface Area ($SA$): saare faces ka total area (square units mein measure hota hai: cm², m²)

HOW hum ye formulas banate hain? Hum first principles se derive karte hain: 2D shapes ko stacking, rotating, ya scaling karke.

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1. Cube

Volume Derivation

WHY? Volume = base area × height. Cube ke liye, base ek square $a\times a$ hai, aur hum is square ko $a$ units oopar tak stack karte hain.

Derivation: $V_{\text{cube}}=(\text{base area})\times (\text{height})=a^2\times a=a^3$

Surface Area Derivation

WHY? Ek cube mein 6 faces hote hain, har ek $a^2$ area ka square hota hai.

Derivation: $S{A}_{\text{cube}}=6\times (\text{area of one face})=6a^2$

Solution:

• Volume: $V=a^3=5^3=125$ cm³
  • Ye step kyun? Hum edge length ko cube karte hain kyunki volume 3D hai (length × width × height, sab equal hain).
• Surface Area: $SA=6a^2=6(5^2)=6(25)=150$ cm²
  • Ye step kyun? 6 faces mein se har ek $5\times 5=25$ cm² ka square hai.

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2. Cuboid (Rectangular Box)

Volume Derivation

WHY? Rectangular slices stack karo. Har slice ka area $l\times w$ hai, $h$ height tak stack kiya gaya hai.

Derivation: $V_{\text{cuboid}}=l\times w\times h$

Surface Area Derivation

WHY? Ek cuboid mein opposite faces ke 3 pairs hote hain:

• Top & bottom: har ek $l\times w$
• Front & back: har ek $l\times h$
• Left & right: har ek $w\times h$

Derivation: $SA=2(lw)+2(lh)+2(wh)=2(lw+lh+wh)$

Solution:

• Volume: $V=8\times 5\times 3=120$ cm³
  • Kyun? 3D space paane ke liye teeno dimensions multiply karo.
• Surface Area: $SA=2(8\cdot 5+8\cdot 3+5\cdot 3)=2(40+24+15)=2(79)=158$ cm²
  • Kyun? Saare 6 faces (3 pairs mein) ke areas add karo.

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3. Cylinder

Volume Derivation

WHY? Circular slices stack karo. Har slice ka area $\pi r^2$ hai, $h$ height tak stack kiya gaya hai.

Derivation: $V_{\text{cylinder}}=(\text{base area})\times h=\pi r^2\times h$

Surface Area Derivation

WHY? Surface area = do circular bases + curved surface. Agar hum curved surface ko "unroll" karein, to wo ek rectangle ban jaata hai jiska width = base ki circumference = $2\pi r$ aur height = $h$ hoti hai.

Derivation:

• Do bases ka area: $2\times \pi r^2=2\pi r^2$
• Curved surface area: $2\pi r\times h$
• Total: $SA=2\pi r^2+2\pi rh=2\pi r(r+h)$

Solution:

• Volume: $V=\pi r^{2}h=\pi (4^2)(10)=160\pi \approx 502.65$ cm³
  • Kyun? Base area $\pi r^2$ times height.
• Surface Area: $SA=2\pi r(r+h)=2\pi (4)(4+10)=2\pi (4)(14)=112\pi \approx 351.86$ cm²
  • Kyun? Do circles plus unrolled curved surface.

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4. Cone

Volume Derivation

WHY? Ek cone circular base wale pyramid jaisa hai. Jaise hum oopar jaate hain, circular slices zero tak shrink ho jaati hain. Calculus (ya Cavalieri's principle) use karke, cone ka volume = same base aur height wale cylinder ka $\frac{1}{3}$ hota hai.

Derivation (Intuitive): Socho ki ek cylinder mein paani bharna hai 3 identical cones se — wo usse exactly bhar dete hain. To: $V_{\text{cone}}=\frac{1}{3}{V}_{\text{cylinder}}=\frac{1}{3}\pi r^{2}h$

Surface Area Derivation

WHY? Surface = base + curved surface. Curved surface, jab unroll kiya jaata hai, ek sector of a circle banta hai jiska radius $l$ (slant height) aur arc length $2\pi r$ (base circumference) hota hai.

Derivation:

• Base area: $\pi r^2$
• Curved surface area: $\pi rl$ (sector formula: $\frac{1}{2}\times \text{arc length}\times \text{radius}=\frac{1}{2}(2\pi r)(l)=\pi rl$)
• Total: $SA=\pi r^2+\pi rl=\pi r(r+l)$

Solution:

• Slant height: $l=\sqrt{3^2+4^2}=\sqrt{9+16}=\sqrt{25}=5$ cm
  • Kyun? Pythagorean theorem: $l^2=r^2+h^2$.
• Volume: $V=\frac{1}{3}\pi r^{2}h=\frac{1}{3}\pi (3^2)(4)=\frac{1}{3}\pi (9)(4)=12\pi \approx 37.7$ cm³
  • Kyun? Cone cylinder ka $\frac{1}{3}$ hota hai.
• Surface Area: $SA=\pi r(r+l)=\pi (3)(3+5)=3\pi (8)=24\pi \approx 75.4$ cm²
  • Kyun? Base circle plus curved surface.

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5. Sphere

Volume Derivation

WHY? Calculus use karke, hum circular cross-sections integrate karte hain. Intuitively, ek sphere ek semicircle ko uske diameter ke around rotate karke banta hai.

Derivation (via integration, simplified): Center se height $y$ par, cross-section ek circle hai jiska radius $x=\sqrt{r^2-y^2}$ hai (Pythagorean theorem). Is slice ka area: $A(y)=\pi x^2=\pi (r^2-y^2)$. $y=-r$ se $y=r$ tak integrate karo: $V={\int }_{-r}^r\pi (r^2-y^2)dy=\pi {\left[r^{2}y-\frac{y^3}{3}\right]}_{-r}^r=\pi \left(2r^3-\frac{2r^3}{3}\right)=\pi \cdot \frac{4r^3}{3}=\frac{4}{3}\pi r^3$

Surface Area Derivation

WHY? Calculus use karke (ya Archimedes' method with cylinders), surface area exactly ek great circle $\pi r^2$ ke area ka 4 guna hota hai.

Derivation (intuitive): Sphere ko tiny patches mein peel karo aur flatten karo — total area $4\pi r^2$ hoti hai.

Solution:

• Volume: $V=\frac{4}{3}\pi r^3=\frac{4}{3}\pi (6^3)=\frac{4}{3}\pi (216)=288\pi \approx 904.78$ cm³
  • Kyun? Volume formula seedha use karo.
• Surface Area: $SA=4\pi r^2=4\pi (6^2)=4\pi (36)=144\pi \approx 452.39$ cm²
  • Kyun? Ek great circle ke area ka chaar guna.

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6. Prism

Volume Derivation

WHY? Identical cross-sections (base shape) ko height $h$ tak stack karo.

Derivation: $V_{\text{prism}}=(\text{base area})\times h=A_{\text{base}}\times h$

Surface Area Derivation

WHY? Surface = do bases + lateral faces.

Derivation:

• Do bases ka area: $2A_{\text{base}}$
• Lateral area: (base ki perimeter) $\times h=P_{\text{base}}\times h$
• Total: $SA=2A_{\text{base}}+P_{\text{base}}\cdot h$

Solution:

• Base area: $A=\frac{1}{2}b{h}_t=\frac{1}{2}(6)(4)=12$ cm²
  • Kyun? Triangle ka area.
• Base perimeter: $P=6+5+5=16$ cm
• Volume: $V=A\times h=12\times 10=120$ cm³
  • Kyun? Triangular slices stack karo.
• Surface Area: $SA=2(12)+16(10)=24+160=184$ cm²
  • Kyun? Do triangular bases plus teen rectangular sides.

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7. Pyramid

Volume Derivation

WHY? Cone ki tarah, ek pyramid same base aur height wale prism ka $\frac{1}{3}$ hota hai. Ye Cavalieri's principle ya calculus se aata hai.

Derivation: $V_{\text{pyramid}}=\frac{1}{3}(\text{base area})\times h=\frac{1}{3}{A}_{\text{base}}\times h$

Surface Area Derivation

WHY? Surface = base + lateral triangular faces.

Derivation: $SA=A_{\text{base}}+\sum (\text{areas of triangular faces})$ Ek regular pyramid ke liye (saare lateral faces identical), agar slant height $l$ hai: $SA=A_{\text{base}}+\frac{1}{2}{P}_{\text{base}}\cdot l$

Solution:

• Base area: $A=a^2=6^2=36$ cm²
• Volume: $V=\frac{1}{3}Ah=\frac{1}{3}(36)(8)=\frac{288}{3}=96$ cm³
  • Kyun? Pyramid ek prism ka $\frac{1}{3}$ hota hai.

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Common Mistakes

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Summary Table

Shape	Volume	Surface Area	Key Feature
Cube	$a^3$	$6a^2$	Saare edges equal
Cuboid	$lwh$	$2(lw+lh+wh)$	6 rectangular faces
Cylinder	$\pi r^{2}h$	$2\pi r(r+h)$	Circular bases
Cone	$\frac{1}{3}\pi r^{2}h$	$\pi r(r+l)$	Apex tak taper, $l=\sqrt{r^2+h^2}$
Sphere	$\frac{4}{3}\pi r^3$	$4\pi r^2$	Saare points center se equidistant
Prism	$A_{\text{base}}\cdot h$	$2A_{\text{base}}+P_{\text{base}}\cdot h$	Parallel polygonal bases
Pyramid	$\frac{1}{3}{A}_{\text{base}}\cdot h$	$A_{\text{base}}+\frac{1}{2}{P}_{\text{base}}\cdot l$	Apex tak taper

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

Socho tumhare paas building blocks hain. Cube ek dice hai — saari sides square aur equal hain. Cuboid ek brick hai — ek taraf se zyada lamba. Cylinder ek can hai — upar aur neeche circles hain, beech mein tube connect karta hai. Cone ek ice cream cone hai — upar se wide shuru hota hai, end mein pointy ho jaata hai. Sphere ek ball hai — har jagah perfectly round. Prism ek candy bar jaisa hai jiska ek hi shape poore through hota hai. Pyramid Egyptian pyramids jaisi hai — flat bottom, upar ek point par aata hai.

Volume = andar kitni space hai (kitna paani fit hoga). Surface area = kitna wrapping paper lagega ise cover karne ke liye. Cubes aur boxes ke liye, hum volume paane ke liye length × width × height multiply karte hain. Cones aur pyramids ke liye, hum uska 1/3 lete hain kyunki wo pointy hain aur utni space nahi lete. Spheres ke liye, ek special formula hai $\frac{4}{3}$ ke saath jo fancy math se aata hai. Key yeh hai: flat sides = dimensions multiply karo; round shapes = π use karo; pointy tops = 3 se divide karo.

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Connections

• Pythagorean Theorem — cones aur pyramids mein slant height nikalne ke liye use hota hai
• Area of2D Shapes — 3D shapes ke bases (squares, circles, triangles)
• Circle Geometry — cylinders aur cones ke liye $\pi r^2$ aur circumference samajhna
• Units and Measurement — volume cm³ mein, surface area cm² mein
• Integration — sphere aur cone volumes ki advanced derivation
• Real-World Applications — packaging, architecture, engineering

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#flashcards/maths

What is the volume formula for a cube with edge length $a$? :: $V=a^3$

What is the slant height $l$ of a cone in terms of $r$ and $h$? :: $l=\sqrt{r^2+h^2}$

Concept Map