Cheat Sheet — Surface Areas & Volumes

📦 Cuboid (l × b × h)

TOTAL SURFACE AREA

TSA = 2(lb + bh + hl)

LATERAL / CURVED SURFACE AREA

LSA = 2(l + b) × h

Only the four walls — excludes top and bottom.

VOLUME

V = l × b × h

DIAGONAL

d = √(l² + b² + h²)

Open box (no lid): SA = 2(bh + hl) + lb = TSA − lb.

Room painting: Walls only = LSA. Walls + ceiling = LSA + lb.

🎲 Cube (side = a)

TOTAL SURFACE AREA

TSA = 6a²

LATERAL SURFACE AREA

LSA = 4a²

VOLUME

V = a³

DIAGONAL

d = a√3

Cube is a special cuboid where l = b = h = a. All 6 faces are equal squares.

🛢️ Right Circular Cylinder (r, h)

CURVED SURFACE AREA

CSA = 2πrh

Think: unroll the cylinder → rectangle of width 2πr and height h.

TOTAL SURFACE AREA

TSA = 2πr(h + r)

= CSA + 2 × base area = 2πrh + 2πr²

VOLUME

V = πr²h

Hollow cylinder: V = π(R² − r²)h where R = outer, r = inner radius.

Open cylinder (no lid): SA = 2πrh + πr² (one base only).

🍦 Right Circular Cone (r, h, l)

SLANT HEIGHT

l = √(r² + h²)

CURVED SURFACE AREA

CSA = πrl

TOTAL SURFACE AREA

TSA = πr(l + r)

= CSA + base area = πrl + πr²

VOLUME

V = ⅓ πr²h

= ⅓ × volume of cylinder with same r and h

💡 Why ⅓? Exactly 3 cones fill a cylinder of same radius and height. Experimental proof: pour water from cone to cylinder 3 times.

🔮 Sphere (radius r)

SURFACE AREA

SA = 4πr²

= 4 × area of a great circle. No CSA/LSA distinction for a sphere.

VOLUME

V = ⁴⁄₃ πr³

Given diameter d: use r = d/2 in all formulas.

Earth/ball problems: SA = 4πr². Cost = SA × rate per unit area.

🥣 Hemisphere (radius r)

CURVED SURFACE AREA (dome only)

CSA = 2πr²

TOTAL SURFACE AREA (dome + flat base)

TSA = 3πr²

= 2πr² + πr² (CSA + one flat circle)

VOLUME

V = ⅔ πr³

Hollow hemisphere: V = ⅔ π(R³ − r³)

🪣 Frustum of a Cone (r₁, r₂, h)

What is a Frustum? When a cone is cut by a plane parallel to the base, the portion between the base and the cut is a frustum. r₁ and r₂ are the radii of the two circular ends.

SLANT HEIGHT

l = √[h² + (r₁ − r₂)²]

CURVED SURFACE AREA

CSA = π(r₁ + r₂)l

TOTAL SURFACE AREA

TSA = π(r₁+r₂)l + πr₁² + πr₂²

VOLUME

V = ⅓ πh(r₁² + r₂² + r₁r₂)

Bucket problems: A bucket is a frustum with r₁ (bottom) < r₂ (top).

🧩 Combination of Solids

Key Principle Volume = Sum of individual volumes. TSA = Sum of visible surfaces (subtract overlapping/hidden areas).

Combination	TSA	Volume
Cone on Cylinder	CSA(cyl) + CSA(cone) + πr²(base)	πr²h₁ + ⅓πr²h₂
Hemisphere on Cylinder	CSA(cyl) + 2πr² + πr²(base)	πr²h + ⅔πr³
Cone on Hemisphere	πrl + 2πr²	⅓πr²h + ⅔πr³
Capsule (cyl + 2 hemi)	2πrh + 4πr²	πr²h + ⁴⁄₃πr³

Scooped out: hollow carved = original TSA + inner surface − 2 × contact area.

♻ Conversion of Solids

Golden Rule When a solid is melted and recast, volume is conserved. Surface area changes.

MELTING INTO n IDENTICAL OBJECTS

V(original) = n × V(small)

Solve for n or for the unknown dimension.

SPHERE → n SMALL SPHERES

n = (R/r)³

Shortcut: number = ratio of radii cubed.

WATER FLOW PROBLEMS

Volume per min = πr²(pipe) × speed

Time = Volume(vessel) ÷ Volume per minute. Ensure consistent units!

📊 Master Formula Table

Shape	CSA / LSA	TSA	Volume
Cuboid	2(l+b)h	2(lb+bh+hl)	lbh
Cube	4a²	6a²	a³
Cylinder	2πrh	2πr(h+r)	πr²h
Cone	πrl	πr(l+r)	⅓πr²h
Sphere	—	4πr²	⁴⁄₃πr³
Hemisphere	2πr²	3πr²	⅔πr³
Frustum	π(r₁+r₂)l	π(r₁+r₂)l+πr₁²+πr₂²	⅓πh(r₁²+r₂²+r₁r₂)

✏️ Worked Examples

Tent (cone on cylinder): r=7m, cyl h=5m, cone h=24m
l = √(7²+24²) = 25m
Canvas = CSA(cyl) + CSA(cone) = 2π(7)(5) + π(7)(25) = 220 + 550 = 770 m²

Sphere melted → wire (cylinder): Sphere r=3cm, wire r=0.2cm
⁴⁄₃π(3)³ = π(0.2)²h → h = 4×27/(3×0.04) = 900 cm = 9 m

Water flow: pipe r=0.7cm, speed=10 m/min, cone r=20cm, h=24cm
Flow/min = π(0.7)²(1000) = 1540 cm³/min
V(cone) = ⅓π(400)(24) ≈ 10057 cm³
Time ≈ 6.53 min ≈ 6 min 32 sec

🧠 Memory Tricks & Mnemonics

Cone = ⅓ Cylinder

Volume of cone = ⅓ × cylinder with same r, h. Three cones fill one cylinder.

Hemisphere = ⅔ Cylinder

V(hemisphere) = ⅔πr³. Think: 2/3 of the cylinder πr²·r where h = r.

Sphere SA = 4 Great Circles

4πr² = 4 × πr². The surface of a sphere equals the area of 4 great circles.

Hemisphere TSA: "2 + 1 = 3"

CSA = 2πr² (dome) + πr² (flat base) = TSA = 3πr². Remember 2+1=3.

Slant Height: Pythagoras!

l² = r² + h² for cone. l² = h² + (r₁−r₂)² for frustum. Always use l in CSA.

Melting = Volume constant

Solid melted & recast → Volume stays same, surface area changes. Set V₁ = V₂.

Combination TSA: subtract hidden

When two solids join, the contact face is hidden on both sides. Subtract 2 × contact area from the raw sum.

Units check!

SA in cm²/m², Volume in cm³/m³. 1 litre = 1000 cm³. 1 m³ = 10⁶ cm³. Always convert before computing.

⚠️ Common Board Exam Mistakes

Mistake	Correct Approach
Using h instead of l in cone CSA	CSA = πrl (slant height), NOT πrh. Always find l = √(r²+h²) first.
Forgetting to add base in TSA	TSA includes ALL surfaces. Check: does the shape have a flat base? Add πr² or lb for each base.
Using diameter instead of radius	Read carefully: if "diameter = 14 cm" is given, use r = 7 cm in all formulas.
Wrong π value	Use π = 22/7 when r is a multiple of 7. Use π = 3.14 otherwise. Follow "unless stated otherwise".
Inconsistent units	Convert all measurements to the same unit BEFORE calculating. cm + m together = disaster.
Adding SAs in combination problems	Remember to subtract the hidden/contact surfaces when solids are joined!