Solid Shapes • Surface Area • Volume • Combination of Solids
THE GOAL • HANDWRITTEN EXAM NOTES
00. Exam Focus 🎯
Current CBSE Focus:
For the current CBSE Class 10 Mathematics syllabus, the
Surface Areas and Volumes section explicitly includes:
✓ Cube
✓ Cuboid
✓ Right Circular Cylinder
NCERT Concept Extension:
The NCERT chapter also discusses combinations of solids and
related concepts involving cone, sphere and hemisphere.
💡 Is page ka main goal hai ki student figure dekhkar
solid identify kare, correct formula choose kare aur
units ke saath answer de.
01. Cuboid ⭐⭐⭐⭐⭐
A cuboid is a three-dimensional solid having:
Length = l Breadth = b Height = h
Volume = lbh
LSA = 2h(l + b)
TSA = 2(lb + bh + hl)
🧠 Volume = solid ke andar kitni space hai.
LSA = sirf 4 side faces.
TSA = all 6 faces.
✏️ Example
Find the volume and total surface area of a cuboid having
l = 8 cm, b = 5 cm and h = 3 cm.
V = lbh
= 8 × 5 × 3
= 120 cm³
TSA = 2(lb + bh + hl)
= 2(40 + 15 + 24)
= 158 cm²
✓ Volume = 120 cm³
✓ TSA = 158 cm²
02. Cube ⭐⭐⭐⭐⭐
A cube has all its edges equal.
Let side = a.
LSA = 4a²
TSA = 6a²
Volume = a³
✏️ Example
Side of a cube = 7 cm.
TSA = 6a²
= 6 × 7²
= 294 cm²
V = a³ = 7³ = 343 cm³
✓ TSA = 294 cm²
✓ Volume = 343 cm³
03. Right Circular Cylinder ⭐⭐⭐⭐⭐
A cylinder has:
Radius = r Height = h
CSA = 2πrh
TSA = 2πr(h + r)
Volume = πr²h
💡 Cylinder ka curved part = CSA.
Dono circular bases ko bhi include karoge to = TSA.
✏️ Example
A cylinder has radius 7 cm and height 10 cm.
Find its volume.
V = πr²h
= π × 7² × 10
Using
227
for π:
= 1540 cm³
✓ Volume = 1540 cm³
04. CSA vs TSA — Never Confuse
Solid
Curved / Lateral Surface
Total Surface
Cube
LSA = 4a²
TSA = 6a²
Cuboid
LSA = 2h(l+b)
TSA = 2(lb+bh+hl)
Cylinder
CSA = 2πrh
TSA = 2πr(h+r)
⚠️ Most common mistake:
Question agar "curved surface area" pooch raha hai,
TSA mat calculate karna.
Question agar "total surface area" pooch raha hai,
required bases ko include karna hai.
05. Units — Very Important ⭐⭐⭐⭐⭐
1 m = 100 cm
1 m² = 10,000 cm²
1 m³ = 1,000,000 cm³
1 L = 1000 cm³
1 mL = 1 cm³
🧠 Surface area ka answer square unit mein.
Volume ka answer cubic unit mein.
Area → cm², m²
Volume → cm³, m³
06. Finding Missing Dimension
Kabhi question mein volume diya hota hai aur
height/radius/side find karni hoti hai.
Formula ko rearrange karo.
For cylinder:
V = πr²h
Therefore:
h =
Vπr²
✏️ Example
Volume of a cylinder = 1540 cm³,
radius = 7 cm. Find height.
h =
1540π × 7²
Using π = 22/7:
h = 10 cm
✓ Height = 10 cm
07. Cone — NCERT Concept Reference
📘 Note: Cone is part of the broader NCERT Chapter 12
treatment, while the current CBSE 2025–26 syllabus explicitly
lists cube, cuboid and right circular cylinder under this unit.
l² = r² + h²
l = √(r² + h²)
CSA = πrl
TSA = πr(l+r)
V =
13
πr²h
✏️ Example
For r = 3 cm and h = 4 cm:
l = √(3² + 4²)
= √25 = 5 cm
Then:
CSA = π × 3 × 5 = 15π cm²
✓ Slant height = 5 cm
✓ CSA = 15π cm²
08. Sphere — NCERT Concept Reference
Surface Area = 4πr²
Volume =
43
πr³
09. Hemisphere — NCERT Concept Reference
CSA = 2πr²
TSA = 3πr²
Volume =
23
πr³
⚠️ Hemisphere mein:
CSA = curved part only.
TSA = curved part + circular base.
10. Combination of Solids ⭐⭐⭐⭐⭐
A solid made by joining two or more basic solids is called a
combination of solids.
Total Volume
=
Volume₁ + Volume₂ + ...
Required Surface Area
=
Only exposed surfaces
💡 सबसे important:
Volume mein joined solids ke volumes add hote hain.
Lekin surface area mein jo surfaces andar join ho gaye hain,
unko count nahi karna.
11. Cylinder + Hemisphere
Volume:
V =
πr²h
+
23
πr³
Exposed Surface Area:
CSA of cylinder
+
CSA of hemisphere
+
top circular base
✏️ Example
A solid consists of a cylinder of radius 3 cm and height 10 cm
with a hemisphere of the same radius attached to one end.
V
=
π(3²)(10)
+
23
π(3³)
= 90π + 18π
= 108π cm³
✓ Volume = 108π cm³
12. Cylinder + Cone
Total Volume
=
πr²h₂
+
13
πr²h₁
💡 Same radius ho to r²π common le sakte ho.
13. Hollow Solid / Material Removed
When a portion of a solid is removed or a hollow is created,
find the required quantity by subtraction.
Remaining Volume
=
Original Volume − Removed Volume
Material Volume
=
Outer Volume − Inner Volume
✏️ Example
A cylinder has outer radius R and inner radius r, with the
same height h.
Volume of material
=
πR²h − πr²h
= πh(R² − r²)
✓ This is the standard subtraction approach for hollow solids.
14. Melting & Recasting ⭐⭐⭐⭐⭐
If one solid is melted and recast into another solid, the
volume of material remains unchanged.
Volume of old solid
=
Volume of new solid
✏️ Example
A metal cube is melted and recast into a cuboid.
Then:
a³ = lbh
If the new dimensions are known, the missing dimension can be
found from this equation.
🧠 Melting/recasting mein surface area change ho sakta hai,
but material ka volume same rehta hai.
15. Capacity & Volume ⭐⭐⭐⭐⭐
1 L = 1000 cm³
1 mL = 1 cm³
1 m³ = 1000 L
✏️ Example
A tank has volume 2 m³.
2 m³ = 2 × 1000 L
= 2000 L
✓ Capacity = 2000 L
16. Important Unit Conversions
Quantity
Conversion
Length
1 m = 100 cm
Area
1 m² = 10,000 cm²
Volume
1 m³ = 1,000,000 cm³
Capacity
1 L = 1000 cm³
Capacity
1 mL = 1 cm³
⚠️ Length conversion ko directly area/volume mein apply mat
karna.
For example:
1 m = 100 cm
but
1 m² ≠ 100 cm²
Instead:
1 m² = 100² = 10,000 cm².
17. Value of π
π =
227
or
π ≈ 3.14
💡 Question mein agar π ki specific value di ho,
usi value ko use karo.
Otherwise required approximation ke according
22/7 or 3.14 use kiya ja sakta hai.