Is it correct to say: Using diameter instead of radius?

All formulas use r. If diameter d is given, use r = d/2 first.

Is it correct to say: Forgetting to compute slant height for a cone?

l = √(r² + h²). CSA needs l, not h.

Is it correct to say: Adding base area of hemisphere to CSA and calling it TSA of a hemisphere on cube?

Where the hemisphere sits on the cube, that circular area is internal — do NOT count.

Is it correct to say: Not converting units (cm ↔ m ↔ L)?

1 m³ = 10^6 cm³ ; 1 L = 1000 cm³. Convert before substituting.

Is it correct to say: Confusing volume of hemisphere with volume of sphere?

V_hemi = (2/3)πr³ = ½ × sphere volume, NOT (4/3)πr³.

Is it correct to say: Ignoring 'no wastage' assumption in recasting problems?

State 'volume is conserved as no metal is lost' — worth 1 justification mark.

Mensuration — Surface Areas and Volumes

Surface areas and volumes of cylinder, cone, sphere, hemisphere and frustum ; combined solids — Maharashtra SSC Geometry

📐 11 formulas✏️ 3 examples🎯 6 practice⚖️ 5-6 marks🏫 Maharashtra State Board📚 Class 10✓ 2025–26 syllabus

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Board Exam Tips

→One 4-mark combined-solid problem is a fixture — cone on cylinder, hemisphere on cube, etc.
→Volume-conservation problems (metal melted and recast) — a favourite 3-mark item.
→Marathi terms: पृष्ठफळ (surface area), घनफळ (volume), शंकु (cone), गोल (sphere), अर्धगोल (hemisphere).
→Always write formulas before substituting numbers — worth 1 method mark.
→Use π = 22/7 unless the paper specifies otherwise. Check the last line.

📐 Formulas(11)

Cylinder★ Board fav

V = π r^{2} h, C S A = 2 π r h, T S A = 2 π r (r + h)

Symbol	Meaning
$r$	Radius of the circular base
$h$	Height of the cylinder

Cone★ Board fav

V = \frac{1}{3} π r^{2} h, C S A = π r l, T S A = π r (r + l)

Symbol	Meaning
$l$	Slant height
$h$	Perpendicular height

Slant Height of Cone

l = r^{2} + h^{2}

Sphere

V = \frac{4}{3} π r^{3}, S A = 4 π r^{2}

Hemisphere★ Board fav

V = \frac{2}{3} π r^{3}, C S A = 2 π r^{2}, T S A = 3 π r^{2}

Frustum of a Cone (Volume)

V = \frac{π h}{3} (R^{2} + R r + r^{2})

Symbol	Meaning
$R$	Radius of larger (bottom) base
$r$	Radius of smaller (top) base
$h$	Height between the two bases

Frustum — Slant Height and CSA

l = h^{2} + (R - r)^{2}, C S A = π (R + r) l

Cube

V = a^{3}, T S A = 6 a^{2}

Cuboid

V = l bh, T S A = 2 (l b + bh + h l)

Volume Conservation (Recasting)★ Board fav

V_{old} = V_{new}

Combined Solid Surface Area

S A_{total} = \sum exposed surfaces only

✏️ Solved Examples

1Solved Exampleeasy3 steps

Find the total surface area of a cylinder of radius 7 cm and height 10 cm. (Use π = 22/7.)

Apply TSA formula

T S A = 2 π r (r + h) = 2 \times \frac{22}{7} \times 7 \times (7 + 10)

2Solved Exampleboard4 steps

A metallic sphere of radius 3 cm is melted and recast into a solid cylinder of radius 3 cm. Find the height of the cylinder.

Volume of sphere

V_{s} = \frac{4}{3} π (3)^{3} = 36 π cm^{3}

3Solved ExampleHOTS4 steps

A tent is in the shape of a cylinder of radius 4 m and height 2.1 m surmounted by a cone of the same radius and slant height 2.8 m. Find the area of canvas needed. (Use π = 22/7.)

Canvas covers CSA of cylinder + CSA of cone (no base needed for tent)

Area = 2 π r h + π r l

⚠️ Traps & Common Mistakes

⚠️Common Mistakes6

1
Using diameter instead of radius
✓All formulas use r. If diameter d is given, use r = d/2 first.
2
Forgetting to compute slant height for a cone
✓l = √(r² + h²). CSA needs l, not h.
3
Adding base area of hemisphere to CSA and calling it TSA of a hemisphere on cube
✓Where the hemisphere sits on the cube, that circular area is internal — do NOT count.
4
Not converting units (cm ↔ m ↔ L)
✓1 m³ = 10^6 cm³ ; 1 L = 1000 cm³. Convert before substituting.
5
Confusing volume of hemisphere with volume of sphere
✓V_hemi = (2/3)πr³ = ½ × sphere volume, NOT (4/3)πr³.
6
Ignoring 'no wastage' assumption in recasting problems
✓State 'volume is conserved as no metal is lost' — worth 1 justification mark.

🎯 Practice Yourself

🎯Practice Yourself6 questions

Q1
Find the volume of a cone of radius 6 cm and height 8 cm. (Use π = 22/7.)
Q2
The radius of a hemispherical bowl is 21 cm. Find its curved surface area.
Q3
A cuboidal water tank has dimensions 6 m × 5 m × 4.5 m. How many litres does it hold?
Q4
A sphere is melted and recast into 27 smaller spheres of equal radius. If the original radius is 9 cm, find the radius of each smaller sphere.
Q5
A bucket in the form of a frustum has radii 20 cm and 12 cm and height 15 cm. Find its capacity in litres.
Q6
Total surface area of a cube is 216 cm². Find its volume.

📝 Notes

Mensuration — Surface Areas and Volumes

Two students can look at the same combined solid and disagree about which surface to add. The trick is to imagine the object from every side and count only the surfaces you can see.

SSC angle

Maharashtra SSC Geometry allots 5-6 marks here:

2-mark: direct area/volume of a single solid.
3-mark: combined solid or a recasting problem.
4-mark HOTS: tent, ice-cream cone, capsule, or bucket.

Volume conservation

When a solid is melted or reshaped, the total volume of material stays the same (unless the paper explicitly says otherwise). This principle solves almost every recasting problem in one line:

V(old) = V(new)

Combined solids — the golden rule

Add only the exposed surfaces. The circular face where a hemisphere meets a cylinder, or the base of a cone attached to a hemisphere, is internal and not counted.

Key unit conversions

1 L = 1000 cm³
1 m³ = 10^6 cm³ = 1000 L
1 hectare = 10 000 m²

Board vs CBSE

Both boards use the same formulas. Maharashtra SSC Geometry emphasises real-world word problems (tent, tank, bucket, ice-cream cone) — draw a labelled diagram before you start writing formulas.