NOTES

📘 Surface Areas and Volumes Class 10 Notes (Chapter 12)

🔷 Introduction

Surface areas and volumes form an important chapter in mensuration. In simple words, this chapter helps us find the area outside a solid shape and the space inside it. Therefore, students learn how to solve real-life measurement problems.

Moreover, engineers, architects, and designers use these formulas every day. As a result, this chapter has many practical uses.

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🔷 What is Surface Area?

Surface area means the total area of the outer surface of a solid object.

For example, when you paint a box, you cover its outer surface. Thus, you calculate surface area.

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🔷 What is Volume?

Volume means the space inside a solid object.

For example, when you fill a bottle with water, the bottle holds volume. Therefore, volume measures capacity.

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🔷 Cuboid

A cuboid has:

• Length (l)
• Breadth (b)
• Height (h)

Examples:

• Books
• Boxes
• Bricks

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Formulas of Cuboid

Total Surface Area

$$TSA=2(lb+bh+hl)$$TSA=2(lb+bh+hl)

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Lateral Surface Area

$$LSA=2h(l+b)$$LSA=2h(l+b)

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Volume

$$V=l\times b\times h$$V=l×b×h

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🔷 Cube

A cube has all sides equal.

Examples:

• Dice
• Ice cubes
• Gift boxes

Let side = a

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Formulas of Cube

Total Surface Area

$$TSA=6a^2$$TSA=6a2

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Lateral Surface Area

$$LSA=4a^2$$LSA=4a2

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Volume

$$V=a^3$$V=a3

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🔷 Cylinder

A cylinder has two circular bases and one curved surface.

Examples:

• Water bottle
• Pipe
• Can

Let:

• Radius = r
• Height = h

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Formulas of Cylinder

Curved Surface Area

$$CSA=2\pi rh$$CSA=2πrh

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Total Surface Area

$$TSA=2\pi r(h+r)$$TSA=2πr(h+r)

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Volume

$$V=\pi r^2 h$$V=πr2h

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🔷 Cone

A cone has one circular base and one vertex.

Examples:

• Ice cream cone
• Party cap

Let:

• Radius = r
• Height = h
• Slant height = l

Formula:$$l=\sqrt{r^2+h^2}$$l=r2+h2​

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Formulas of Cone

Curved Surface Area

$$CSA=\pi rl$$CSA=πrl

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Total Surface Area

$$TSA=\pi r(l+r)$$TSA=πr(l+r)

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Volume

$$V=\frac13\pi r^2 h$$V=31​πr2h

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🔷 Sphere

A sphere is a perfectly round solid shape.

Examples:

• Football
• Orange
• Ball

Let radius = r

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Formulas of Sphere

Surface Area

$$4\pi r^2$$4πr2

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Volume

$$\frac43\pi r^3$$34​πr3

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🔷 Hemisphere

A hemisphere is half of a sphere.

Examples:

• Bowl
• Dome

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Formulas of Hemisphere

Curved Surface Area

$$CSA=2\pi r^2$$CSA=2πr2

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Total Surface Area

$$TSA=3\pi r^2$$TSA=3πr2

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Volume

$$V=\frac23\pi r^3$$V=32​πr3

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🔷 Frustum of a Cone

When we cut the top of a cone, the remaining shape becomes a frustum.

Let:

• Radius = R
• Small radius = r
• Height = h
• Slant height = l

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Formulas of Frustum

Curved Surface Area

$$CSA=\pi l(R+r)$$CSA=πl(R+r)

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Total Surface Area

$$TSA=\pi l(R+r)+\pi(R^2+r^2)$$TSA=πl(R+r)+π(R2+r2)

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Volume

$$V=\frac13\pi h(R^2+r^2+Rr)$$V=31​πh(R2+r2+Rr)

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🔷 Solved Example

Example

Find the volume of a cube of side 5 cm.

Solution

Given:$$a=5cm$$a=5cm

Now use:$$V=a^3$$V=a3 $$V=5^3$$V=53 $$V=125cm^3$$V=125cm3

Answer:

Volume = 125 cm³

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🔷 Important Tips

Remember these points carefully.

• Area uses square units.
• Volume uses cubic units.
• Always write units.
• Convert units before solving.
• Use π = 22/7 or 3.14.

Therefore, careful calculation gives correct answers.

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🔷 Real-Life Uses

People use these formulas in:

• Construction
• Architecture
• Packaging
• Water tanks
• Sports equipment
• Machine design

Thus, mensuration helps in many industries.