Surface Areas and Volumes

Board: CBSE | Class: Class 10

Comprehensive study notes for Surface Areas and Volumes by Ajay Yadav (Math King of Katargam). Master every concept with clear explanations, solved examples, and practice problems.

Key Concepts

Combination of Solids

Many real objects are combinations of basic solids (cylinder + cone, cylinder + hemisphere, etc.). Find total surface area by summing curved surfaces (excluding joining surfaces). Volume is additive.

Conversion of Solids

When a solid is melted and recast into another shape, volume remains constant. Use volume equality to find unknown dimensions.

Frustum of a Cone

When a cone is cut parallel to base, the lower part is a frustum. Slant height l = √h² + (R-r²). CSA = π(R+r)l. TSA = π(R+r)l + πR² + πr². Volume = (πh/3)(R² + r² + Rr).

Important Formulas

Cylinder Volume	V = πr²h
Cone Volume	V = (1/3)πr²h
Sphere Volume	V = (4/3)πr³
Hemisphere Volume	V = (2/3)πr³
Frustum Volume	V = (πh/3)(R² + r² + Rr)
Frustum CSA	CSA = π(R+r)l, l = √h² + (R-r²)

Solved Examples

Example 1: A cone of radius 6 cm and height 8 cm is melted into a sphere. Find radius.

Solution: Cone volume = (1/3)π(36)(8) = 96π. Sphere volume = (4/3)πr³ = 96π. r³ = 72. r = 4.16 cm.

Example 2: Find the volume of a frustum with R=6, r=3, h=4.

Solution: V = (π×4/3)(36 + 9 + 18) = (4π/3)(63) = 264 cm³.

Example 3: A toy is a cone on a hemisphere (same radius 3.5 cm, total height 15.5 cm). Find total surface area.

Solution: Cone height = 12, l = √144+12.25 = 12.5. CSA(cone)=πrl = 137.5. CSA(hemisphere)=2πr² = 77. TSA = 214.5 cm².

Practice Questions

1. A sphere of radius 6 cm is melted into a cylinder of radius 4 cm. Find height.
2. Find the number of coins of diameter 1.5 cm and thickness 0.2 cm that can be melted from a cuboid of dimensions 5.5 cm × 10 cm × 3.5 cm.
3. A bucket has top radius 20 cm, bottom radius 10 cm, height 30 cm. Find its capacity.
4. A solid is a cylinder with hemispherical ends. Total length 20 cm, diameter 7 cm. Find TSA.
5. A cone of height 24 cm and base radius 6 cm is made of clay. A child reshapes it into a sphere. Find radius.

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