1. Volume of a Cube

A cube is a special type of cuboid where all sides are equal (l=b=h=a).

Step-by-Step Derivation

• We start with the volume formula of a cuboid: V=length×breadth×heigh
• Since all sides of a cube are equal (a), we substitute: V=a×a×a
• Simplifying, we get: V=a³​

✅ Interpretation: This means the cube’s volume is the product of three equal dimensions.

2. Volume of a Cuboid

A cuboid has three different dimensions: length (l), breadth (b), and height (h).

Step-by-Step Derivation

• A cuboid is made up of layers of rectangular areas stacked on top of each other.
• The base area of the cuboid is a rectangle with area: Abase=l×b
• If we stack these rectangular layers up to height h, the total volume becomes: V=Base Area×Height=(l×b)×h
• So, the final formula is: V=l×b×h​

✅ Interpretation: This formula calculates the space occupied by the cuboid.

3. Volume of a Cylinder

A cylinder is formed by extending a circle along a height h.

Step-by-Step Derivation

• The base of a cylinder is a circle with radius r
• The area of a circle is: Abase=πr²
• If this circular base is extended to height h, the volume is:
• $$V=\text{Base Area}\times \text{Height}$$
• Substituting the area of the base: V=πr²×h
• Thus, the final formula is: V=πr²h​

✅ Interpretation: A cylinder is like a stack of infinitely thin circular layers, forming its total volume.

Final Summary of Volume Formulas