# Volume of a Cuboid - Definition, How to find the Diameter and Deriving the volume of a Cuboid

## Definition

A cuboid is the three-dimensional version of a rectangle. Well, partially. It is a convex polyhedron (a convex shape with many finite points that are not in the same plane), with six quadrilateral faces, eight vertices, and twelve edges. The polyhedral graph of a cuboid is the same as that of a cube.

There are many types of cuboids, like rectangular hexahedron, right rectangular prism, right cuboid, rectangular parallelepiped, and rectangular cuboid.

## Volume of cuboidThe space or cavity inside a cuboid is called its volume.

The volume of a cuboid = l x b x h units3

where l = length of the cuboid

b = breadth of the cuboid

h = height of the cuboid

That means we can fill lbh unit cubes of water inside a box of length l, breadth b, and height h.

## Deriving the volume of a cube from the volume of a cuboid

We know that the volume of a cuboid is length multiplied by breadth multiplied by height. Imagine all the sides of a cuboid are the same.

Then, l x b x h = a x a x a         (considering all the sides as length a) --------------------------- (1)

Therefore, the volume of the cuboid = a3 = volume of the cube

Hence, proved.

Do you know?

What is the most extensive length of a rod that can be inserted inside a cuboid?

Hint: Use Pythagoras theorem.

Fun activity: Let us find out how many cubes can fit inside a giant cuboid!

Consider some cubes with side S. All the cubes are symmetric and identical in shape and size. Now, consider a cuboid with sides L, B, H.

Volume of a cube = S x S x S

Volume of a cuboid = LBH

Number of cubes inside a cuboid = Volume of the smaller element / Volume of the more prominent element

= Volume of a cube / Volume of a cuboid

= S x S x S / LBH

The number which we will get will be the number of cubes that can fit inside the cuboid.

## How to find the diameter of the possible giant sphere inscribed in a cube or cuboid?Volume of a cube = a3 ----------------------------------------------- (2)

Where a is the side of a cube

Volume of a sphere = 4/3 π r3 ------------------------------------------- (3)

Where r is the radius of the sphere

Equating (2) and (3), we can find the sphere radius that can fit inside a cube.

Fun Fact: The size of this sphere will be 47.6% smaller than the size of the cube. How? Find out for yourself!

Consider the minor side of the cuboid to calculate the size of a sphere. Also, many such spheres can be placed inside the cuboid because of its varying sides.