Question

Derivation of volume and total surface area of cube cuboid and pyramid

Answer 1

For Cube
Since cube has 6 faces of say side s
area of 1 side = s*s= s^2
Area of 6 faces = 6s^2

Volume = l*b*h
= s*s*s = s^3

For Cuboid
For Surface are
Let sides be
l=a
b=b
h=h
area of 1 side = ab
2 sides area = 2ab

area of side 2 = bh
2 sides area = 2bh

area of side 3= ah
2 side area = 2ah

Total area = 2(ab+bh+ah)
Area of the base rectangular section = a * b ( a,b are length & breadth)

If there are total ‘h' such horizontal cross sections.. ie height = h

So total area covered by all these sections =

= h x( ab)

=> VOLUME OF CUBOID = a * b * h


For pyramid
Area of pyramid
= 1/2 * perimeter of base area * slant height + base area


Area of the base equilateral triangular horizontal cross section = (√3/4) * a² ( ‘a' is its side length)

Now, if we calculate the areas of ‘h' such similar triangular sections , and add them. Total area will be the volume of triangular PRISM. = h * (√3/4) a²

And Pyramid is one third of the volume of prism on the same base with same height. Just like circular based Cylinder & Cone.( This can be proved by finding their volumes by water capacity experiment)

And in pyramids each horizontal section placed above the previous ones are similar but its sides go on decreasing proportionally until the top most section sides become zero.

So VOLUME ( TRIANGULAR PYRAMID) =

=(1/3) * h *(√3/4 )a²