how can we derive or prove these formulas?

Question

I NEED TO KNOW HOW TO PROVE THE lateral surface area, the total surface area and volume of pyramid. pls tell me derivation.
also i need models of pyramid which i can make using cardboard paper.
pls derive formulas in terms of b,h,s and p

b is area of base
h is height of pyramid
s is slant height
p is perimeter of pyramid

Answer 1

You don't say what kind of pyramid this is or what the shape of the base is(triangle square hexagon ,etc). Also I'm guessing that p = perimeter of the base of the pyramid: otherwise, it makes no sense. Also, I assume the bases are regular polygons and the the height of the pyramid is perpendicular to the base at the base's center.

Let's assume the base is a square with side x.
Then x=p/4
Then the lateral area will be made up of four triangles whose base is p/4 and whose altitude is the slant height.So the area of each ot these triangles is 1/2(p/4)(s) = ps/8
Since there are 4 such triangles, the total lateral area = ps/2

The total surface area = b+ps/2

In general, If the base is a regular polygon of n sides, then the length of one side of the base will be p/n. Then there would be n triangles each with an area of (sp/2n)*n = ps/2 as shown above for the square base. So, in general = total surface area = b+ps/2.

Now the volume = (1/3)*(bh)
This formula can be proven by integrating the cross-sectional area formed by cutting the pyramid at right angles to the z-axis, from 0 to h. This is a calculus problem.

To make a cardboard cutout, draw a square then on eac side of the square draw a triangle with height =s. Then cut this figure out and fold the triangles up using the sides as a fold line. You will have formed a pyrmaid.

Answer 2

It is simple.
At the base area is b
At the apex area is zero.
The decrease in area is uniform.
Volume=[Area at bottom +area at apex]/2*height
=b*h/2
Similar way we can work the surface area.
=average perimeter*slant height
=p*s/2