Question

What is the formula for the surface area of any pyramid?

Answer 1

This, for example

Answer 2

Surface Area = B + 1/2 * P * s
B = base area
P = perimeter of the base
s = slant height

Answer 3

But for more general cases the slant height is not constant. Do you take the average slant height? How do you prove that?

Answer 4

Do you work with Calculus?

Answer 5

For a random case you have to move form a different prospective.
Use will have to use integration of the curved surface that is solved by calculus. do you want me to tell you the method.

Answer 6

You will need a function which defines the slant height along the base perimeter, or some other variable. Then the slant height can be taken into account as you integrate along the base perimeter.

Answer 7

Yes, I do

Answer 8

This is the actual process.

Answer 9

' the slant height along the base perimeter': what do you mean by that?

Answer 10

that means that u have to intregrate it along the perimeter

Answer 11

For what purpose? I still don't get it. With respect to what? Surely the slant height is a constant for a single point of the perimeter?

Answer 12

@punnus 's method will integrate over the base area, and hence the function which defines the slope (in both directions x and y) must be known.

Answer 13

It all depends on how the geometry (and slope) of the pyramid is defined.

Answer 14

punnus- so by your link, there is no simple function of the perimeter you could use to find it.

Answer 15

Yes, mathmate, I should have defined that in more detail eariler

Answer 16

*earlier

Answer 17

Choose any point on the perimeter (which is all on the same plane). The surface will always coincide with a line from that point on the perimeter to the apex of the pyramid.

Answer 18

Well the question you have asked is not that simple .
You have made every thing as variable and when there are more variales then we proceed via a general method and not a specialised method

Answer 19

No, sorry, you're right- use dSA=d(perimeter)*d(slant height(perimeter)), and integrate. I understand that now- but it's still very messy

Answer 20

*dSA=d(perimeter)*(slant height(perimeter))

Answer 21

If you know the base perimeter as a function of x and y, then you can integrate over the the line integral ds and accumulate the area of the elemental triangles as a function of x, y and the height of the pyramid.