How does one find the surface area of a pyramid?

Question

I would like to know how to find the surface area of a pyramid with any kind of base. If I already know the area of the base and the height of the pyramid, is it enough to find the whole surface area? What about if I have the perimeter of the base as well? I know you need to have the slant height, but can we derive this from a combination of already known measurements? Thanks for the help!

The base of the pyramid shouldn't make a difference. We can assume that the base will always be a regular n-gon (square, triangle, hexagon, octagon, etc...).

Also, we can assume that it is a right pyramid.

Answer 1

It depends on the shape of the base. What base are you looking for?
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Now let me rephrase your question. In a regular pyramid, if the height is h, and the base is a regular n-gon with side length = a, what is the formula for the surface area?

If this is the question you asked, I'll show you how to get that formula.
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First, you need to find apothem (b).
b = (1/2)a/tan(180/n)

Then you can find the slant height (l) of each face.
l = √(b^2 + h^2)

The base area can be calculated by
B = n(1/2)a(b) = n (1/2)a^2/tan(180/n)

Therefore, you can get the formula

Area
=f(a, h, n)
= (1/2)( Perimeter of the base)(l) + B
= (1/2)(n a)√[((1/2)a/tan(180/n))^2 + h^2] + n (1/2)a^2/tan(180/n)

Answer 2

The area of the base A(b) = l^2 is simply the area of a square with equal sides (l). As that is on the ground for Egyptian pyramids, I don't think I'd call that a "surface" area. If you care to include it, just add it into what's coming up next.

If we presume the pyramid is made up of isosceles same-sized triangles, then each of the four triangles has an area A(t) = 1/2 hl; where h is the height of a triangle. H = h sin(theta); where H = the height of the pyramid at its center point on the base and theta = the slant angle of a side relative to the base. You can find theta = arc tan(H/l); where both H and l are measured or known.

Therefore, the surface (without base) area is 4A(t) = 4(1/2)*(H/sin(theta))*l = 2Hl/sin(theta) because four same-sized, isosceles triangle sides were presumed for the pyramid. [Don't forget to add in the base area if you consider that as a surface too.]

Answer 3

Base and height should be enough, but you didn't state whether its a 3 or 4 sided pyramid. i.e. is the base a square or a triangle?

SA = B + n(1/2sl)

SA = 1/2as + (3)(1/2sl)

SA = 1/2as + 3/2sl

This is for triangles bases... all the info you need is on the source page. Just scroll down. Hexagonal pyramids???

To derive the slant height from the height and base you need the measurement from the centre of the base to the nearest point on any of the sides (lets call that z). Then use those two measurements and the slant height is the hypotenuse of the right-angled triangle they make. i.e. z(squared) + h(squared) = slant height (squared)

As far as I know there's no other way.

Answer 4

The area of a pyramid can always be found by adding the faces individually. With skewed pyramids having irregular bases, this may be the only way.

Answer 5

ok, u find the base area first, then u find the area of one of the sides and u multiply that by the amount of sides there are.
Hence, u add the base area to ur second result and u get the surface area of a pyramid.

Answer 6

The base of a pyramid may not be a regular shape.
Even if it is, the pyramid may not be a "RIGHT pyramid" (ie apex is directly over centre of base). (Apex may not be directly over the base at all!)

Therefore unless you have it specified that the pyramid is a right pyramid with a regular base there is no overall formula.
Sorry!

Answer 7

i comprehend E might desire to be suitable. Cos the fringe of base equivalent to p ability that the two factors might desire to be p/4 because of the fact it relatively is a sq.. So the component of the climate would be a million/2 x p/4 x s x 4 = playstation /2. upload to that the BA and you gets the floor section.

Answer 8

4(Bh)/2