confusion in Area?

Question

hi, i dont understand the definition of Area.

ok...let me tell you an example...

http://www.sfzc.org/Media/photos/empty_bowl.jpg

please see that image....if i ask whats the Area of that object ....do you consider the empty Top circular area also ?
do you mean Area in this case 2*pi*radius^2 + pi*radius^2(top)

is it ok ?

Q2:

see, this image...

http://www.gomath.com/geometry/pyramid.jpg

suppose, the above pyramid is

(a)solid pyramid
(b)empty pyramid

does the Area (a)= Area(b) ?

mathsmart
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you mean, all empty area should be excluded ?

for example, in the 1st image link....do u want to exclude top area for calculating the total area.

-----------------------
unstable


ok...can u plz tell.
What is the total surface area of the bowl?
i want to see, whether the top circular area to be included or not.

steve:
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>'empty' or not. The external area does not depend >on the solidity.

so, (1) will you consider the base area when the pyramid is empty ?

(2)) will you consider the base area when the pyramid is sloid ?

Answer 1

Usually when someone says area with respect to a solid, they are talking about surface area. More specifically, with shapes like cylinders, prisms, cones, and pyramids, the area of all the faces and surfaces excluding the base(s) is called the lateral area of the solid. The area of all the faces and surfaces including the base(s) is called the total surface area. If you are asked for surface area and it is not specified whether it is lateral or total, I would say you should report the total surface area.

In the case of a cylinder that is open on top, the total surface area would be given by (pi)(r^2)(h) + (pi)(r^2). That would give the lateral area and the bottom base, but not the top because the top is not part of the shape.

In the case of the pyramid, the lateral area and the total surface area are not affected by whether the pyramid is solid or empty.

Answer 2

Area isn't automatically defined by a geometric shape. You have to specify what area you are talking about. for example, in your first link, the area could be the outside surface area only, or the sum of the inside and outside surface areas. The key word is surface. If there is no surface, there is no area, so the open top would definitely not count. The first part of your formula would not give the area excluding the top. The link shows a rounded cup shape, so no simple formula like that would come close to the right answer. If the figure was purely a circular cylinder, the formula (including the bottom) would be pi r^2 + 2 pi r h

Fig 2: A 'pyramid' is a soild shape regardless of whether it is 'empty' or not. The external area does not depend on the solidity.

PS: Watch out for the first term in Mathsmart's equation: It has the units of volume, not area (r^2*h). The 2 should go in front of the pi like mine above..........

Answer 3

Area is essentially a 2-dimensional quantity and has no meaning for a 3-dimensional objects that you have posted.

For 3-dimensional objects it is more meaningful to ask "surface areas" of defined parameters upon the surface of the object. So take your first example of the bowl, you cannot simply ask what is the area of the bowl as the question is meaningless. But you can ask the following questions (that would be more meaningful):
(a) What is the area occupied by the base of the bowl?
(b) What is the area of the mouth of the bowl?
(c) What is the total surface area of the bowl?

Similarly for your question 2. There are additional parameters missing which may explain why you are confused.

If you imagine if I give you a triangle and ask you a question like what is the length (one-dimensional quality) of the triangle (2-dimensional object) without giving you additional reference points you would be similarly hard-pressed to answer it.