how do you find the area of a single lateral face of a pyramid, without having the slant height?

Question

the height of the pyramid is 3 and the base area is 49, it says the answers supposed to be 17.5, but i can't get that

Answer 1

If you don't know anything else about the pyramid, then the problem can't be solved. If you know that the base is regular, then you can do it.

If the base is regular, then you can figure the apothem of the base by using the area = (1/2)* apothem * perimeter formula (and using the fact that all three sides of the base are the same). Once you have the apothem, a quick sketch will show you that the apothem and the height of the pyramid form the legs of a right triangle, with the slant height of the faces as the hypotenuse.

Answer 2

The face of a pyramid is a triangle. The area of a triangle is 1/2 X base X height. The base is 49, but that is the area, so the question is whether the base is a triangle or a square. If it is a square (four faces on the pyramid) then the lenght of a side of the square is 7 because 7 X 7 = 49. So 1/2 X base = 3 1/2. If the area of the face is 17 1/2, then that means that the height would have to be 17.5 / 3.5 = 5. You say the height is 3, but the height you are talking about is to the top of the pyramid, and the side of the pyramid is leaning in towards the point at the top.

It sounds like it would be just about impossible for the area of the side to be 17.5 given the conditions you mentioned, unless the base of the pyramid is a triangle (3 faces) instead of square (4 faces).