How do I find the side of the base, the lateral area, and the surface area of a square base pyramid with only the height and slant height?

Question

How do I find the side of the base, the lateral area, and the surface area of a square base pyramid with only the height and slant height? H=7cm and slanted height is 25cm. I need to know how to find the surface area, lateral area, and length of the side of the base :(

Answer 1

The key to solving this triangle is understanding where the top of the pyramid is in relation to the square base.

If the square is of length s, the top is height h and located above a point 1/2s from each side. Then, you can see the relationship between the slant height, the pyramid height, and s. We have a triangle with hypotenuse slant height, one side equal to height, and one side equal to 1/2 s. The slant height is 25, and the height is 7, so 1/2s = 24, and s = 48.

Then, each triangle on the lateral sides of the triangle has base s and altitude equal to slant height, so the area is 1/2 s (slant height) = 1/2(48)25. There are 4 of these sides, so the lateral area is 4(1/2)(48)25 = 2400.

As the base is 48 x 48 and has total area 2304, the total surface area is 2400 + 2304 = 2704. Also, the relative size of the lateral area and base makes sense, as the lateral area will always be greater than the base but, as the pyramid only has height 7 versus side length of 48, we shouldn't expect it to be a lot greater.

Not asked, but provided: the volume of a pyramid is 1/3 bh, and the base in this case is s², so the volume is 1/3s²h