To determine which formulas can be used to find the surface area of a regular pyramid with a square base, where the given parameters are [tex]\( p \)[/tex] (the perimeter of the base), [tex]\( s \)[/tex] (the slant height), [tex]\( BA \)[/tex] (the base area), and [tex]\( LA \)[/tex] (the lateral area), let's review the correct formulas:
1. Formula A: [tex]\(SA = BA + \frac{1}{2} p s\)[/tex]
- The surface area [tex]\(SA\)[/tex] of a regular pyramid with a square base can be found by adding the base area [tex]\(BA\)[/tex] to half the product of the perimeter [tex]\(p\)[/tex] and the slant height [tex]\(s\)[/tex].
2. Formula B: [tex]\(SA = \frac{1}{2} BA + \frac{1}{2} p s\)[/tex]
- This formula is incorrect because the correct contribution from the base area should be [tex]\(BA\)[/tex], not [tex]\(\frac{1}{2} BA\)[/tex].
3. Formula C: [tex]\(SA = BA \cdot LA\)[/tex]
- This formula is incorrect because the surface area of a pyramid is not found by multiplying the base area by the lateral area.
4. Formula D: [tex]\(SA = BA + LA\)[/tex]
- This formula is correct as it correctly identifies that the surface area of a pyramid is the sum of the base area and the lateral area.
5. Formula E: [tex]\(SA = BA - \angle A\)[/tex]
- This formula is clearly incorrect as there is no meaningful geometric calculation involving subtracting an angle from the base area when determining the surface area of a pyramid.
Therefore, the correct formulas to find the surface area of a regular pyramid with a square base where the perimeter of the base is equal to [tex]\( p \)[/tex], the slant height is [tex]\( s \)[/tex], the base area is [tex]\( BA \)[/tex], and the lateral area is [tex]\( LA \)[/tex] are:
- Formula A: [tex]\( SA = BA + \frac{1}{2} p s \)[/tex]
- Formula D: [tex]\( SA = BA + LA \)[/tex]
So, the correct answers are options A and D.
