To find the surface area ([tex]\(SA\)[/tex]) of a regular pyramid with a square base, we need to understand the components that make up the surface area. The surface area of such a pyramid is the sum of its base area ([tex]\(BA\)[/tex]) and its lateral area ([tex]\(LA\)[/tex]).
Here, let’s explore each given formula to determine whether it accurately calculates the surface area:
A. [tex]\( SA = BA - LA \)[/tex]
- This formula subtracts the lateral area from the base area, which isn’t logically correct. The surface area should be the sum of the base area and the lateral area.
- Incorrect
B. [tex]\( SA = BA + \frac{1}{2} p s \)[/tex]
- This formula is correct. The lateral area ([tex]\(LA\)[/tex]) of the pyramid is given by [tex]\( \frac{1}{2} p s \)[/tex] where [tex]\(p\)[/tex] is the perimeter of the base and [tex]\(s\)[/tex] is the slant height.
- Hence, adding the base area ([tex]\(BA\)[/tex]) to the lateral area ([tex]\( \frac{1}{2} p s \)[/tex]) correctly gives us the surface area.
- Correct
C. [tex]\( SA = \frac{1}{2} BA + \frac{1}{2} p s \)[/tex]
- This formula incorrectly calculates half the base area, which does not align with the requirement of the total surface area computation.
- Incorrect
D. [tex]\( SA = BA \cdot LA \)[/tex]
- This formula multiplies the base area by the lateral area, which does not represent the correct computation for the surface area of a pyramid.
- Incorrect
E. [tex]\( SA = BA + LA \)[/tex]
- This formula is correct. It states that the surface area is the sum of the base area ([tex]\(BA\)[/tex]) and the lateral area ([tex]\(LA\)[/tex]), which aligns perfectly with how the surface area of a pyramid is computed.
- Correct
Thus, the two formulas that can be used to find the surface area of a regular pyramid with a square base given the base area ([tex]\(BA\)[/tex]), perimeter of the base ([tex]\(p\)[/tex]), and slant height ([tex]\(s\)[/tex]) are:
- B. [tex]\( SA = BA + \frac{1}{2} p s \)[/tex]
- E. [tex]\( SA = BA + LA \)[/tex]
