To determine the correct formula for the surface area of a regular pyramid, we need to understand how the surface area is composed. The surface area (SA) of a pyramid is the sum of the base area (BA) and the lateral area (LA).
1. Base Area (BA): This is the area of the polygon that forms the base of the pyramid.
2. Lateral Area (LA): This is the sum of the areas of all the triangular faces that connect the base to the apex (summit) of the pyramid.
A generally accepted formula to calculate the lateral area (LA) of a regular pyramid is:
[tex]\[ LA = \frac{1}{2}ps \][/tex]
where [tex]\( p \)[/tex] is the perimeter of the base and [tex]\( s \)[/tex] is the slant height.
The total surface area (SA) of the pyramid is then the sum of the base area and the lateral area:
[tex]\[ SA = BA + LA \][/tex]
Substituting the formula for LA, we get:
[tex]\[ SA = BA + \frac{1}{2}ps \][/tex]
Let's examine each of the given options:
A. [tex]\( SA = 2BA + \frac{1}{2}ps \)[/tex]
- This formula incorrectly doubles the base area, which is not part of the standard formula for the surface area. So, this option is not correct.
B. [tex]\( SA = BA + 2LA \)[/tex]
- This formula doubles the lateral area, which is incorrect because the lateral area should not be multiplied by 2. Thus, this option is not correct.
C. [tex]\( SA = BA + LA \)[/tex]
- The symbol [tex]\( \angle A \)[/tex] used in the problem statement is unclear and does not relate properly to the components needed for the pyramid's surface area calculation. This option is not correct.
D. [tex]\( SA = 8 \cdot LA \)[/tex]
- This formula multiples the lateral area by 8, which doesn't follow the correct formula for a pyramid's surface area calculation. So, this option is not correct.
E. [tex]\( SA = BA + \frac{1}{2}ps \)[/tex]
- This matches our derived formula for the surface area. This option is correct.
Therefore, the correct option(s) to check for the surface area of a regular pyramid is/are:
[tex]\[ \boxed{E} \][/tex]
In summary, the valid formula for the surface area of a regular pyramid, where [tex]\( p \)[/tex] is the perimeter of the base, [tex]\( s \)[/tex] is the slant height, and [tex]\( BA \)[/tex] is the base area, is given by:
[tex]\[ SA = BA + \frac{1}{2} p s \][/tex]
Which corresponds to option [tex]\( \boxed{E} \)[/tex].
