To find the surface area [tex]\( SA \)[/tex] of a regular pyramid with a square base, let's first consider what each element in the formula refers to:
- [tex]\( BA \)[/tex] is the base area of the pyramid.
- [tex]\( p \)[/tex] is the perimeter of the base.
- [tex]\( s \)[/tex] is the slant height of the pyramid.
- [tex]\( LA \)[/tex] is the lateral area of the pyramid, which is the area of all the triangular faces excluding the base.
The formula for the surface area of a regular pyramid with a square base can be given as:
[tex]\[ SA = BA + LA \][/tex]
Where [tex]\( LA \)[/tex] (the lateral area) for a pyramid with a square base can be calculated using:
[tex]\[ LA = \frac{1}{2} \times p \times s \][/tex]
This is because the lateral area is made up of the triangular faces. Each triangular face has a base equal to one side of the square base and a height equal to the slant height [tex]\( s \)[/tex].
Combining these, we get:
[tex]\[ SA = BA + \frac{1}{2} \times p \times s \][/tex]
Now let's analyze the options provided:
A. [tex]\( SA = BA \cdot \angle A \)[/tex]
- This formula doesn't make sense for finding the surface area. The base area multiplied by an angle does not yield the surface area.
B. [tex]\( SA = BA - \angle A \)[/tex]
- Subtracting an angle from the base area does not logically relate to finding the surface area of a pyramid.
C. [tex]\( SA = BA + \angle A \)[/tex]
- Adding an angle to the base area is also not a valid formula for the surface area.
D. [tex]\( SA = \frac{1}{2} BA + \frac{1}{2} p s \)[/tex]
- This formula incorrectly uses only half of the base area, which is not the standard formula for the surface area of a pyramid with a square base.
E. [tex]\( SA = BA + \frac{1}{2} p s \)[/tex]
- This is the correct formula. It represents the sum of the base area and the lateral area, which is calculated as half the product of the perimeter of the base and the slant height.
Therefore, the formulas that can be used to find the surface area of a regular pyramid with a square base are represented by option:
E. [tex]\( SA = BA + \frac{1}{2} p s \)[/tex]
