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Which formulas can be used 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], [tex]s[/tex] is the slant height, [tex]B A[/tex] is the base area, and [tex]L A[/tex] is the lateral area?

Check all that apply.

A. [tex]S A = B A - L A[/tex]

B. [tex]S A = B A + \frac{1}{2} p s[/tex]

C. [tex]S A = \frac{1}{2} B A + \frac{1}{2} p s[/tex]

D. [tex]S A = B A \cdot L A[/tex]

E. [tex]S A = B A + L A[/tex]


Sagot :

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]
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