Question

consider a regular pyramid where ( p ) is the perimeter of the base, ( s ) is the slant height, ( ba ) is the base area, and ( la ) is the lateral area. what is a valid formula for the surface area?
check all that apply.

a. ( sa = ba + 2la )
b. ( sa = ba + \frac{1}{2}ps )
c. ( sa = 2ba + \frac{1}{2}ps )
d. ( sa = ba + la )
e. ( sa = 8cdot la )

Explanation:

Step1: Recall the formula for surface area of a pyramid

The surface area (\(SA\)) of a pyramid is the sum of the base area (\(BA\)) and the lateral area (\(LA\)). So \(SA=BA + LA\).

Step2: Recall the formula for lateral area of a regular pyramid

The lateral area (\(LA\)) of a regular pyramid is given by \(LA=\frac{1}{2}ps\) (where \(p\) is the perimeter of the base and \(s\) is the slant height). Substituting \(LA=\frac{1}{2}ps\) into \(SA = BA+LA\), we get \(SA=BA+\frac{1}{2}ps\).

For option A: \(SA = BA + 2LA\) is incorrect because \(SA=BA+LA\) (not \(2LA\)).
For option C: \(SA = 2BA+\frac{1}{2}ps\) is incorrect. There is only one base in a pyramid, so the coefficient of \(BA\) should be \(1\) (not \(2\)).
For option E: \(SA = 8\cdot LA\) has no basis in the surface - area formula of a pyramid.

Answer:

B. \( SA = BA+\frac{1}{2}ps \), D. \( SA = BA + LA \)