Answer :
To determine the correct formulas for finding the surface area [tex]\( S A \)[/tex] of a right prism, we need to recall the structure and properties of right prisms.
The surface area of a right prism presents the sum of the areas of all its faces. This can be broken down as follows:
- The lateral area [tex]\( LA \)[/tex]: This is the sum of the areas of all the rectangular faces that connect the two bases of the prism. For a right prism, the lateral area can be calculated as [tex]\( p \times h \)[/tex], where [tex]\( p \)[/tex] is the perimeter of the base and [tex]\( h \)[/tex] is the height of the prism.
- The area of the bases [tex]\( BA \)[/tex]: This is the sum of the areas of the two parallel bases. Since there are two bases, the total area of the bases is [tex]\( 2 \times B A \)[/tex], where [tex]\( B A \)[/tex] is the area of one base.
Thus, the formula for the total surface area of a right prism is:
[tex]\[ S A = 2BA + LA \][/tex]
Given the provided variables and above formulas, we can evaluate each option:
A. [tex]\( S A=\frac{1}{2} B A+L A \)[/tex]
This formula is incorrect because it incorrectly uses half of the base area instead of twice, which does not align with the correct surface area formula.
B. [tex]\( S A=B A+\angle A \)[/tex]
This formula does not make sense in the context of a right prism, as it includes an angle [tex]\( \angle A \)[/tex] which is not part of the given variables ([tex]\( p \)[/tex], [tex]\( h \)[/tex], [tex]\( B A \)[/tex], [tex]\( L A \)[/tex]) and does not correctly represent the surface area.
C. [tex]\( S A=p+L A \)[/tex]
This formula is incorrect because the parameter [tex]\( p \)[/tex] (perimeter) cannot be added directly to the lateral area [tex]\( L A \)[/tex]. The correct formulation should involve the product of [tex]\( p \)[/tex] and [tex]\( h \)[/tex] to account for the lateral area.
D. [tex]\( S A=B A-L A \)[/tex]
This formula is incorrect because subtracting [tex]\( L A \)[/tex] (lateral area) from [tex]\( B A \)[/tex] (base area) would not represent the total surface area of a right prism.
E. [tex]\( S A=B A+p h \)[/tex]
This formula is correct. The total surface area of the right prism is the area of the bases plus the lateral area. The lateral area is given by [tex]\( p \times h \)[/tex], and the area of the bases is given by [tex]\( 2BA \)[/tex]. Therefore, the formula [tex]\( SA = BA + p \cdot h \)[/tex] accurately represents the total surface area when considering one base area [tex]\( BA \)[/tex] and the perimeter-height product for the laterals.
Hence, the correct formula from the given options is:
E. [tex]\( S A=B A+p h \)[/tex]
The surface area of a right prism presents the sum of the areas of all its faces. This can be broken down as follows:
- The lateral area [tex]\( LA \)[/tex]: This is the sum of the areas of all the rectangular faces that connect the two bases of the prism. For a right prism, the lateral area can be calculated as [tex]\( p \times h \)[/tex], where [tex]\( p \)[/tex] is the perimeter of the base and [tex]\( h \)[/tex] is the height of the prism.
- The area of the bases [tex]\( BA \)[/tex]: This is the sum of the areas of the two parallel bases. Since there are two bases, the total area of the bases is [tex]\( 2 \times B A \)[/tex], where [tex]\( B A \)[/tex] is the area of one base.
Thus, the formula for the total surface area of a right prism is:
[tex]\[ S A = 2BA + LA \][/tex]
Given the provided variables and above formulas, we can evaluate each option:
A. [tex]\( S A=\frac{1}{2} B A+L A \)[/tex]
This formula is incorrect because it incorrectly uses half of the base area instead of twice, which does not align with the correct surface area formula.
B. [tex]\( S A=B A+\angle A \)[/tex]
This formula does not make sense in the context of a right prism, as it includes an angle [tex]\( \angle A \)[/tex] which is not part of the given variables ([tex]\( p \)[/tex], [tex]\( h \)[/tex], [tex]\( B A \)[/tex], [tex]\( L A \)[/tex]) and does not correctly represent the surface area.
C. [tex]\( S A=p+L A \)[/tex]
This formula is incorrect because the parameter [tex]\( p \)[/tex] (perimeter) cannot be added directly to the lateral area [tex]\( L A \)[/tex]. The correct formulation should involve the product of [tex]\( p \)[/tex] and [tex]\( h \)[/tex] to account for the lateral area.
D. [tex]\( S A=B A-L A \)[/tex]
This formula is incorrect because subtracting [tex]\( L A \)[/tex] (lateral area) from [tex]\( B A \)[/tex] (base area) would not represent the total surface area of a right prism.
E. [tex]\( S A=B A+p h \)[/tex]
This formula is correct. The total surface area of the right prism is the area of the bases plus the lateral area. The lateral area is given by [tex]\( p \times h \)[/tex], and the area of the bases is given by [tex]\( 2BA \)[/tex]. Therefore, the formula [tex]\( SA = BA + p \cdot h \)[/tex] accurately represents the total surface area when considering one base area [tex]\( BA \)[/tex] and the perimeter-height product for the laterals.
Hence, the correct formula from the given options is:
E. [tex]\( S A=B A+p h \)[/tex]
