To determine which formulas correctly calculate the surface area of a right cone, recall the necessary components for the surface area:
1. Lateral Area (LA): [tex]\( LA = \pi r s \)[/tex], where [tex]\( r \)[/tex] is the radius and [tex]\( s \)[/tex] is the slant height.
2. Base Area (BA): [tex]\( BA = \pi r^2 \)[/tex], where [tex]\( r \)[/tex] is the radius.
The total surface area of a right cone is the sum of the lateral area and the base area:
[tex]\[ \text{Surface Area} = \pi r s + \pi r^2 \][/tex]
Now let's analyze each option step by step:
- Option A: [tex]\( 2 L A + \pi r^2 \)[/tex]
Substituting [tex]\( LA = \pi r s \)[/tex]:
[tex]\[ 2 (\pi r s) + \pi r^2 = 2 \pi r s + \pi r^2 \][/tex]
This formula has an extra [tex]\( \pi r s \)[/tex], which does not represent the surface area correctly.
- Option B: [tex]\( BA + 2 \pi r^2 \)[/tex]
Substituting [tex]\( BA = \pi r^2 \)[/tex]:
[tex]\[ \pi r^2 + 2 \pi r^2 = 3 \pi r^2 \][/tex]
This formula includes an extra [tex]\( \pi r^2 \)[/tex], which is incorrect.
- Option C: [tex]\( \pi r^2 + \pi r s \)[/tex]
This directly matches the components of the surface area:
[tex]\[ \pi r^2 + \pi r s \][/tex]
This formula accurately represents the surface area.
- Option D: [tex]\( 2 \pi r^2 + 2 \pi r h \)[/tex]
This includes [tex]\( h \)[/tex] (height) instead of [tex]\( s \)[/tex] (slant height):
[tex]\[ 2 \pi r^2 + 2 \pi r h \][/tex]
This is incorrect as [tex]\( h \)[/tex] is not used in the surface area calculation for the cone.
- Option E: [tex]\( BA + \angle A \)[/tex]
This formula is nonsensical in the context of surface area calculation because the angle ([tex]\(\angle A\)[/tex]) is not involved in the surface area of a cone.
Thus, the correct option is:
[tex]\[ \boxed{C} \][/tex]
