Answer :
To find the lateral area of a right cone, we use the formula for the lateral area [tex]\( \text{LA} \)[/tex]. The correct formula is given by:
[tex]\[ \text{LA} = \pi r s \][/tex]
where:
- [tex]\( r \)[/tex] is the radius of the base of the cone.
- [tex]\( s \)[/tex] is the slant height of the cone.
Analyzing the given options:
Option A: [tex]\( \text{LA} = \frac{1}{2} \pi r s \)[/tex]
This is incorrect because the formula for the lateral area of a cone does not include the factor [tex]\(\frac{1}{2}\)[/tex].
Option B: [tex]\( \text{LA} = r s \)[/tex]
This is also incorrect because it lacks the [tex]\(\pi\)[/tex] factor. The complete formula must include [tex]\(\pi\)[/tex].
Option C: [tex]\( \text{LA} = \pi r^2 + \pi r s \)[/tex]
This formula adds the base area ([tex]\(\pi r^2\)[/tex]), which is not part of the lateral area. The lateral area should only account for the side surface area, not the base.
Option D: [tex]\( \text{LA} = \pi r s \)[/tex]
This is the correct formula for the lateral area of a right cone. It correctly incorporates both the radius [tex]\( r \)[/tex] and the slant height [tex]\( s \)[/tex] with the [tex]\(\pi\)[/tex] factor.
Therefore, the correct option is:
D. [tex]\( \text{LA} = \pi r s \)[/tex]
[tex]\[ \text{LA} = \pi r s \][/tex]
where:
- [tex]\( r \)[/tex] is the radius of the base of the cone.
- [tex]\( s \)[/tex] is the slant height of the cone.
Analyzing the given options:
Option A: [tex]\( \text{LA} = \frac{1}{2} \pi r s \)[/tex]
This is incorrect because the formula for the lateral area of a cone does not include the factor [tex]\(\frac{1}{2}\)[/tex].
Option B: [tex]\( \text{LA} = r s \)[/tex]
This is also incorrect because it lacks the [tex]\(\pi\)[/tex] factor. The complete formula must include [tex]\(\pi\)[/tex].
Option C: [tex]\( \text{LA} = \pi r^2 + \pi r s \)[/tex]
This formula adds the base area ([tex]\(\pi r^2\)[/tex]), which is not part of the lateral area. The lateral area should only account for the side surface area, not the base.
Option D: [tex]\( \text{LA} = \pi r s \)[/tex]
This is the correct formula for the lateral area of a right cone. It correctly incorporates both the radius [tex]\( r \)[/tex] and the slant height [tex]\( s \)[/tex] with the [tex]\(\pi\)[/tex] factor.
Therefore, the correct option is:
D. [tex]\( \text{LA} = \pi r s \)[/tex]
