Answer :
To determine the correct formula for finding the lateral area of a right cylinder with height [tex]\( h \)[/tex] and radius [tex]\( r \)[/tex], let's analyze each option provided:
1. Option A: [tex]\( LA = 2\pi rh \)[/tex]
- The lateral area of a cylinder is calculated by unwrapping the cylindrical surface into a rectangle. The height of this rectangle is [tex]\( h \)[/tex] (the height of the cylinder), and the width is the circumference of the base circle, which is given by [tex]\( 2\pi r \)[/tex]. Therefore, the lateral surface area [tex]\( LA \)[/tex] is:
[tex]\[ LA = \text{circumference} \times \text{height} = 2\pi r \times h = 2\pi rh \][/tex]
2. Option B: [tex]\( LA = 2\pi r^2 \)[/tex]
- This formula does not make sense for our context. The term [tex]\( 2\pi r^2 \)[/tex] typically represents the combined area of the two circular bases of the cylinder, not the lateral area.
3. Option C: [tex]\( LA = 2\pi r^2 + 2\pi rh \)[/tex]
- This formula includes both the total surface area of the cylinder (the lateral area plus the area of the two circular bases). Since we are interested only in the lateral area, this option is not correct.
4. Option D: [tex]\( LA = 2\pi r \)[/tex]
- This formula represents just the circumference of the base circle, not an area expression. Consequently, it is not suitable for finding the lateral area of a cylinder.
Thus, we see that the correct formula for the lateral area of a right cylinder with height [tex]\( h \)[/tex] and radius [tex]\( r \)[/tex] is given by Option A:
[tex]\[ LA = 2\pi rh \][/tex]
Therefore, the correct answer is option 1 (A), which aligns with our understanding of the geometry of a cylinder.
1. Option A: [tex]\( LA = 2\pi rh \)[/tex]
- The lateral area of a cylinder is calculated by unwrapping the cylindrical surface into a rectangle. The height of this rectangle is [tex]\( h \)[/tex] (the height of the cylinder), and the width is the circumference of the base circle, which is given by [tex]\( 2\pi r \)[/tex]. Therefore, the lateral surface area [tex]\( LA \)[/tex] is:
[tex]\[ LA = \text{circumference} \times \text{height} = 2\pi r \times h = 2\pi rh \][/tex]
2. Option B: [tex]\( LA = 2\pi r^2 \)[/tex]
- This formula does not make sense for our context. The term [tex]\( 2\pi r^2 \)[/tex] typically represents the combined area of the two circular bases of the cylinder, not the lateral area.
3. Option C: [tex]\( LA = 2\pi r^2 + 2\pi rh \)[/tex]
- This formula includes both the total surface area of the cylinder (the lateral area plus the area of the two circular bases). Since we are interested only in the lateral area, this option is not correct.
4. Option D: [tex]\( LA = 2\pi r \)[/tex]
- This formula represents just the circumference of the base circle, not an area expression. Consequently, it is not suitable for finding the lateral area of a cylinder.
Thus, we see that the correct formula for the lateral area of a right cylinder with height [tex]\( h \)[/tex] and radius [tex]\( r \)[/tex] is given by Option A:
[tex]\[ LA = 2\pi rh \][/tex]
Therefore, the correct answer is option 1 (A), which aligns with our understanding of the geometry of a cylinder.
