Answer :
To determine the lateral area of a right cylinder with height [tex]\( h \)[/tex] and radius [tex]\( r \)[/tex], we should use the correct formula for the lateral area. Let's analyze each given option step-by-step:
A. [tex]\( L A = 2 \pi r^2 \)[/tex]
- The expression [tex]\( 2 \pi r^2 \)[/tex] resembles the formula for the area of two circles, each with radius [tex]\( r \)[/tex], since the area of one circle is [tex]\( \pi r^2 \)[/tex]. Multiplying by 2 gives the total area of the top and bottom circles of the cylinder. However, this does not account for the height [tex]\( h \)[/tex] and therefore does not represent the lateral area of the cylinder.
B. [tex]\( L A = 2 \pi r h \)[/tex]
- The lateral area of a right cylinder is essentially the area of the rectangle that wraps around the sides of the cylinder. When the cylinder is "unrolled" or "unwrapped", the rectangle has a height [tex]\( h \)[/tex] and width equal to the circumference of the base circle, which is [tex]\( 2\pi r \)[/tex]. The area of this rectangle (which is the lateral area of the cylinder) is thus [tex]\( \text{height} \times \text{circumference} = h \times 2 \pi r \)[/tex]. Therefore, [tex]\( L A = 2 \pi r h \)[/tex] is the correct formula for the lateral area of the cylinder.
C. [tex]\( L A = 2 \pi r \)[/tex]
- The expression [tex]\( 2 \pi r \)[/tex] represents the circumference of the base circle of the cylinder. It does not take the height [tex]\( h \)[/tex] into account and thus cannot represent the area of a surface. This formula does not give the lateral area of the cylinder.
D. [tex]\( L A = 2 \pi r^2 + 2 \pi r h \)[/tex]
- This formula seems to combine the area of the two bases ([tex]\( 2 \pi r^2 \)[/tex]) with the lateral area ([tex]\( 2 \pi r h \)[/tex]). While this expression might correctly represent the total surface area of the cylinder (including the two bases and the lateral surface), it is not solely the lateral area.
Given these analyses, the correct formula for the lateral area of a right cylinder is:
B. [tex]\( L A = 2 \pi r h \)[/tex]
Therefore, the correct option is B.
A. [tex]\( L A = 2 \pi r^2 \)[/tex]
- The expression [tex]\( 2 \pi r^2 \)[/tex] resembles the formula for the area of two circles, each with radius [tex]\( r \)[/tex], since the area of one circle is [tex]\( \pi r^2 \)[/tex]. Multiplying by 2 gives the total area of the top and bottom circles of the cylinder. However, this does not account for the height [tex]\( h \)[/tex] and therefore does not represent the lateral area of the cylinder.
B. [tex]\( L A = 2 \pi r h \)[/tex]
- The lateral area of a right cylinder is essentially the area of the rectangle that wraps around the sides of the cylinder. When the cylinder is "unrolled" or "unwrapped", the rectangle has a height [tex]\( h \)[/tex] and width equal to the circumference of the base circle, which is [tex]\( 2\pi r \)[/tex]. The area of this rectangle (which is the lateral area of the cylinder) is thus [tex]\( \text{height} \times \text{circumference} = h \times 2 \pi r \)[/tex]. Therefore, [tex]\( L A = 2 \pi r h \)[/tex] is the correct formula for the lateral area of the cylinder.
C. [tex]\( L A = 2 \pi r \)[/tex]
- The expression [tex]\( 2 \pi r \)[/tex] represents the circumference of the base circle of the cylinder. It does not take the height [tex]\( h \)[/tex] into account and thus cannot represent the area of a surface. This formula does not give the lateral area of the cylinder.
D. [tex]\( L A = 2 \pi r^2 + 2 \pi r h \)[/tex]
- This formula seems to combine the area of the two bases ([tex]\( 2 \pi r^2 \)[/tex]) with the lateral area ([tex]\( 2 \pi r h \)[/tex]). While this expression might correctly represent the total surface area of the cylinder (including the two bases and the lateral surface), it is not solely the lateral area.
Given these analyses, the correct formula for the lateral area of a right cylinder is:
B. [tex]\( L A = 2 \pi r h \)[/tex]
Therefore, the correct option is B.
