To determine the correct formula for the volume of a cylinder, let's carefully analyze each of the options provided.
We know that the volume [tex]\( V \)[/tex] of a cylinder is found by multiplying the area of its base by its height.
1. Area of the base:
- The base of a cylinder is a circle.
- The area [tex]\( A \)[/tex] of a circle is given by the formula [tex]\( A = \pi r^2 \)[/tex], where [tex]\( r \)[/tex] is the radius.
2. Height of the cylinder:
- The height is denoted by [tex]\( h \)[/tex].
To find the volume [tex]\( V \)[/tex] of the cylinder, we must multiply the area of the base by the height [tex]\( h \)[/tex]:
[tex]\[ V = (\text{area of the base}) \times (\text{height}) \][/tex]
[tex]\[ V = \pi r^2 \times h \][/tex]
[tex]\[ V = \pi r^2 h \][/tex]
Now let's match this formula to the given options:
A. [tex]\( V = \pi r h \)[/tex]
- This formula is incorrect because it lacks the square of the radius [tex]\( r \)[/tex].
B. [tex]\( V = \frac{1}{2} \pi r h \)[/tex]
- This formula is incorrect because it not only omits the square of the radius but also includes an unnecessary factor of [tex]\( \frac{1}{2} \)[/tex].
C. [tex]\( V = \pi r^2 h \)[/tex]
- This formula matches the derived formula exactly. It correctly includes the square of the radius and the height.
D. [tex]\( V = 2 \pi r h \)[/tex]
- This formula is incorrect because it includes an additional factor of 2 and misses the square of the radius [tex]\( r \)[/tex].
Hence, the correct formula for the volume of a cylinder is:
[tex]\[ V = \pi r^2 h \][/tex]
From the given options, this matches option C.
Therefore, the correct answer is:
C. [tex]\( V = \pi r^2 h \)[/tex]
