To determine the surface area of a cylinder with a radius of 4.5 cm and a height of 12 cm, we can follow these detailed steps:
1. Finding the Area of the Base:
The base of the cylinder is a circle. The formula to find the area of a circle is:
[tex]\[
A = \pi r^2
\][/tex]
where [tex]\( r \)[/tex] is the radius of the circle.
Given:
[tex]\[
r = 4.5 \, \text{cm}
\][/tex]
Substituting the radius into the formula, we get:
[tex]\[
A = \pi (4.5 \, \text{cm})^2 \approx 63.617 \, \text{cm}^2
\][/tex]
Therefore, the area of one base is approximately 63.617 square centimeters.
2. Finding the Lateral Surface Area:
The lateral surface area of a cylinder can be found using the formula:
[tex]\[
L = 2\pi rh
\][/tex]
where [tex]\( r \)[/tex] is the radius and [tex]\( h \)[/tex] is the height of the cylinder.
Given:
[tex]\[
r = 4.5 \, \text{cm}, \quad h = 12 \, \text{cm}
\][/tex]
Substituting these values into the formula, we get:
[tex]\[
L = 2\pi (4.5 \, \text{cm})(12 \, \text{cm}) \approx 339.292 \, \text{cm}^2
\][/tex]
Therefore, the lateral surface area is approximately 339.292 square centimeters.
3. Finding the Total Surface Area:
The total surface area of the cylinder is the sum of the lateral surface area and the areas of the two bases. The formula for the total surface area is:
[tex]\[
\text{Total Surface Area} = 2A + L
\][/tex]
where [tex]\( A \)[/tex] is the area of one base and [tex]\( L \)[/tex] is the lateral surface area.
Substituting the values we found earlier:
[tex]\[
\text{Total Surface Area} = 2(63.617 \, \text{cm}^2) + 339.292 \, \text{cm}^2 \approx 466.526 \, \text{cm}^2
\][/tex]
Therefore, the total surface area of the cylinder is approximately 466.526 square centimeters.
Explanation of Surface Area:
The surface area of a solid figure like a cylinder is the total area that covers the outer surfaces of the figure. In this case, the surface area includes the area of the two circular bases and the lateral (side) surface that wraps around the cylinder. The surface area measures how much material would be needed, for example, to wrap the cylinder completely with a thin sheet.
