To solve the problem, we need to calculate three things: the lateral area of the cylinder, the area of the two bases together, and the total surface area of the cylinder.
1. Lateral Area of the Cylinder:
The lateral area [tex]\(A_{\text{lateral}}\)[/tex] of a cylinder is given by the formula:
[tex]\[
A_{\text{lateral}} = 2 \pi r h
\][/tex]
where [tex]\(r\)[/tex] is the radius and [tex]\(h\)[/tex] is the height. In this case, the height [tex]\(h\)[/tex] is given as [tex]\(2r\)[/tex].
Substituting [tex]\(h = 2r\)[/tex] into the formula:
[tex]\[
A_{\text{lateral}} = 2 \pi r (2r) = 4 \pi r^2
\][/tex]
Hence, the lateral area of the cylinder is:
[tex]\[
4 \pi r^2 \text{ square inches}
\][/tex]
2. Area of the Two Bases Together:
Each base of the cylinder is a circle with area [tex]\(A_{\text{base}}\)[/tex] given by the formula:
[tex]\[
A_{\text{base}} = \pi r^2
\][/tex]
Since the cylinder has two bases, the total area of the two bases together is:
[tex]\[
A_{\text{bases}} = 2 \times \pi r^2 = 2 \pi r^2
\][/tex]
Hence, the area of the two bases together is:
[tex]\[
2 \pi r^2 \text{ square inches}
\][/tex]
3. Total Surface Area of the Cylinder:
The total surface area [tex]\(A_{\text{total}}\)[/tex] is the sum of the lateral area and the area of the two bases:
[tex]\[
A_{\text{total}} = A_{\text{lateral}} + A_{\text{bases}} = 4 \pi r^2 + 2 \pi r^2 = 6 \pi r^2
\][/tex]
Hence, the total surface area of the cylinder is:
[tex]\[
6 \pi r^2 \text{ square inches}
\][/tex]
Putting it all together:
- The lateral area of the cylinder is [tex]\( \boxed{4} r^2 \pi\)[/tex] square inches.
- The area of the two bases together is [tex]\( \boxed{2} r^2 \pi\)[/tex] square inches.
- The total surface area of the cylinder is [tex]\( \boxed{6} r^2 \pi\)[/tex] square inches.
