A right cylinder has a radius of [tex]\( r \)[/tex] inches and a height of [tex]\( 2r \)[/tex] inches.

1. The lateral area of the cylinder is [tex]\( \square \, r^2 \pi \)[/tex] square inches.
2. The area of the two bases together is [tex]\( \square \, r^2 \pi \)[/tex] square inches.
3. The total surface area of the cylinder is [tex]\( \square \, r^2 \pi \)[/tex] square inches.



Answer :

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.