Answer :
To find the surface area of a right cylinder with a radius [tex]\( r \)[/tex] inches and a height of [tex]\( 2r \)[/tex] inches, we need to break down the calculation into the lateral area and the area of the bases.
### Lateral Area
The lateral surface area of a cylinder is given by the formula:
[tex]\[ \text{Lateral Area} = 2 \pi r h \][/tex]
Here, [tex]\( h \)[/tex] is the height of the cylinder. Given the height [tex]\( h = 2r \)[/tex], we substitute:
[tex]\[ \text{Lateral Area} = 2 \pi r (2r) = 4 \pi r^2 \][/tex]
So, the lateral area is [tex]\( 4 \pi r^2 \)[/tex] square inches.
### Area of the Bases
A cylinder has two bases, and each base is a circle with radius [tex]\( r \)[/tex]. The area of one circle is given by:
[tex]\[ \text{Area of one base} = \pi r^2 \][/tex]
Since there are two bases, the total area of the two bases together is:
[tex]\[ \text{Total Base Area} = 2 \pi r^2 \][/tex]
So, the area of the two bases together is [tex]\( 2 \pi r^2 \)[/tex] square inches.
### Total Surface Area
The total surface area of the cylinder includes both the lateral area and the area of the two bases combined. Therefore, we add the lateral area and the total base area:
[tex]\[ \text{Total Surface Area} = \text{Lateral Area} + \text{Total Base Area} = 4 \pi r^2 + 2 \pi r^2 = 6 \pi r^2 \][/tex]
So, the total surface area is [tex]\( 6 \pi r^2 \)[/tex] square inches.
### Final Answer
- The lateral area of the cylinder is [tex]\( 4 \pi r^2 \)[/tex] square inches.
- The area of the two bases together is [tex]\( 2 \pi r^2 \)[/tex] square inches.
- The total surface area of the cylinder is [tex]\( 6 \pi r^2 \)[/tex] square inches.
### Lateral Area
The lateral surface area of a cylinder is given by the formula:
[tex]\[ \text{Lateral Area} = 2 \pi r h \][/tex]
Here, [tex]\( h \)[/tex] is the height of the cylinder. Given the height [tex]\( h = 2r \)[/tex], we substitute:
[tex]\[ \text{Lateral Area} = 2 \pi r (2r) = 4 \pi r^2 \][/tex]
So, the lateral area is [tex]\( 4 \pi r^2 \)[/tex] square inches.
### Area of the Bases
A cylinder has two bases, and each base is a circle with radius [tex]\( r \)[/tex]. The area of one circle is given by:
[tex]\[ \text{Area of one base} = \pi r^2 \][/tex]
Since there are two bases, the total area of the two bases together is:
[tex]\[ \text{Total Base Area} = 2 \pi r^2 \][/tex]
So, the area of the two bases together is [tex]\( 2 \pi r^2 \)[/tex] square inches.
### Total Surface Area
The total surface area of the cylinder includes both the lateral area and the area of the two bases combined. Therefore, we add the lateral area and the total base area:
[tex]\[ \text{Total Surface Area} = \text{Lateral Area} + \text{Total Base Area} = 4 \pi r^2 + 2 \pi r^2 = 6 \pi r^2 \][/tex]
So, the total surface area is [tex]\( 6 \pi r^2 \)[/tex] square inches.
### Final Answer
- The lateral area of the cylinder is [tex]\( 4 \pi r^2 \)[/tex] square inches.
- The area of the two bases together is [tex]\( 2 \pi r^2 \)[/tex] square inches.
- The total surface area of the cylinder is [tex]\( 6 \pi r^2 \)[/tex] square inches.
