To find the total surface area of the cylinder, including the top and bottom faces, we need to calculate the lateral surface area and the area of the two circular faces and then sum these areas.
1. **Lateral Surface Area (A₁)**:
The formula for the lateral surface area of a cylinder is:
\[
A₁ = 2 \pi r h
\]
where \( r \) is the radius and \( h \) is the height. Plugging in the given values:
\[
A₁ = 2 \times 3.14 \times 4 \times 3 = 75.36 \, \text{square feet}
\]
2. **Area of the Top and Bottom Faces (A₂)**:
Each circular face has an area given by:
\[
A = \pi r^2
\]
Since there are two faces, we double this area:
\[
A₂ = 2 \times \pi r^2 = 2 \times 3.14 \times 4^2 = 2 \times 3.14 \times 16 = 100.48 \, \text{square feet}
\]
3. **Total Surface Area (A_total)**:
Adding the lateral surface area and the areas of the top and bottom faces:
\[
A_{\text{total}} = A₁ + A₂ = 75.36 + 100.48 = 175.84 \, \text{square feet}
\]
Rounded to the nearest tenth, the surface area of the cylinder is:
\[
\boxed{175.8} \, \text{square feet}
\]
Cones, Cylinders, & Spheres Unit Test
3 of 153 of 15 Items
Question
Adrian is painting the outside of a cylinder that he plans to use as a coffee table. The cylinder has a radius of 4 feet and a height of 3 feet. Adrian wants to paint all around the outside of the cylinder, including the top and bottom faces. In order to understand how much paint is needed, he wants to know the surface are of the outside of the cylinder. What is the surface area of the cylinder, measured in square feet? Use 3.14 for pi and round your answer to the nearest tenth.(1 point)
ft2
3 answers
are you sure?
Let's go through the calculations again to confirm.
1. **Lateral Surface Area (A₁)**
The formula for the lateral surface area of a cylinder is:
\[
A₁ = 2 \pi r h
\]
Given:
- \( r = 4 \) feet
- \( h = 3 \) feet
- \( \pi = 3.14 \)
Plugging in the values:
\[
A₁ = 2 \times 3.14 \times 4 \times 3 = 2 \times 3.14 \times 12 = 75.36 \, \text{square feet}
\]
2. **Area of the Top and Bottom Faces (A₂)**
Each circular face has an area given by:
\[
A = \pi r^2
\]
For one face:
\[
A = 3.14 \times 4^2 = 3.14 \times 16 = 50.24 \, \text{square feet}
\]
Since there are two faces:
\[
A₂ = 2 \times 50.24 = 100.48 \, \text{square feet}
\]
3. **Total Surface Area (A_total)**
Adding the lateral surface area and the areas of the top and bottom faces:
\[
A_{\text{total}} = A₁ + A₂ = 75.36 + 100.48 = 175.84 \, \text{square feet}
\]
Rounded to the nearest tenth, the surface area of the cylinder is:
\[
\boxed{175.8} \, \text{square feet}
\]
So, after recalculating, I confirm that the total surface area is indeed **175.8 square feet**.
1. **Lateral Surface Area (A₁)**
The formula for the lateral surface area of a cylinder is:
\[
A₁ = 2 \pi r h
\]
Given:
- \( r = 4 \) feet
- \( h = 3 \) feet
- \( \pi = 3.14 \)
Plugging in the values:
\[
A₁ = 2 \times 3.14 \times 4 \times 3 = 2 \times 3.14 \times 12 = 75.36 \, \text{square feet}
\]
2. **Area of the Top and Bottom Faces (A₂)**
Each circular face has an area given by:
\[
A = \pi r^2
\]
For one face:
\[
A = 3.14 \times 4^2 = 3.14 \times 16 = 50.24 \, \text{square feet}
\]
Since there are two faces:
\[
A₂ = 2 \times 50.24 = 100.48 \, \text{square feet}
\]
3. **Total Surface Area (A_total)**
Adding the lateral surface area and the areas of the top and bottom faces:
\[
A_{\text{total}} = A₁ + A₂ = 75.36 + 100.48 = 175.84 \, \text{square feet}
\]
Rounded to the nearest tenth, the surface area of the cylinder is:
\[
\boxed{175.8} \, \text{square feet}
\]
So, after recalculating, I confirm that the total surface area is indeed **175.8 square feet**.