Question
Consider the shape below.
Determine the total surface area of the shape. Remember to not round the answers. The formula chart is at the bottom of the page.
(7 points)
Radius of circle = 5
centimeters Area of each Circular Face =
square centimeters
Total Area of the Circular Faces =
square centimeters
Circumference of Circle = 20
centimeters Height of Cylinder =
centimeters
Area of the Lateral Face =
square centimeters
Total Surface Area of Cylinder =
square centimeters
Determine the total surface area of the shape. Remember to not round the answers. The formula chart is at the bottom of the page.
(7 points)
Radius of circle = 5
centimeters Area of each Circular Face =
square centimeters
Total Area of the Circular Faces =
square centimeters
Circumference of Circle = 20
centimeters Height of Cylinder =
centimeters
Area of the Lateral Face =
square centimeters
Total Surface Area of Cylinder =
square centimeters
Answers
GPT-4o mini
To find the total surface area of the cylinder, we will use the formulas for the area of the circular faces and the lateral surface area.
### Given:
- Radius of the circle \( r = 5 \) cm
- Height of the cylinder \( h \) (need to calculate this based on the information you may provide or it might be given in a full problem statement)
### Step 1: Calculate the area of each circular face.
The formula for the area of a circle is:
\[
A = \pi r^2
\]
Substituting the value of \( r = 5 \):
\[
A = \pi (5)^2 = \pi (25) = 25\pi \text{ square centimeters}
\]
### Step 2: Calculate the total area of the circular faces.
A cylinder has two circular faces:
\[
\text{Total Area of the Circular Faces} = 2 \times \text{Area of each Circular Face} = 2 \times 25\pi = 50\pi \text{ square centimeters}
\]
### Step 3: Calculate the circumference of the base.
The circumference of a circle is given by:
\[
C = 2\pi r
\]
Substituting the value of \( r = 5 \):
\[
C = 2\pi (5) = 10\pi \text{ centimeters}
\]
### Step 4: Define height of the cylinder (if not provided).
If the height \( h \) is given or assumed, you would substitute that value in the next step. If it's not specified, let's assume that the height is 10 cm (for example purposes).
### Step 5: Calculate the area of the lateral face of the cylinder.
The formula for the lateral surface area of a cylinder is:
\[
A_{\text{lateral}} = \text{Circumference} \times \text{Height} = C \times h
\]
Using \( C = 10\pi \) and \( h = 10 \):
\[
A_{\text{lateral}} = 10\pi \times 10 = 100\pi \text{ square centimeters}
\]
### Step 6: Calculate the total surface area of the cylinder.
The total surface area \( A_{\text{total}} \) is the sum of the lateral surface area and the total area of the circular faces:
\[
A_{\text{total}} = A_{\text{lateral}} + \text{Total Area of the Circular Faces}
\]
Substituting the values:
\[
A_{\text{total}} = 100\pi + 50\pi = 150\pi \text{ square centimeters}
\]
### Final Answers:
1. **Area of each Circular Face**: \( 25\pi \) square centimeters.
2. **Total Area of the Circular Faces**: \( 50\pi \) square centimeters.
3. **Circumference of Circle**: \( 10\pi \) centimeters.
4. **Height of Cylinder**: \( 10 \) centimeters (assuming).
5. **Area of the Lateral Face**: \( 100\pi \) square centimeters.
6. **Total Surface Area of Cylinder**: \( 150\pi \) square centimeters.
If you have a different height for the cylinder, you would replace the height in those calculations to find the correct total surface area.
### Given:
- Radius of the circle \( r = 5 \) cm
- Height of the cylinder \( h \) (need to calculate this based on the information you may provide or it might be given in a full problem statement)
### Step 1: Calculate the area of each circular face.
The formula for the area of a circle is:
\[
A = \pi r^2
\]
Substituting the value of \( r = 5 \):
\[
A = \pi (5)^2 = \pi (25) = 25\pi \text{ square centimeters}
\]
### Step 2: Calculate the total area of the circular faces.
A cylinder has two circular faces:
\[
\text{Total Area of the Circular Faces} = 2 \times \text{Area of each Circular Face} = 2 \times 25\pi = 50\pi \text{ square centimeters}
\]
### Step 3: Calculate the circumference of the base.
The circumference of a circle is given by:
\[
C = 2\pi r
\]
Substituting the value of \( r = 5 \):
\[
C = 2\pi (5) = 10\pi \text{ centimeters}
\]
### Step 4: Define height of the cylinder (if not provided).
If the height \( h \) is given or assumed, you would substitute that value in the next step. If it's not specified, let's assume that the height is 10 cm (for example purposes).
### Step 5: Calculate the area of the lateral face of the cylinder.
The formula for the lateral surface area of a cylinder is:
\[
A_{\text{lateral}} = \text{Circumference} \times \text{Height} = C \times h
\]
Using \( C = 10\pi \) and \( h = 10 \):
\[
A_{\text{lateral}} = 10\pi \times 10 = 100\pi \text{ square centimeters}
\]
### Step 6: Calculate the total surface area of the cylinder.
The total surface area \( A_{\text{total}} \) is the sum of the lateral surface area and the total area of the circular faces:
\[
A_{\text{total}} = A_{\text{lateral}} + \text{Total Area of the Circular Faces}
\]
Substituting the values:
\[
A_{\text{total}} = 100\pi + 50\pi = 150\pi \text{ square centimeters}
\]
### Final Answers:
1. **Area of each Circular Face**: \( 25\pi \) square centimeters.
2. **Total Area of the Circular Faces**: \( 50\pi \) square centimeters.
3. **Circumference of Circle**: \( 10\pi \) centimeters.
4. **Height of Cylinder**: \( 10 \) centimeters (assuming).
5. **Area of the Lateral Face**: \( 100\pi \) square centimeters.
6. **Total Surface Area of Cylinder**: \( 150\pi \) square centimeters.
If you have a different height for the cylinder, you would replace the height in those calculations to find the correct total surface area.