To determine the total surface area of the cylindrical shape described, we will need to use the following formulas:
-
Area of each Circular Face:
\[ A_{circle} = \pi r^2 \]
where \( r \) is the radius of the circle. -
Total Area of the Circular Faces:
Since there are two circular faces (top and bottom), this is given by:
\[ A_{circular,faces} = 2 \cdot A_{circle} = 2 \pi r^2 \] -
Circumference of the Circle:
This can be calculated using:
\[ C = 2 \pi r \] -
Area of the Lateral Face:
This is calculated as:
\[ A_{lateral} = C \cdot h = 2 \pi r h \]
where \( h \) is the height of the cylinder. -
Total Surface Area of the Cylinder:
This is the sum of the areas of the circular faces and the lateral face:
\[ A_{total} = A_{circular,faces} + A_{lateral} = 2 \pi r^2 + 2 \pi r h \]
Now, let's assign the values based on the information provided:
- Let's assume the radius of the circle \( r = 5 \) cm.
- The height of the cylinder \( h = 20 \) cm.
Step 1: Calculate the Area of Each Circular Face
\[ A_{circle} = \pi (5^2) = \pi (25) = 25\pi , \text{cm}^2 \]
Step 2: Calculate the Total Area of the Circular Faces
\[ A_{circular,faces} = 2 \cdot A_{circle} = 2 \cdot 25\pi = 50\pi , \text{cm}^2 \]
Step 3: Calculate the Circumference of the Circle
\[ C = 2\pi (5) = 10\pi , \text{cm} \]
Step 4: Calculate the Area of the Lateral Face
\[ A_{lateral} = C \cdot h = (10\pi)(20) = 200\pi , \text{cm}^2 \]
Step 5: Calculate the Total Surface Area of the Cylinder
\[ A_{total} = A_{circular,faces} + A_{lateral} = 50\pi + 200\pi = 250\pi , \text{cm}^2 \]
Final Answers:
- Radius of circle = 5 centimeters
- Area of each Circular Face = \( 25\pi \) square centimeters
- Total Area of the Circular Faces = \( 50\pi \) square centimeters
- Circumference of Circle = \( 10\pi \) centimeters
- Height of Cylinder = 20 centimeters
- Area of the Lateral Face = \( 200\pi \) square centimeters
- Total Surface Area of Cylinder = \( 250\pi \) square centimeters
Feel free to let me know if you have any other questions or need further assistance!