To find the total surface area of the cylindrical storage box, we will use the following formulas:
- Area of each Circular Face (top and bottom): \[ A_{circle} = \pi r^2 \]
- Total Area of the Circular Faces: \[ A_{total_circles} = 2 \times A_{circle} \]
- Circumference of the Circle: \[ C = 2\pi r \]
- Area of the Lateral Face: \[ A_{lateral} = C \times h \]
- Total Surface Area of the Cylinder: \[ A_{total} = A_{total_circles} + A_{lateral} \]
Now, we’ll calculate each part given the radius and height of the cylinder.
Assuming:
- Radius of circle (\( r \)) = 3 inches
- Height of cylinder (\( h \)) = 10 inches
Step 1: Calculate the Area of each Circular Face
\[ A_{circle} = \pi r^2 = 3.14 \times (3)^2 = 3.14 \times 9 = 28.26 \text{ square inches} \]
Step 2: Calculate the Total Area of the Circular Faces
\[ A_{total_circles} = 2 \times A_{circle} = 2 \times 28.26 = 56.52 \text{ square inches} \]
Step 3: Calculate the Circumference of the Circle
\[ C = 2\pi r = 2 \times 3.14 \times 3 = 18.84 \text{ inches} \]
Step 4: Calculate the Area of the Lateral Face
\[ A_{lateral} = C \times h = 18.84 \times 10 = 188.4 \text{ square inches} \]
Step 5: Calculate the Total Surface Area of the Cylinder
\[ A_{total} = A_{total_circles} + A_{lateral} = 56.52 + 188.4 = 244.92 \text{ square inches} \]
Summary of Calculations
- Radius of circle = 3 inches
- Area of each Circular Face = 28.26 square inches
- Total Area of the Circular Faces = 56.52 square inches
- Circumference of Circle = 18.84 inches
- Height of Cylinder = 10 inches
- Area of the Lateral Face = 188.4 square inches
- Total Surface Area of Cylinder = 244.92 square inches
Now Simone knows that she will need 244.92 square inches of patterned paper to cover the entire box.