what is the total surface area including the inside of a ring which is .375” thick, had an outside diameter of 7 5/16” and an inside diameter of 5 3/16”

To find the total surface area of a ring (which is essentially a hollow cylinder), including both the inside and outside surfaces, as well as the top and bottom surfaces, we can break down the problem into several parts.

### Given:
- **Thickness of the ring** = 0.375 inches
- **Outside diameter** = 7 5/16 inches = 7.3125 inches
- **Inside diameter** = 5 3/16 inches = 5.1875 inches

### To Find:
The total surface area, including the inside, outside, and top/bottom surfaces.

### Step 1: Calculate the radii
- **Outside radius (Rₒ)** = \( \frac{7.3125}{2} \) inches = 3.65625 inches
- **Inside radius (Rᵢ)** = \( \frac{5.1875}{2} \) inches = 2.59375 inches

### Step 2: Surface Area of the Cylindrical Part (Inside and Outside Surface)
The lateral surface area of a cylinder is given by \( A = 2\pi r h \), where \( r \) is the radius and \( h \) is the height (thickness).

- **Outside surface area** = \( 2\pi Rₒ \times \text{thickness} \)
- **Inside surface area** = \( 2\pi Rᵢ \times \text{thickness} \)

### Step 3: Surface Area of the Top and Bottom Faces
The area of each face (top and bottom) is the area of the outer circle minus the area of the inner circle.

- **Area of each face** = \( \pi (Rₒ^2 - Rᵢ^2) \)

### Step 4: Calculate Total Surface Area
Total surface area = (Outside surface area) + (Inside surface area) + 2 × (Area of top/bottom face)

Let's plug in the numbers:

1. **Outside surface area**:
   \[
   2\pi \times 3.65625 \times 0.375 \approx 8.618 \text{ square inches}
   \]

2. **Inside surface area**:
   \[
   2\pi \times 2.59375 \times 0.375 \approx 6.115 \text{ square inches}
   \]

3. **Area of top/bottom face**:
   \[
   \pi \times (3.65625^2 - 2.59375^2) \approx 12.812 \text{ square inches}
   \]
   Since there are two faces, the total area for both faces is \( 2 \times 12.812 = 25.624 \text{ square inches} \).

4. **Total Surface Area**:
   \[
   8.618 + 6.115 + 25.624 \approx 40.357 \text{ square inches}
   \]

### Final Answer:
The total surface area, including the inside and outside surfaces, as well as the top and bottom faces, is approximately **40.36 square inches**.

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Do you want details or have any questions? 

Here are some related questions:
1. How would the surface area change if the thickness increased?
2. What is the volume of the ring?
3. How would you calculate the surface area of a solid cylinder?
4. What is the effect of doubling the outside diameter on the surface area?
5. How does the thickness impact the ring's overall mass?
6. What is the difference in surface area between a ring and a solid disk of the same diameter?
7. How can you use calculus to derive the formula for surface area?
8. What would be the surface area if the ring were a square cross-section instead of circular?

**Tip:** For accuracy, always convert fractional inches to decimals before calculations.

Learn how to calculate the total surface area of a ring with dimensions including thickness, outside diameter, and inside diameter. This detailed solution covers the surface area formula for a cylindrical ring, suitable for advanced high school students and beyond.

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