To find the total surface area of a solid hemisphere, we need to consider both the curved surface area and the flat circular base of the hemisphere. Let's break down the steps to calculate the total surface area. ### Step 1: Understand the Components of the Surface Area 1. **Curved Surface Area (CSA)**: This is the area of the curved part of the hemisphere. 2. **Base Area**: This is the area of the circular base of the hemisphere. ### Step 2: Formulas 1. **Curved Surface Area of a Hemisphere**: The formula for the curved surface area of a hemisphere is given by: \[ \text{CSA} = 2\pi r^2 \] where \( r \) is the radius of the hemisphere. 2. **Area of the Base**: The area of the circular base is given by: \[ \text{Base Area} = \pi r^2 \] 3. **Total Surface Area (TSA)**: The total surface area of the hemisphere is the sum of the curved surface area and the base area: \[ \text{TSA} = \text{CSA} + \text{Base Area} = 2\pi r^2 + \pi r^2 = 3\pi r^2 \] ### Step 3: Plug in the Values Given that the radius \( r = 7 \) cm, we can now calculate the total surface area. 1. **Calculate \( r^2 \)**: \[ r^2 = 7^2 = 49 \text{ cm}^2 \] 2. **Calculate the Total Surface Area**: \[ \text{TSA} = 3\pi r^2 = 3\pi \times 49 \] \[ \text{TSA} = 147\pi \text{ cm}^2 \] 3. **Approximate \( \pi \)**: Using \( \pi \approx 3.14 \): \[ \text{TSA} \approx 147 \times 3.14 \approx 461.58 \text{ cm}^2 \] ### Step 4: Round to the Nearest Whole Number Rounding \( 461.58 \) cm² gives us approximately \( 462 \) cm². ### Conclusion Thus, the total surface area of the solid hemisphere with a radius of 7 cm is approximately **462 cm²**. ### Explanation of Options - **Option A (400 cm²)**: This is incorrect because it underestimates the total surface area. The calculations show that the total surface area is significantly higher than this value. - **Option B (462 cm²)**: This is the correct answer, as derived from the calculations above. - **Option C (66 cm²)**: This is incorrect as it is far too low. It does not account for the size of the hemisphere and the formulas used. - **Option D (308 cm²)**: This is also incorrect. Like option A, it underestimates the total surface area. ### Common Pitfalls - **Forgetting to include the base area**: Some students might only calculate the curved surface area and forget to add the base area. - **Incorrectly calculating \( \pi \)**: Using an inaccurate value for \( \pi \) can lead to significant errors in the final answer. - **Rounding too early**: Always perform calculations fully before rounding to ensure accuracy. ### Revision Summary - The total surface area of a solid hemisphere is calculated using the formula \( \text{TSA} = 3\pi r^2 \). - For a radius of 7 cm, the total surface area is approximately 462 cm². - Always include both the curved surface area and the base area in your calculations. - Be careful with rounding and ensure you use an accurate value for \( \pi \).