If the radius of a sphere is doubled, how does the surface area change compared to the original surface area?

To determine how the surface area of a sphere changes when the radius is doubled, we can follow these steps: 1. **Identify the formula for the surface area of a sphere**: The surface area \( A \) of a sphere is given by the formula: \[ A = 4\pi r^2 \] where \( r \) is the radius of the sphere. 2. **Calculate the original surface area**: Let the original radius be \( r \). The original surface area \( A_1 \) is: \[ A_1 = 4\pi r^2 \] 3. **Determine the new radius**: If the radius is doubled, the new radius \( r' \) is: \[ r' = 2r \] 4. **Calculate the new surface area**: The new surface area \( A_2 \) with the doubled radius is: \[ A_2 = 4\pi (r')^2 = 4\pi (2r)^2 = 4\pi (4r^2) = 16\pi r^2 \] 5. **Compare the new surface area to the original surface area**: To find the factor by which the surface area has changed, we can take the ratio of the new surface area to the original surface area: \[ \text{Change Factor} = \frac{A_2}{A_1} = \frac{16\pi r^2}{4\pi r^2} = \frac{16}{4} = 4 \] Thus, when the radius of a sphere is doubled, the surface area increases by a factor of 4 compared to the original surface area.