13. Maggie is creating a cushion for a circular stool. The stool has a diameter of finches. If she needs fabric to extend' 2 inches all the way around the cushion, how much fabric will she need?

To determine how much fabric Maggie will need for the cushion, we first need to calculate the dimensions of the cushion, including the additional 2 inches of fabric that extends all the way around. 1. **Find the radius of the stool:** The diameter of the stool is given as \( d \) inches. The radius \( r \) is half of the diameter: \[ r = \frac{d}{2} \] 2. **Add the extension for the fabric:** Since the fabric extends 2 inches all the way around, we need to add 2 inches to the radius on each side. Therefore, the new radius \( R \) for the cushion will be: \[ R = r + 2 = \frac{d}{2} + 2 \] 3. **Calculate the area of the cushion:** The area \( A \) of a circle is given by the formula: \[ A = \pi R^2 \] Substituting \( R \): \[ A = \pi \left(\frac{d}{2} + 2\right)^2 \] 4. **Expand the expression:** \[ A = \pi \left(\left(\frac{d}{2}\right)^2 + 2 \cdot \frac{d}{2} \cdot 2 + 2^2\right) \] \[ = \pi \left(\frac{d^2}{4} + 2d + 4\right) \] 5. **Final area expression:** Thus, the area of the fabric needed for the cushion is: \[ A = \pi \left(\frac{d^2}{4} + 2d + 4\right) \] This formula gives the area of the fabric needed for the cushion based on the diameter \( d \) of the stool. If you have a specific value for \( d \), you can substitute it into the formula to find the exact area of fabric required.