The problem reads:

"From the circle of radius 12 centimeters, a sector of central angle 90° is cut out and made into a cone. What is the base radius of this cone?"

Solution:

1. Calculate the length of the arc of the sector: The arc length of a sector is given by the formula: $\text{Arc length} = \theta \times r$ where:

   • $\theta$ is the angle in radians.
   • $r$ is the radius of the circle.

   Convert the given angle $90^\circ$ to radians: $\theta = \frac{90^\circ \times \pi}{180^\circ} = \frac{\pi}{2} \, \text{radians.}$

   Substituting the values: $\text{Arc length} = \frac{\pi}{2} \times 12 = 6\pi \, \text{cm.}$

   The arc length becomes the circumference of the base of the cone.

2. Relate arc length to cone's base circumference: The circumference of a cone's base is given by: $C = 2\pi R$ where $R$ is the base radius of the cone.

   Equating the arc length to the circumference: $6\pi = 2\pi R.$

   Solve for $R$: $R = \frac{6\pi}{2\pi} = 3 \, \text{cm.}$

3. Conclusion: The base radius of the cone is: $\boxed{3 \, \text{cm.}}$

Let me know if you'd like further explanation or have any questions!

Related Questions:

1. How is the height of the cone derived from the given parameters?
2. What is the slant height of the cone, and how is it calculated here?
3. Can this process be extended for other central angles? If so, how?
4. What happens to the base radius if the sector angle is halved?
5. How can we calculate the cone's volume in this problem?

Tip:

Always ensure angles are converted to radians when working with arc lengths in formulas involving trigonometry or geometry!