A sphere with diameter $50$ cm intersects a plane $14$ cm from the center of the sphere. What is the number of square centimeters in the area of the circle formed?

A plane intersecting a sphere forms a circle whose radius is found using the Pythagorean theorem. The radius of the sphere is 25 cm, and the distance from the center to the plane is 14 cm, making the radius of the circle $\sqrt{25^2 - 14^2} = \sqrt{625 - 196} = \sqrt{429}$. The area of this circle is $\pi r^2$, which equals $\pi(\sqrt{429})^2 = 429\pi$. The area is $429\pi$ square centimeters.

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