A sector of a circle of radius 8cm subtends a 90 degree angle at the center of the circle. If the sector is folded without overlap to form the curved surface of a cone, find the: [i] base radius of the cone [ii] height of the cone.

Hello!
Denote the radius of the original circle as R and the base radius of the cone as r. Then the length of the original circle is 2piR , and the given sector is a one quarter of the entire circle, hence the length of its curved part is (piR)/2.
And this length becomes the length of the entire base of the cone. This means (piR)/2 = 2pir, thus r = R/4 = 2 cm. This is the answer for [i].
The height of the cone, denote it h, forms a right triangle with a base radius of the cone and its slanted height. The slanted height is obviously R, and h^2 + r^2 = R^2, so h = sqrt(R^2-r^2) = sqrt(60) = 2sqrt(15) (cm). This is the answer for [ii].