A sector of a circle of radius 5cm is recast into a right circular cone of height 4cm. What is the volume of the resulting cone?

Question: A sector of a circle of radius 5cm is recast into a right circular cone of height 4cm. What is the volume of the resulting cone?

Solution:

When a sector of a circle is recast into a cone, the radius of the circle (R) becomes the slant height (l) of the cone.
So, slant height, l = 5 cm.
Given the height of the cone, h = 4 cm.
We know, l² = r² + h² [where r is the base radius of the cone]
⇒ 5² = r² + 4²
⇒ 25 = r² + 16
⇒ r² = 25 - 16
⇒ r² = 9
⇒ r = 3 cm.

The volume of a cone (V) = (1/3)πr²h
⇒ V = (1/3) × π × 3² × 4
⇒ V = (1/3) × π × 9 × 4
⇒ V = 3 × π × 4
∴ V = 12π

∴ The volume of the resulting cone is 12π cm³.