Answer to Question #197972 in Geometry for Angelo

Question

A circular piece of cardboard with a diameter of 1 m will be made into a conical hat 40 cm high by cutting a sector off and joining the edges to form a cone. Determine the angle subtended by the sector removed.

Answer 1

Given diameter of circular piece is 1 metre = 100 centimetres

So, radius = Diameter/2 = 100/2 = 50 centimetres

Given height of conical hat = 40 centimetres

From Pythagoras theorem,

r² = h² + x²

x² = (50)² - (40)² = 2500-1600 = 900

x = 30 centimetres

Let C1 be circumference of circular piece

C2 be circumference of base of conical hat

C be length of arc

C = C1 - C2 = 2Ï€r - 2Ï€x = 2Ï€(50) - 2Ï€(30) = 40Ï€

C = r*θ

40π = 50*θ

θ = (40/50)π*(360°/2π) = 144°