



 
Hi Russell, One measurement in a cone or truncated cone that can be confusing is the height. I took your measurement, 6.5 inches, as the vertical height rather than the slant height. In my diagram below the distance from E to D is the slant height of the truncated cone and the distance from A to D is the slant height of the cone. I extended d the sides of the cat food dispenser to the point A in the diagram. The length x inches is the distance from B to A. I want to find the length AD and I do it by using similar triangles and Pythagoras' theorem. (Russell you used mixed fraction notation in your question but I am using decimal fractions simple because they are easier to type.) Triangles DCA and EBA are similar so
thus
Hence the length of AC is 9.75 + 6.5 = 16.25 inches. Triangle DCA is a right triangle so, using Pythagoras' theorem
In a similar fashion I found that AE = 9.86 inches. Now slice along the line segment AD and roll the cone out flat to form a segment of the circle of radius (These diagrams are not to scale.) All that remains is to find the measure of the angle at the centre of the sector, that is t^{o} in the diagram. The length of the arc DF is the circumference of the circle at the top of the cat food dispenser, that is a circle of radius
Thus
I hope this helps,
 


Math Central is supported by the University of Regina and The Pacific Institute for the Mathematical Sciences. 