







Archimedes, Euclid and "Circular Reasoning" 
20151115 

From Ron: I have read about Archimedes and his work with sphere in cylinder and cone in cylinder and the volume relationships. Did he or any others also extend this to regular based polygon based regular like pillars, and columns? The ratio of 1/3 to 1 whole holds true with all regular based columns as example: a regular pyramid having a regular hexagon base inside a regular hexagon column of equal height. Answered by Chris Fisher. 





Archimedes Burning Mirror 
20120717 

From Frakeetta: Archimedes Burning Mirror
There is a story about Archimedes that he used a “burning mirror” in the shape of a paraboloid of revolution to set fire to enemy ships in the harbor. What would be the equation of the parabola that one would rotate to form the appropriate paraboloid if it were to be designed to set fire to a ship 100m from the mirror? How large would the burning mirror need to be? What is the likelihood that this story is true? Answered by Robert Dawson. 





How much work is done? 
20111015 

From Jean: "A conical buoy that weighs B pounds floats upright in water with its
vertex "a" feet below the surface. A crane on a dock lifts the buoy
until its vertex just clears the surface. How much work is done ?" Answered by Penny Nom. 





Archimedes' formula for parabolic arches 
20090123 

From La: Use calculus to verify Archimedes' formula for y=9x^2. Prove Archimedes' formula for a general parabolic arch. Answered by Harley Weston. 





The perimeter of a regular polygon 
20070918 

From Ashwynn: why does the area of regular polygons with a perimeter of 1000m increase as the number of sides increase? Answered by Stephen La Rocque. 





The volume of a sphere. Why 4/3? 
20050530 

From Lauren: You know when you find the volume of sphere? I know the formula is V= 4/3 pi r^{3} but why do they use 4/3? Answered by Penny Nom. 





Volume of a sphere 
20000521 

From Kevin Partridge: Does anyone have a way to physically demonstrate how to explain the volume formula for a sphere? Or perhaps how to derive the formula without calculus? Answered by Harley Weston. 





A roll of paper 
20000115 

From Richard: I have a roll of paper, wrapped around a corrugate core, whos diameter is 10.750 in. The outer diameter of the roll is approx. 60 in. The thickness of the paper is .014 in. I am trying to find out how much linear feet of paper is left on the roll, given only the diameter of paper remaining on the core. Answered by Chris Fisher and Harley Weston. 





Approximating pi. 
19961104 

From Ben Dixon: How do you calculate Pi? Do you have to somehow combine the equation for a circle with the formula for the circumference? Answered by Chris Fisher. 





Le salinon d'Archimèdre 
19990311 

From Don Craig: I am trying to find the English translation of "Le salinon d'Archimèdre" and would appreciate any help. This is a figure, presumably studied by Archimedes, created from 4 semicircles. Since I can't draw it for you, I will try to describe it with the help of the 5 collinear, horizontal points below. . . . . . A B C D E A semicircle is constructed on AE as diameter (let's say above AE). Two more semicircles are then constructed with diameters AB and DE on the same side of the line AE as the first semicircle (above it). Finally, a fourth semicircle is constructed on diameter BD, this time on the opposite side of the line AE from the others (i.e. below the line). These semicircles and the region enclosed by them constitute what is called in French "Le salinon d'Archimèdre". If you know the English name of this curve I would appreciate it if you let me know. Answered by Harley Weston. 

