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Asia-Pacific Forum on Science
Learning and Teaching, Volume 5, Issue 2, Article 6 (Aug., 2004) MAN Yiu Kwong Solving geometry problems via mechanical principles
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Asia-Pacific Forum on Science
Learning and Teaching, Volume 5, Issue 2, Article 6 (Aug., 2004) MAN Yiu Kwong Solving geometry problems via mechanical principles
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A historical example
A famous example on using mechanical principle to solve geometry problems was provided by Archimedes (287-212 BC). Using the principle of lever, he was able to derive the correct formula for the volume of a sphere (Man & Lo, 2002). Let us illustrate how such an approach works. In Figure 1, N denotes the origin of a plane and TS is a horizontal line through N. Let TN = NS = 2r. First, we draw a circle with radius r, a square ABCD with side 2r and an isosceles triangle NEF with height 2r and base 4r. Then, the whole figure is rotated about TS to generate a sphere, a cylinder and a cone. Next, an arbitrary vertical thin slice of thickness Δx at a distance x from N, is cut from the three solids in turn. By simple calculations, we have:
The volume of slice from the sphere = π(2xr - x2) Δx
The volume of slice from the cylinder = πr2Δx
The volume of slice from the cone = πx2Δx
Figure 1
Consider TS as a lever with pivot at N. By hanging the slices from the sphere and the cone at the point T, the total moment generated is:
[π(2xr - x2)Δx + π x 2Δx] × 2r = 4π r2x Δx .
It is equal to four times the moment generated by the slice of the cylinder on the opposite side of the lever. Since this relation holds for arbitrary slices at any position x on NS , so we have:
(volume of sphere + volume of cone) × 2r = (volume of cylinder) × 4r.
Let V denotes the volume of sphere. Then, we obtain:
,
and hence
. We can see that it is a correct formula for the volume of sphere and the approach adopted is quite ingenious, without using any sophisticated mathematics.
Copyright (C) 2004 HKIEd APFSLT. Volume 5, Issue 2, Article 6 (Aug., 2004). All Rights Reserved.
