If you have a truncated cone such that its upper base has a radius of a and the radius of its [larger] lower base is b, and a height h (between bases), how could you figure out its surface area using geometric reasoning?
(In reply to
re(3): general equation: by Richard)
To show
pi*b*sqrt(b^2+((h*b/(ba))^2)pi*a*sqrt(a^2+((h*a/(ba))^2)
= pi*(a+b)*sqrt((ba)^2+h^2)
write
b^2 + (h^2*b^2)/(ba)^2 = b^2*((ba)^2+h^2)/(ba)^2 and
a^2 + (h^2*a^2)/(ba)^2 = a^2*((ba)^2+h^2)/(ba)^2. Hence
pi*b*sqrt(b^2+((h*b/(ba))^2)pi*a*sqrt(a^2+((h*a/(ba))^2)
= (pi* (b^2a^2)/(ba))*sqrt((ba)^2+h^2)
= pi*(a+b)*sqrt((ba)^2+h^2).
Thus the area of the curved part of the truncated cone can be expressed in the simpler form
pi*(a+b)*sqrt((ba)^2+h^2) given by Ady TZIDON as well as in the form
pi*b*sqrt(b^2+((h*b/(ba))^2)pi*a*sqrt(a^2+((h*a/(ba))^2)
geometrically obtained by removing a small full cone from a larger one. Is there a simple geometric interpretation for the simpler formula?

Posted by Richard
on 20040126 19:06:43 