Calculating the lateral area of a cone
On my book, it says
...thinking of the lateral surface as swept out by revolving a generator (i.e. slant height) about the axis: the lateral area equals the length of this generator multiplied by the distance traveled by its midpoint, $ s*2pi*(\frac{1}{2}r) = pi*r*s $ ...
However, I can't understand why midpoint is involved. I looked up two proofs online (one of them), but both of them are different from my book's proof, and actually they seem more reasonable to me. Is my book wrong?
Update: my question is that how do you get the left hand side of the equation $ s*2pi*(\frac{1}{2}r) = pi*r*s $. Namely, why do you use $ \frac{1}{2}r $ and what's the point of using midpoint?
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$\begingroup$I'm reading the same book and, no surprise, had the same question. I've given it some thought and here's my best guess:
Think about what's happening when we revolve the generator (or slant length, s) around the axis. At the very top of the cone sits a single point, which doesn't move at all. At the very bottom of the cone (on its edge, which is one of the endpoints of the generator) sits a point which rotates around the entire base of the cone. The distance traveled by this point is 2*pi*r.
Now, how far does the average point on the generator travel on its trip around the axis? Intuition suggests it's the average of the distance traveled by each endpoint. That equals (0 + 2*pi*r)/2, which equals (1/2)(2*pi*r).
So, the points composing the generator move an average distance of pi(r) as they "sweep out" the lateral area of the cone. This implies that the lateral area is equal to the length of the generator multiplied by pi(r), which as shown above is the distance traveled by the generator's midpoint.
One thing to note: the author says that "the lateral area equals the length of this generator multiplied by the distance traveled by its midpoint." He then asserts (without proof) that the midpoint of the generator lies at the point on the cone where the cross-sectional radius is equal to 1/2 the radius of the cone's base. See if you can convince yourself if this assertion is justified.
$\endgroup$ 3 $\begingroup$See cone is formed by joining the end of the sector of a circle.
Let us take a circle of radius small r and the angle between the two radius is theta(this is the sector) so the area of the sector is theta divided by 360 degrees multiplied by pi*r^2
Next we should eliminate the theta, for that I am taking the circle at the base of the cone of radius r'(this circle is formed by joining the ends of the sector). Now the circumference of the circle at the base of the cone is (2*pi*r')
This circumference will be equal to that sector's circumference that is theta/360 multiplied by (2*pi*r)
Now equate,
2*pi*r' = theta/360 multiplied by (2*pi*r)
Now we can get theta angle to be 360*r' divided by r .
Now substitute the theta value in the first equation
It will be 360*r'/r divided by 360 multiplied by pir^2 This will be equal to pir*r'
Here r is equal to the slant height (s) that implies pisr'
Hence proved
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