Use Pappus theorem:
V = volume generated
A = surface area generated
At = generating area
L = generating length of curve
C = distance traveled by the centroid
You can consider the volume generated as a cone, there will be two cones involved with common bases. The radius of each cone is h and the height is s/2.
$V = 2 \times \frac{1}{3}\pi h^2 (\frac{1}{2}s)$
Use Pappus theorem:
V = volume generated
A = surface area generated
At = generating area
L = generating length of curve
C = distance traveled by the centroid
$h = \sqrt{s^2 - (\frac{1}{2}s)^2} = \frac{\sqrt{3}}{2}s$
$\bar{y} = \frac{1}{3}h = \frac{1}{3}(\frac{\sqrt{3}}{2}s) = \frac{\sqrt{3}}{6}s$ ← centroid of area
$A_t = \frac{1}{2}sh = \frac{1}{2}s(\frac{\sqrt{3}}{2}s) = \frac{\sqrt{3}}{4}s^2$
$V = A_t \times C = A_t \times 2\pi \bar{y}$
$V = \frac{\sqrt{3}}{4}s^2 \times 2\pi (\frac{\sqrt{3}}{6}s)$
$V = \frac{1}{4}\pi s^3$
$L = 2s$ ← the third side, being the axis of rotation, cannot generate a surface
$\bar{y} = \frac{1}{2}h = \frac{1}{2}(\frac{\sqrt{3}}{2}s) = \frac{\sqrt{3}}{4}s$ ← centroid of lines
$A = L \times C = L \times 2\pi \bar{y}$
$A = 2s \times 2\pi (\frac{\sqrt{3}}{4}s)$
$A = \sqrt{3}\pi s^2$
You can consider the volume generated as a cone, there will be two cones involved with common bases. The radius of each cone is h and the height is s/2.
$V = 2 \times \frac{1}{3}\pi h^2 (\frac{1}{2}s)$
$V = 2 \times \frac{1}{3}\pi (\frac{\sqrt{3}}{2}s)^2 (\frac{1}{2}s)$
$V = \frac{1}{4}\pi s^3$
For the surface area generated, it is the lateral area of two cones
$A = 2 \times \pi hs = 2 \times \pi (\frac{\sqrt{3}}{2}s)s$
$A = \sqrt{3}\pi s^2$
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