Volume of sphere split into eight sections?

Question

An orange is sliced into eight equal slices. The total surface area of each slice is $54 \pi.$ Find the volume of each slice.

Answer 1

If the radius of the orange is r then the volume of the orange is $$(4/3)\pi r^3$$ and the volume of each slice is $$ (1/6)\pi r^3$$

Assuming that the orange has been sliced longitudinally, the total surface area of each slice is the sum of three surfaces.

Two semi circular and a cured one. The total area of the two semi circular surfaces is $\pi r^2.$

The surface of the curved one is $$ (1/8)(4\pi r^2)=(1/2)(\pi r^2)$$

Thus the total surface area of each slice is $$ (3/2)(\pi r^2)$$

Solving $$ (3/2)(\pi r^2) = 54\pi$$ we come up with $$r=6$$

Thus the volume of each slice is $$36\pi = 113.09733..$$

Answer 2

If the slices are eight spherical wedges, the area of each slice is two semicircles the radius of the sphere plus one eighth of the exterior area of the sphere. That gives $$\pi r^2 + \frac 18\cdot4\pi r^2=54\pi\\\frac 32r^2=54\\r=6$$ The volume of each is then one eighth of the volume of the sphere $$V=\frac 18 \cdot \frac 43 \pi r^3=36\pi$$

If the sphere is split into eight octants, the surface area of each is three quarters of a circle of radius $r$ plus one eighth of the surface of the sphere. That gives $$\frac 34 \pi r^2 + \frac 12 \pi r^2=54\pi\\r^2=\frac {216}5\\r=\frac {6\sqrt{30}}5\approx 6.5723\\V=\frac {216}5\sqrt{\frac 65}\pi \approx 148.67$$

Answer 3

Here's how I would approach this problem. Again assuming the orange is sliced longitudinally, the area of the lune (that would be the skin of the orange) is given by

$$S_l=2R^2\theta$$

The area of the flat sides, of course, is simply

$$S_s=\pi R^2$$

Thus, the total area of the orange section is

$$S=S_l+S_s=R^2(\pi+2\theta)$$

Now, for 8 slices, $\theta=\pi/4$ so that $S=\frac32\pi R^2=54\pi$, or $R=6$. Then the volume is given by

$$V=\frac18\cdot\frac43\pi R^3=\frac16\pi R^3=36\pi\approx 113.097$$