The circumference is given by $\displaystyle 2\pi r$ which yields a radius of 4.  The volume is given by $\displaystyle \frac{4}{3}\pi r^{3}$

$\displaystyle \frac{4}{3}\cdot \pi \cdot r^3\rightarrow \frac{4}{3}\cdot \pi \cdot (7)^3$

Simplified, it is $\displaystyle \frac{1372}{3}\; \pi \; cm^3$

The formula for the volume of a sphere is:

$\displaystyle V=\frac{4}{3}\pi r^{3}$

where $\displaystyle r$ = radius.  The diameter is 6 in, so the radius will be 3 in. 

The formula to find the volume of a sphere is: $\displaystyle V=\frac{4}{3} \cdot \pi \cdot r^3$

Finding the volume is simple. All that we need is the radius! 

The only information the problem provides is the circumference. In order to find the radius, we have to think how circumference relates to radius.

Since the equation for circumference is $\displaystyle C=\pi \cdot d$, where d stands for diameter, and radius is half of the diameter, the two have diameter in common. 

The first step in solving this problem is to determine the diameter from the circumference:

$\displaystyle C= \pi \cdot d$

$\displaystyle 27 \cdot \pi = \pi \cdot d$

$\displaystyle \frac{27 \cdot \pi}{ \pi} = \frac{\pi \cdot d}{\pi}$

$\displaystyle 27 = d$

Because the diameter is $\displaystyle 27\:cm$, the radius must be $\displaystyle 13.5\:cm$. 

Now we are ready to solve for the volume after substituting in our $\displaystyle r$ value.

$\displaystyle V= \frac{4}{3} \cdot \pi \cdot (13.5)^3$

$\displaystyle V= \frac{4}{3} \cdot \pi \cdot (2,460.375)$

$\displaystyle V = 3280.5 \cdot \pi$

$\displaystyle V =10,305.99 \approx10,306\:cm^3$

The cube has a side length of 20 inches. Since the sphere is inscribed within the cube its diameter measures 20 inches; the radius will be 10 inches.

The volume of a sphere is given by 

$\displaystyle V=\frac{4}{3}\pi r^3$.

$\displaystyle V=\frac{4}{3}\pi (10)^3=\mathbf{4,189\hspace{1mm}in^3}$

Recall how to find the volume of a sphere:

$\displaystyle \text{Volume of Sphere}=\frac{4}{3}\pi\times\text{radius}^3$

Now since we only have half a sphere, divide the volume by $\displaystyle 2$.

$\displaystyle \text{Volume of Figure}=\frac{1}{2}(\frac{4}{3}\pi\times\text{radius})^2$

$\displaystyle \text{Volume of Figure}=\frac{2}{3}\pi\times\text{radius}^3$

Plug in the given radius to find the volume of the figure.

$\displaystyle \text{Volume of Figure}=\frac{2}{3}\pi(2)^3=\frac{16}{3}\pi=16.76$

Make sure to round to $\displaystyle 2$ places after the decimal.

Recall how to find the volume of a sphere:

$\displaystyle \text{Volume of Sphere}=\frac{4}{3}\pi\times\text{radius}^3$

Now since we only have half a sphere, divide the volume by $\displaystyle 2$.

$\displaystyle \text{Volume of Figure}=\frac{1}{2}(\frac{4}{3}\pi\times\text{radius})^2$

$\displaystyle \text{Volume of Figure}=\frac{2}{3}\pi\times\text{radius}^3$

Plug in the given radius to find the volume of the figure.

$\displaystyle \text{Volume of Figure}=\frac{2}{3}\pi(3)^3=18\pi=56.55$

Make sure to round to $\displaystyle 2$ places after the decimal.

Recall how to find the volume of a sphere:

$\displaystyle \text{Volume of Sphere}=\frac{4}{3}\pi\times\text{radius}^3$

Now since we only have half a sphere, divide the volume by $\displaystyle 2$.

$\displaystyle \text{Volume of Figure}=\frac{1}{2}(\frac{4}{3}\pi\times\text{radius})^2$

$\displaystyle \text{Volume of Figure}=\frac{2}{3}\pi\times\text{radius}^3$

Plug in the given radius to find the volume of the figure.

$\displaystyle \text{Volume of Figure}=\frac{2}{3}\pi(4)^3=\frac{128}{3}\pi=134.04$

Make sure to round to $\displaystyle 2$ places after the decimal.

Recall how to find the volume of a sphere:

$\displaystyle \text{Volume of Sphere}=\frac{4}{3}\pi\times\text{radius}^3$

Now since we only have half a sphere, divide the volume by $\displaystyle 2$.

$\displaystyle \text{Volume of Figure}=\frac{1}{2}(\frac{4}{3}\pi\times\text{radius})^2$

$\displaystyle \text{Volume of Figure}=\frac{2}{3}\pi\times\text{radius}^3$

Plug in the given radius to find the volume of the figure.

$\displaystyle \text{Volume of Figure}=\frac{2}{3}\pi(\frac{1}{2})^3=\frac{1}{12}\pi=0.26$

Make sure to round to $\displaystyle 2$ places after the decimal.

Recall how to find the volume of a sphere:

$\displaystyle \text{Volume of Sphere}=\frac{4}{3}\pi\times\text{radius}^3$

Now since we only have half a sphere, divide the volume by $\displaystyle 2$.

$\displaystyle \text{Volume of Figure}=\frac{1}{2}(\frac{4}{3}\pi\times\text{radius})^2$

$\displaystyle \text{Volume of Figure}=\frac{2}{3}\pi\times\text{radius}^3$

Plug in the given radius to find the volume of the figure.

$\displaystyle \text{Volume of Figure}=\frac{2}{3}\pi(5)^3=\frac{250}{3}\pi=261.80$

Make sure to round to $\displaystyle 2$ places after the decimal.