I am having some issue calculating the mass of a shape. I am having trouble with the volume of the shape * density.
The material density is $7.85 \rm g/cc$ or $7.85 \rm g/cm^3$
1 rectangular prism: $L=600\rm mm$ $W=15\rm mm$ $H=600\rm mm$ $v = 5.4x10^6\rm mm^3$
subtract $6$ cylinders: $R=8 \rm mm$ $v = 6*(3.016x10^3\rm mm^3)$
subtract $4$ rectangular prism: $\rm L=280\ mm\ W=15\ mm\ H=20\ mm \ v = 4*(8.4x10^4\ mm^3)$
subtract $4$ triangles $\rm B=280\ mm\ H=280 \ mm\ W=15\ mm\ v = 4*(5.88x10^5 \ mm^3)$
subtract $1$ hexagon $\rm S=70\ mm\ H=15\ mm\ v = 19.0959x10^4\ mm^3$
$\rm v = 5.4x10^6\ mm^3 - 6*(3.016x10^3\ mm^3) - 4*(8.4x10^4\ mm^3) - 4*(5.88x10^5\ mm^3) - 19.0959x10^4\ mm^3$
$\rm =5.4x10^6\ mm^3 - 0.018096x10^6\ mm^3 - 0.336x10^6\ mm^3 - 2.352x10^6\ mm^3 - 0.190959x10^6\ mm^3$
$\rm v = 2.502945x10^6\ mm^3 = 2,502,945\ mm^3$
The density of this material is $\rm 7.85 \ g/cm^3$
This is where I am running into serious issues.
When I multiply the volume x density I get numbers that doesn't seem right;
I am pretty sure that I am failing to do some conversion between $\rm cm^3$ to $\ mm^3$ or something along those lines but I am just not sure.
$\rm 1cm = 10mm$ I know that but I just can't seem to get the correct answer mathematically.
Can anyone explain how to get the correct answer from this position $\rm v = 2.502945x10^6\ mm^3 = 2,502,945\ mm^3$ density = $\rm 7.85\ g/cm^3$
Note that $1 \text{cm} = 10 \text{mm}$ which implies that $1 \text{cm}^3 = 10^3 \text{mm}^3$. If you are having trouble seeing this, draw a cube of $1 \text{cm}^3$. Each of its sides is $10 \text{mm}$. From this it is clear what the volume is in $\text{mm}^3$.
So you have that
$$ \frac{7.85 \text{ grams}}{1 \text{ cm}^3}= \frac{7.85 \text{ grams}}{1 \text{ cm}^3} \cdot \frac{1 \text{ cm}^3}{10^3 \text{ mm}^3} = 0.00785 \frac{\text{g}}{\text{mm}^3} $$
And the result you are looking for is
$$ \text{Mass} = \left (0.00785 \frac{\text{g}}{\text{mm}^3} \right) \left(2.502945 \cdot 10^6 \text{ mm}^3 \right) $$