Building a raised garden bed 101

I'm deviating a little from the typical technology projects I like to talk about and going over a weekend project I did with my Dad.  It turned out to be a three weekend project with Erica helping to finish the project and plant the seeds. It was a fun project and involved a bit of math so I can justify documenting it here. We decided to build some raised garden beds over the weekend.  Garden beds are useful to us for a for a few reasons. We have a small area for a garden in our backyard. Every year this area gets over run by cats, veggies and weeds.  It will make it easier to turn the soil over time and we can also easily keep other garden pests out of our garden. A garden bed will also let us get the most nutrition out of our soil.

So what is a raised garden bed? Here is a picture of our garden beds after we installed them:


Each garden bed is 1'x4'x8'. That means for it to be square we need the diagonal to be \[\sqrt{(a)^2+(b)^2} = c^2 =  \sqrt{(4*12)^2+(8*12)^2} = 107.33126"\]

However, we are measuring 103" diagonal, so that means we are not a perfect 90 degrees! It would be good to know all the angles...since we know all the lengths we can find the angles:

Law of Cosines
\[a^2 = c^2+b^2-2cb(cos(A))\]
\[b^2 = a^2+c^2-2ac(cos(B))\]
\[c^2 = a^2+b^2-2ab(cos(C))\]

solving for the angle:
\[A =\dfrac{\arccos(a^2-c^2-b^2)}{-2cb}\]
\[B =\dfrac{\arccos(b^2-a^2-c^2)}{-2ac}\]
\[C =\dfrac{\arccos(c^2-a^2-b^2)}{-2ab}\]

a = 4*12 = 48"
b = 8*12 = 96"
c = 103" (but it should be 107.331" according to the math above)

$A = 27.6285\,^{\circ}$
$B = 68.0444\,^{\circ}$
$C = 84.3271\,^{\circ}$
to check:  $27.6285\,^{\circ} + 68.0444\,^{\circ} + 84.3271\,^{\circ} = 180\,^{\circ}$

Enough of the theory of garden boxes. Lets go over how to actually build one. Here's what you'll need:
Tools:
Drill
Measuring tape
Circular saw

Bill of materials:
 We purchased this material from Lowes. The lumber is fair quality but for the convenience of location it is unmatched.

Total Space: 96"*48"*12" = 55296 in $^3$ or 32 ft $^3$
Total cost: $100.00

Total time: 3 weekends total.

Step 1: Measure and clear a spot and make sure there is sufficient space for the garden bed

 Step 2: Get the lumber

 Step 3: Measure twice cut once and make sure it all fits

  Step 4: Attach it all together and verify the $90^{\circ}$ angles...or close to $90^{\circ}$ :-)

The blue arrows are where we connected the wide planks. The red arrows are where we attached the square posts to the frame. This allowed an amount of redundancy.

Step 5: Install and fill with dirt

Step 6: Finishing touches We sprayed weed killer around the outside of the garden beds and put down some landscaping cloth. Then we bought 8 bags of rubberized mulch (from CostCo) and spread it down around the boxes. After we planted the seeds we attached the images of the seeds to the side of the garden bed so we would know what we were growing.