Terumah- Volume (again) and Surface Area

"They shall make an Aron of shittim wood, two and a half amot its length; an amah and a half its width; and an amah and a half its height. You shall cover it with pure gold, from inside and from outside you shall cover it, and you shall make on it a golden diadem all around...You shall make a lid of pure gold, two and a half amot its length, and an amah and a half its width" ~Shemot 25;10-11, 17

In Parshat Terumah, we are essentially given the blueprints for building the mishkan, or traveling tabernacle, that the Jewish people built while traveling in the dessert. This parsha is filled with so many mathematical possibilities, but I'd like to focus on a follow-up topic from back in Parshat Vayeishev. 

When investigating with 3-dimensional figures, the major calculations that students investigate are volume and surface area. Volume, as discussed in Parshat Vayeishev, is the space, in cubic units, that an object occupies or the amount of space inside the shell of a shape that can be filled. We learned that the calculation for the volume of a rectangular prism (or box shape) is to multiply the length x width x height of the prism.

Surface area is the calculation of the 2-dimensional area that covers the entire shape. For a rectangular prism, that would be the total area of all 6 sides of the prism added together. While volume is measured in cubic units, surface area is calculated in square units (the difference between a 3-D measurement for volume and a 2-D measurement for area). Lateral surface area, a variant of surface area, is the calculation of the area that covers the middle section of the prism, but does not include the top and the bottom. As a reminder, the area of a rectangular 2-dimensional shape is calculated by multiplying the length x width.

An example that I like to use to help understand the meaning of volume, surface area, and lateral surface area:

If I have a rectangular shaped room...
...volume is the amount of space inside the room if I filled it completely from floor to ceiling  and wall to wall.
...surface area is the amount of wall space that I would cover if I wanted to paint the entire room- all four walls, the floor, and the ceiling.
...lateral surface area is the amount of wall space that I would cover if I wanted to paint or wallpaper just around the four walls.

Connection to the parsha:
Given the information that we are given for building the mishkan, can we calculate volume, surface area, and lateral surface area? If so, what are each of these three measurements?

We are given the 3 necessary dimensions for accurately building the mishkan, so we do have enough information to make our calculations.

Length- 2.5 amot
Width- 1.5 amot
Height- 1.5 amot

We also know that the mishkan had a lid made for it as well, with a length of 2.5 amot and a width of 1.5 amot, which matches perfectly to fit right on top of the mishkan and close it to create a closed rectangular prism.

Volume- how much space was inside the mishkan?
2.5 x 1.5 x 1.5 = 5.625 cubic amot

Surface area- the parsha states that the mishkan was to be covered in gold, both the box and the lid. So, how much gold was needed to cover the mishkan on all sides?
The area of the top and bottom of the mishkan was 2.5 x 1.5 = 3.75 square amot for each (or 3.75 x 2 = 7.5 square amot total for those two sides).
The area of the front and back of the mishkan were 2.5 x 1.5 = 3.75 square amot for each (or 3.75 x 2 = 7.5 square amot total for those two sides).
The area of the left and right sides of the mishkan were 1.5 x 1.5 = 2.25 square amot for each (or 2.25 x 2 = 4.5 square amot total for those two sides).

When you add the square area of all six sides together, you get 7.5 + 7.5 + 4.5 = 19.5 square amot of gold to cover the entire mishkan.

Lateral surface area would be the same calculation as above, but we would leave out the area of the top and bottom, which would leave us with 7.5 + 4.5 = 12 square amot. (Note that another way to calculate lateral surface area is to calculate the perimeter, or length around the outside of the base shape, and multiply that length times the height of the prism.)

An extra question- The parsha states that the bottom of the mishkan was made out of shittim wood before being covered in gold, but the top is just made of gold. How much wood was necessary to build the wooden framework of the mishkan?

Answer- This is a combination of the calculation for surface area and lateral surface area. Really, the calculation is for lateral surface area with the addition of just the bottom area (but not the top area). From our lateral surface area calculations, we know that the area of the four sides is 12 square amot. We also know that the area of just the bottom part was 3.75 square amot. So, we add 12 + 3.75 = 15.75 square amot of shittim wood for the base framework of the mishkan.