Solids
2 +1 = 3. Simple enough. Take any two-dimensional polygon
and add one more dimension to it (height), and you
have a three-dimensional solid.
A right circular cylinder is
just a circle with height. A cube is a square increased
upward by the length of one of its sides. A rectangular solid—which most
humans would call a box—is a rectangle with height
added to it.
There are three main item types associated with
three-dimensional figures:
-
Volume items, designed to blow the minds of students
used to area and perimeter
- Hidden
shape items, like the well-concealed right triangle UPR hiding
within the cube
- Surface
area items, which sometimes throw out the term face
The volume items shouldn’t cause you much trouble. The
volume formulas are in the reference portion of the Math test, so
check them over if you’re unsure.
The sneaky hidden-shape items are a little tougher
but not insurmountable by any means. The box and cube are swarming
with triangles, sure, but they are also chock-full of right angles.
This means all the triangles are right triangles, so you can use
the good ol’ Pythagorean Theorem.
If the box in our diagram were somehow alive, the surface
area would be its skin. The surface area of a rectangular
solid equals the sum of the area of the six rectangles that make
up its sides. For a cube, the surface area would be the sum of the
area of its six sides. These sides are sometimes referred to as faces.
Written in geometry language:
Surface area of a rectangular solid=
2(length)(width) + 2(length)(height) + 2(height)(width)
Surface area of a cube = 
There are formulas for figuring out the surface area of
other solids, but the SAT will only ask you to calculate the surface
area of a rectangular solid or a cube.
