Solid As A Rock
Solids are three-dimensional shapes, with the dimension
of depth added to length and height. With solids, there’s good news
and bad news. The bad news is that solids can be difficult to visualize.
But the good news more than makes up for it: The only solids on
the new SAT are cubes, rectangular solids, and right cylinders.
Learn to visualize these three shapes now, before the test, and
you’ll be fine.
A rectangular solid is a prism with a rectangular base
and edges that are perpendicular to its base. It looks a lot like
a cardboard box.
A rectangular solid has three important dimensions: length
(l), width (w),
and height (h). If you know these
three measurements, you can find the solid’s surface area, volume,
and diagonal length.
Volume of a Rectangular Solid
The formula for the volume of a rectangular
solid takes the formula for area of a rectangle and adds another
dimension. The area of a rectangle is A = lh (area equals
length times height). The formula for the volume of a rectangular
solid adds on width:
Here’s a good old-fashioned example:
is the volume of the figure presented below?
Just plug the values into the volume formula and you’re
good to go: V = (3x)(2x)(x) =
Surface Area of a Rectangular Solid
The surface area of a solid is the area of its outermost
skin. In the case of rectangular solids, imagine a cardboard box
all closed up. The surface of that closed box is made of six rectangles:
the sum of the areas of the six rectangles is the surface area of
the box. To make things even easier, the six rectangles come in
three congruent pairs. We’ve marked the congruent pairs by shades
of gray in the image below: One pair is clear, one pair is light
gray, and one pair is dark gray.
Two faces have areas of l w
, two faces have
areas of l h
, and two faces
have areas of w h
. The surface
area of the entire solid is the sum of the areas of the congruent
Wanna practice? Alright. What’s the surface area of this
Plug in. Plug in. Plug in.
Meat Cleaver Problems
The SAT won’t just hand you surface area questions on
a silver platter. It’ll make you work for them. One of the ways
the SAT likes to make you work goes like this. A question will describe
a solid, give you all of its measurements, and then tell you that the
box has been cut in half. You’ll then have to find the combined
surface area of the two new boxes. For example, pictured below is
a rectangular solid that has a length of 8, a depth
of 4, and a height of 4. Then, out of
the blue, a giant cleaver comes down and cuts the solid into two
As you can see from the diagram, when the rectangle was
cut in two, two new surfaces suddenly appeared (these are the darkened
surfaces). But notice that the total volume of the two cubes has
remained the same. So here’s a rule: Whenever a solid is cut
into smaller pieces, its surface area increases, but its volume
is unchanged. The SAT loves to test this little factoid.
Diagonal Length of a Rectangular Solid
The diagonal of a rectangular solid, d,
is the line segment whose endpoints are opposite corners of the
solid. Every rectangular solid has four diagonals, each with the
same length, that connect each pair of opposite vertices. Here’s
one diagonal drawn in:
It’s possible that an SAT question will test to see if
you can find the length of a diagonal. Now you can:
where l is the length, w is
the width, and h is the height. The
formula is like a pumped up Vin Diesel version of the Pythagorean
theorem. Check it out in action:
is the length of diagonal AH in the
rectangular solid below if AC = 5, GH =
6, and CG = 3?
The question gives the length, width, and height of the
rectangular solid, so you can just plug those numbers into the formula:
A cube is a square brought into 3-D. The length, width,
and height of a cube are equal, and each of its six faces is a square.
Volume of a Cube
The formula for finding the volume of a cube is essentially
the same as the formula for the volume of a rectangular volume.
However, since a cube’s length, width, and height are all equal,
the formula for the volume of a cube is
where s is the length of
one edge of the cube.
Surface Area of a Cube
Since a cube is just a rectangular solid whose sides are
all equal, the formula for finding the surface area of a cube is
the same as the formula for finding the surface area of a rectangular
solid, except with s substituted in
for l, w,
Diagonal Length of a Cube
The formula for the diagonal of a cube is also adapted
from the formula for the diagonal length of a rectangular solid,
with s substituted for l, w,
Right Circular Cylinders
A right circular cylinder looks like one of those cardboard
things that toilet paper comes on, except it isn’t hollow. In fact,
one way to think of a right circular cylinder is as a rectangle
curved around so that its ends meet.
A right circular cylinder has two connected congruent
circular bases and looks like this:
The height of a cylinder, h,
is the length of the line segment whose endpoints are the centers
of the bases. The radius of a cylinder, r,
is the radius of its base. For the new SAT, all you need to know
about a right circular cylinder is how to calculate its volume.
Volume of a Cylinder
The volume of a cylinder is the product of the area of
its base and its height. Because a cylinder has a circular base,
the volume of a cylinder is equal to the area of the circle that
is the base times the height:
Try to find the volume of the cylinder below:
This cylinder has a radius of 4 and a height
of 6. Using the volume formula,
Sketchy Word Problems
The SAT has been known to ask word problems about solids.
Because solids are so difficult to visualize, these problems can
seem brutally difficult. So here’s the rule: Always sketch
out what the question is describing. Once you see what
the question’s talking about, you seldom have to do much more than
plug the right numbers into the right equation.