


Geometry
Geometry has a long and storied history that goes back thousands of years,
which we’re sure you’re dying to hear about. But unless you’re taking the GRE to
go for your masters in math history, this has no possible relevance to your
life. So we’ll skip the history lesson and get right to the facts.
Lines and Angles
An angle is composed of two lines and is a measure of the spread
between them. The point where the two lines meet is called the
vertex. On the GRE, angles are measured in degrees.
Here’s an example:
The ° after the 45 is a degree symbol. This angle measures 45 degrees.
Acute, Obtuse, and Right Angles
Angles that measure less than 90° are called acute
angles. Angles that measure more than 90° are called
obtuse angles. A special angle that the GRE test
makers love is called a right angle. Right angles
always measure exactly 90° and are indicated by a little square where
the angle measure would normally be, like this:
Straight Angles
Multiple angles that meet at a single point on a line are called
straight angles. The sum of the angles meeting at a single point on a
straight line is always equal to 180°. You may see something like this,
asking you to solve for n:
The two angles (one marked by 40° and the one marked by
n°) meet at the same point on the line. Since the sum
of the angles on the line must be 180°, you can plug 40° and
n° into a formula like this:
40° + n° = 180°
Subtracting 40° from both sides gives n° = 140°,
or n = 140.
Vertical Angles
When two lines intersect, the angles that lie opposite each other,
called vertical angles, are always equal.
Angles and are vertical angles and are
therefore equal. Angles and
are also vertical, equal
angles.
Parallel Lines
A more complicated version of this figure that you may see on the
GRE involves two parallel lines intersected by a third line, called a
transversal. Parallel means that
the lines run in exactly the same direction and never intersect. Here’s
an example:
line 1 is parallel to line 2
Don’t assume lines are parallel just because they look like they
are—the question will always tell you if two lines are meant to be
parallel. Even though this figure contains many angles, it turns out
that it only has two kinds of angles: big angles
(obtuse) and little angles (acute). All the big angles are equal to each
other, and all the little angles are equal to each other. Furthermore,
any
big angle + any
little angle = 180°. This is true for any figure with
two parallel lines intersected by a third line. In the following figure,
we label every angle to show you what we mean:
As described above, the eight angles created by these two
intersections have special relationships to each other:
 Angles 1, 4, 5, and 8 are equal to one another. Angle 1 is vertical to angle 4, and angle 5 is vertical to angle 8.
 Angles 2, 3, 6, and 7 are equal to one another. Angle 2 is vertical to angle 3, and angle 6 is vertical to angle 7.
 The sum of any two adjacent angles, such as 1 and 2 or 7 and 8, equals 180º because they form a straight angle lying on a line.
 The sum of any big angle + any little angle = 180°, since the big and little angles in this figure combine into straight lines and all the big angles are equal and all the little angles are equal. So a big and little angle don’t need to be next to each other to add to 180°; any big plus any little will add to 180°. For example, since angles 1 and 2 sum to 180º, and since angles 2 and 7 are equal, the sum of angles 1 and 7 also equals 180º.
By using these rules, you can figure out the degrees of angles
that may seem unrelated. For example:
If line 1 is parallel to line 2, what is the value of
y in the figure above?
Again, this figure has only two kinds of angles: big angles
(obtuse) and little angles (acute). We know that 55° is a little angle,
so y must be a big angle. Since big
angle + little angle = 180°, you can write:
y° + 55° = 180°
Solving for y:
y° = 180° – 55°
y = 125
Perpendicular Lines
Two lines that meet at a right angle are called
perpendicular lines. If you’re told two lines are
perpendicular, just think 90°. We’d love to tell you more about these
beauties, but there’s really not much else to say.
Polygon Basics
A polygon is a twodimensional figure with three or more straight
sides. Polygons are named according to the number of sides they have.
Number of Sides  Name 
3  triangle 
4  quadrilateral 
5  pentagon 
6  hexagon 
7  heptagon 
8  octagon 
9  nonagon 
10  decagon 
12  dodecagon 
n  ngon 
All polygons, no matter how many sides they possess, share certain
characteristics:
 The sum of the interior angles of a polygon with n sides is (n – 2)180°. For instance, the sum of the interior angles of an octagon is (8 – 2)180°= 6(180°) = 1080°.
 A polygon with equal sides and equal interior angles is a regular polygon.
 The sum of the exterior angles of any polygon is 360°.
 The perimeter of a polygon is the sum of the lengths of its sides.
 The area of a polygon is the measure of the area of the region enclosed by the polygon. Each polygon tested on the GRE has its own unique area formula, which we’ll cover below.
For the most part, the polygons tested in GRE math include triangles
and quadrilaterals. All triangles have three sides, but there are special
types of triangles that we’ll cover next. Then we’ll move on to four common
quadrilaterals (foursided figures) that appear on the test: rectangles,
squares, parallelograms, and trapezoids.
We’ll then leave the world of polygons and make our way to circles and
then conclude this geometry section with a discussion of the
threedimensional solids you may see on your test: rectangular solids,
cubes, and right circular cylinders.
Triangles
Of all the geometric shapes, triangles are among the most commonly
tested on the GRE. But since the GRE tends to test the same triangles over
and over, you just need to master a few rules and a few diagrams. We’ll look
at some special triangles shortly, but first we’ll explain four very special
rules.
The Four Rules of Triangles
Commit these four rules to memory.
1. The Rule of Interior Angles.
The sum of the interior angles of a triangle always equals
180°. Interior angles are those on the inside. Whenever you’re
given two angles of a triangle, you can use this formula to calculate
the third angle. For example:
What is the value of a in the figure
above?
Since the sum of the angles of a triangle equals 180°, you can set
up an equation:
a° + 100° + 60° = 180°
Isolating the variable and solving for a gives
a = 20.
2. The Rule of Exterior Angles. An exterior angle of
a triangle is the angle formed by extending one of the sides of the
triangle past a vertex (or the intersection of two sides of a figure).
In the figure below, d is the exterior angle.
Since, together, d and c form a
straight angle, they add up to 180°: d + c
= 180°. According to the first rule of triangles, the three
angles of a triangle always add up to 180°, so a
+ b + c = 180°. Since
d + c = 180° and a
+ b + c = 180°, d
must be equal to a + b (the remote
interior angles). This generalizes to all triangles as the following
rules: The exterior angle of a triangle plus the interior angle
with which it shares a vertex is always 180°. The exterior angle is
also equal to the sum of the measures of the remote interior angles.
3. The Rule of the Sides. The length of any side of a
triangle must be greater than the difference and less than the sum of
the other two sides. In other words:
difference of other two sides < one side < sum of
other two sides
Although this rule won’t allow you to determine a precise length
of the missing side, it will allow you to determine a range of values
for the missing side, which is exactly what the test makers would ask
for in such a problem. Here’s an example:
What is the range of values for x in the triangle above?
Since the difference of 10 and 4 is 6, and the sum of 10 and 4 is
14, we can determine the range of values of x:
6 < x < 14
Keep in mind that x must be
inside this range. That is, x could
not be 6 or 14.
4. The Rule of Proportion. In every triangle,
the longest side is opposite the largest angle and the shortest side
is opposite the smallest angle. Take a look at the
following figure and try to guess which angle is largest.
In this figure, side a is clearly the longest
side and is the largest
angle. Meanwhile, side c is the shortest side and
is the smallest angle.
So c < b <
a and This
proportionality of side lengths and angles holds true for all triangles.
Use this rule to solve the question below:

According to the rule of proportion, the longest side of a
triangle is opposite the largest angle, and the shortest side of a
triangle is opposite the smallest angle. The question tells us that
angle C < angle A <
angle B. So, the largest angle in triangle
ABC is angle B, which is opposite the
side of length 8. We know too that the smallest angle is angle
C, since angle C < angle
A. This means that the third side, with a length
of x, measures between 6 and 8 units in length. The
only choice that fits this criterion is 7, choice
C.
Isosceles Triangles
An isosceles triangle has two equal sides and two equal angles,
like this:
The tick marks indicate that sides a and
b are equal, and the curved lines inside the triangle
indicate that angle A equals angle B.
Notice that the two equal angles are the ones opposite the two equal
sides. Let’s see how we might put this knowledge to use on the test.
Check out this next triangle:
With two equal sides, this is an isosceles triangle. Even though
you’re not explicitly told that it has two equal angles, any triangle
with two equal sides must also have two equal angles. This means that
x must be 70°, because angles opposite equal sides
are equal. Knowing that the sum of the angles of any triangle is 180°,
we can also calculate y: y + 70 + 70 =
180, or y = 40.
Equilateral Triangles
An equilateral triangle has three equal sides and three equal
angles, like this:
The tick marks tip us off that the three sides are equal. We can
precisely calculate the angles because the 180° of an equilateral
triangle broken into three equal angles yields 60° for each. As soon as
you’re given one side of an equilateral triangle, you’ll immediately
know the other two sides, because all three have the same measure. If
you ever see a triangle with three equal sides, you’ll immediately know
its angles measure 60°. Conversely, if you see that a triangle’s three
interior angles all measure 60°, then you’ll know its sides must all be
equal.
Right Triangles
A right triangle is any triangle that contains a right angle. The
side opposite the right angle is called the hypotenuse.
The other two sides are called legs. The angles
opposite the legs of a right triangle add up to 90°. That makes sense,
because the right angle itself is 90° and every triangle contains 180°
total, so the other two angles must combine for the other 90°. Right
triangles are so special that they even have their own rule, known as
the Pythagorean theorem.
The Pythagorean Theorem
The ancient Greeks spent a lot of time philosophizing, eating
grapes, and riding around on donkeys. They also enjoyed the
occasional mathematical epiphany. One day, Pythagoras discovered
that the sum of the squares of the two legs of a right triangle is
equal to the square of the hypotenuse. “Eureka!” he shouted, and the
GRE had a new topic to test. And the Greek police had a new person
to arrest, since Pythagoras had this epiphany while sitting in the
bath and immediately jumped out to run down the street and proclaim
it to the world, entirely in the buff. Or maybe
that was Archimedes . . . ? Hey, it was a long time ago. Suffice it
to say that some naked Greek guy thought up something cool.
Since Pythagoras took the trouble to invent his theorem, the
least we can do is learn it. Besides, it’s one of the most famous
theorems in all of math, and it is tested with regularity on the
GRE, to boot. Here it is:
In a right triangle, a2 +
b2 = c2, where
c is the length of the hypotenuse and
a and b are the lengths of the
two legs.
And here’s a simple application:
What is the value of b in the triangle above?
The little square in the lower left corner lets you know that
this is a right triangle, so you’re clear to use the Pythagorean
theorem. Substituting the known lengths into the formula gives:
3^{2} +
b^{2} =
5^{2}
9 + b^{2} = 25
b^{2} = 16
b = 4
This is therefore the worldfamous 345 right triangle.
Thanks to the Pythagorean theorem, if you know the measures of any
two sides of a right triangle, you can always find the third.
“Eureka!” indeed.
Pythagorean Triples.
Because right triangles obey the Pythagorean theorem, only a
specific few have side lengths that are all integers. For example, a
right triangle with legs of length 3 and 5 has a hypotenuse of
length . Positive
integers that obey the Pythagorean theorem are called Pythagorean
triples, and these are the ones you’re likely to see on your test as
the lengths of the sides of right triangles. Here are some common
ones:
{3, 4, 5}
{5, 12, 13}
{7, 24, 25}
{8, 15, 17}
In addition to these Pythagorean triples, you should also
watch out for their multiples. For example, {6, 8, 10} is a
Pythagorean triple, since it is a multiple of {3, 4, 5}, derived
from simply doubling each value. This knowledge can significantly
shorten your work in a problem like this:
What is the value of z in the triangle above?
Sure, you could calculate it out using the Pythagorean
theorem, but who wants to square 120 and 130 and
then work the results into the formula? Pythagoras himself would
probably say, “Ah, screw it . . .” and head off for a chat with
Socrates. (Actually, Pythagoras died about twenty years before
Socrates was born, but you get the point.)
But armed with our Pythagorean triples, it’s no problem for
us. The hypotenuse is 130, and one of the legs is 120. The ratio
between these sides is 130:120, or 13:12. This exactly matches the
{5, 12, 13} Pythagorean triple. So we’re missing the 5 part of the
triple for the other leg. However, since the sides of the triangle
in the question are 10 times longer than those in the {5, 12, 13}
triple, the missing side must be 5 × 10 = 50.
306090 Right Triangles
As you can see, right triangles are pretty darn special. But
there are two extraspecial ones that appear with astounding
frequency on the GRE. They are 306090 right triangles and 454590
right triangles. When you see one of these, instead of working out
the Pythagorean theorem, you’ll be able to apply standard ratios
that exist between the length of the sides of these triangles.
The guy who named 306090 triangles didn’t have much of an
imagination— or maybe he just didn’t have a cool name like
“Pythagoras.” The name derives from the fact that these triangles
have angles of 30°, 60°, and 90°. So, what’s so special about that?
This: The side lengths of 306090 triangles always follow a
specific pattern. If the short leg opposite the 30° angle has
length x, then the hypotenuse has length
2x, and the long leg, opposite the 60° angle,
has length . Therefore:
The sides of every 306090 triangle will follow
the ratio 1::2
Thanks to this constant ratio, if you know the length of just
one side of the triangle, you’ll immediately be able to calculate
the lengths of the other two. If, for example, you know that the
side opposite the 30º angle is 2 meters, then by using the ratio you
can determine that the hypotenuse is 4 meters, and the leg opposite
the 60º angle is
meters.
454590 Right Triangles
A 454590 right triangle is a triangle with two angles of 45°
and one right angle. It’s sometimes called an isosceles
right triangle, since it’s both isosceles and right.
Like the 306090 triangle, the lengths of the sides of a 454590
triangle also follow a specific pattern. If the legs are of length
x (the legs will always be equal), then the
hypotenuse has length .
The sides of every 454590 triangle will follow
the ration of
This ratio will help you when faced with triangles like this:
This right triangle has two equal sides, which means the two
angles other than the right angle must be 45° each. So we have a
454590 right triangle, which means we can employ the ratio. But instead of being
1 and 1, the lengths of the legs are 5 and 5. Since the lengths of
the sides in the triangle above are five times the lengths in the
right triangle, the
hypotenuse must be ,
or.
Area of a Triangle
The formula for the area of a triangle is:
area = or
Keep in mind that the base and height of a triangle are
not just any two sides of a triangle. The base and
height must be perpendicular, which means they must meet at a right
angle.
Let’s try an example. What’s the area of this triangle?
Note that the area is not , because those two sides do not
meet at a right angle. To calculate the area, you must first determine
b. You’ll probably notice that this is the 345
right triangle you saw earlier, so b = 4. Now you have
two perpendicular sides, so you can correctly calculate the area as
follows:
Triangles are surely important, but they aren’t the only geometric
figures you’ll come across on the GRE. We move now to quadrilaterals,
which are foursided figures. The first two we cover you’re no doubt
familiar with, no matter how long you’ve been out of high school. The
other two you may have forgotten about long ago.
Rectangles
A rectangle is a quadrilateral in which the opposite sides are
parallel and the interior angles are all right angles. The opposite sides of
a rectangle are equal, as indicated in the figure below:
Area of a Rectangle
The formula for the area of a rectangle is:
area = base × height or simply a =
bh
Since the base is the length of the rectangle and the height is
the width, just multiply the length by the width to get the area of a
rectangle.
Diagonals of a Rectangle
The two diagonals of a rectangle are always equal to each other,
and either diagonal through the rectangle cuts the rectangle into two
equal right triangles. In the figure below, the diagonal BD cuts
rectangle ABCD into congruent right triangles BAD and
BCD. Congruent means that those triangles are exactly
identical.
Since the diagonal of the rectangle forms right triangles that
include the diagonal and two sides of the rectangle, if you know two of
these values, you can always calculate the third with the Pythagorean
theorem. For example, if you know the side lengths of the rectangle, you
can calculate the length of the diagonal. If you know the diagonal and
one side length, you can calculate the other side length. Also, keep in
mind that the diagonal might cut the rectangle into a 306090 triangle,
in which case you could use the 1:: 2 ratio to make your
calculating job even easier.
Squares
“Hey, buddy, don’t you be no square . . .”
Elvis Presley, Jailhouse Rock
We don’t know why, but sometime around the 1950s the word
square became synonymous with “uncool.” Of course,
that’s the same decade that brought us words like daddyO
and dances called “The Hand Jive.”
We think that squares are in fact nifty little geometric creatures.
They, along with circles, are perhaps the most symmetrical shapes in the
universe—nothing to sneeze at. A square is so symmetrical because its angles
are all 90º and all four of its sides are equal in length. Rectangles are
fairly common, but a perfect square is something to behold. Like a
rectangle, a square’s opposite sides are parallel and it contains four right
angles. But squares oneup rectangles by virtue of their equal sides.
As if you’ve never seen one of these in your life, here’s what a
square looks like:
Area of a Square
The formula for the area of a square is:
area = s2
In this formula, s is the length of a side. Since the sides of a
square are all equal, all you need is one side to figure out a square’s
area.
Diagonals of a Square
The square has two more special qualities:
 Diagonals bisect each other at right angles and are equal in length.
 Diagonals bisect the vertex angles to create 45º angles. (This means that one diagonal will cut the square into two 454590 triangles, while two diagonals break the square into four 454590 triangles.)
Because a diagonal drawn into the square forms two congruent
454590 triangles, if you know the length of one side of the square,
you can always calculate the length of the diagonal:
Since d is the hypotenuse of the 454590 triangle that has legs
of length 5, according to the ratio, you know that. Similarly, if you know only the
length of the diagonal, you can use the same ratio to work backward to
calculate the length of the sides.
Parallelograms
A parallelogram is a quadrilateral whose opposite sides are parallel.
That means that rectangles and squares qualify as parallelograms, but so do
foursided figures that don’t contain right angles.
In a parallelogram, opposite sides are equal in length. That means
that in the figure above, BC = AD and AB = DC. Opposite angles are equal:
∠ABC = ∠ADC and ∠BAD = ∠BCD. Adjacent angles are
supplementary, which means they add up to 180°. Here, an example is
∠ABC + ∠BCD = 180°.
Area of a Parallelogram
The area of a parallelogram is given by the formula:
area = bh
In this formula, b is the length of the base, and
h is the height. As shown in the figure below, the height of a
parallelogram is represented by a perpendicular line dropped from one
side of the figure to the side designated as the base.
Diagonals of a Parallelogram
 The diagonals of a parallelogram bisect (split) each other: BE = ED and AE = EC
 One diagonal splits a parallelogram into two congruent triangles: ∆ABD = ∆BCD
 Two diagonals split a parallelogram into two pairs of congruent triangles: ∆AEB = ∆DEC and ∆BEC = ∆AED
Trapezoids
A trapezoid may sound like a new Star Wars character, but it’s
actually the name of a quadrilateral with one pair of parallel sides and one
pair of nonparallel sides. Here’s an example:
In this trapezoid, AB is parallel to CD (shown by the arrow marks),
whereas AC and BD are not parallel.
Area of a Trapezoid
The formula for the area of a trapezoid is a bit more complex than
the area formulas you’ve seen for the other quadrilaterals in this
section. We’ll set it off so you can take a good look at it:
In this formula, s1 and s2 are the lengths of the parallel sides
(also called the bases of the trapezoid), and h is the height. In a
trapezoid, the height is the perpendicular distance from one base to the
other.
If you come across a trapezoid question on the GRE, you may need
to use your knowledge of triangles to solve it. Here’s an example of
what we mean:
Find the area of the figure above.
First of all, we’re not told that the figure is a trapezoid, but
we can infer as much from the information given. Since both the line
labeled 6 and the line labeled 10 form right angles with the line
connecting them, the 6 and 10 lines must be parallel. Meanwhile, the
other two lines (the left and right sides of the figure) cannot be
parallel because one connects to the bottom line at a right angle, while
the other connects with that line at a 45° angle. So we can deduce that
the figure is a trapezoid, which means the trapezoid area formula is in
play.
The bases of the trapezoid are the parallel sides, and we’re told
their lengths are 6 and 10. So far so good, but to find the area, we
also need to find the trapezoid’s height, which isn’t given.
To do that, split the trapezoid into a rectangle and a 454590
triangle by drawing in the height.
Once you’ve drawn in the height, you can split the base that’s
equal to 10 into two parts: The base of the rectangle is 6, and the leg
of the triangle is 4. Since the triangle is 454590, the two legs must
be equal. This leg, though, is also the height of the trapezoid. So the
height of the trapezoid is 4. Now you can plug the numbers into the
formula:
Another way to find the area of the trapezoid is to find the areas
of the triangle and the rectangle, then add them together:
area = (4 × 4) +
(6 × 4)
(16) + 24
8 + 24 = 32
Circles
A circle is the collection of points equidistant from a given point,
called the center. Circles are named after their center points—that’s just
easier than giving them names like Ralph and Betty. All circles contain
360°. The distance from the center to any point on the circle is called the
radius (r). Radius is the most important measurement in
a circle, because if you know a circle’s radius, you can figure out all its
other characteristics, such as its area, diameter, and circumference. We’ll
cover all that in the next few pages. The diameter (d) of a circle stretches
between endpoints on the circle, passing through the center. A chord also
extends from endpoint to endpoint on the circle, but it does not necessarily
pass through the center. In the following figure, point C is the center of
the circle, r is the radius, and AB is a chord.
Tangent Lines
Tangents are lines that intersect a circle at only one point. Just
like everything else in geometry, tangent lines are defined by certain
fixed rules.
Here’s the first: A radius whose endpoint is the intersection
point of the tangent line and the circle is always perpendicular to the
tangent line, as shown in the following figure:
And the second rule: Every point in space outside the circle can
extend exactly two tangent lines to the circle. The distances from the
origin of the two tangents to the points of tangency are always equal.
In the figure below, XY = XZ.
Central Angles and Inscribed Angles
An angle whose vertex is the center of the circle is called a
central angle.
The degree of the circle (the slice of pie) cut by a central angle
is equal to the measure of the angle. If a central angle is 25º, then it
cuts a 25º arc in the circle.
An inscribed angle is an angle formed by two
chords originating from a single point.
An inscribed angle will always cut out an arc in the circle that
is twice the size of the degree of the inscribed angle. For example, if
an inscribed angle is 40º, it will cut an arc of 80º in the circle.
If an inscribed angle and a central angle cut out the same arc in
a circle, the central angle will be twice as large as the inscribed
angle.
Circumference of a Circle
The circumference is the perimeter of the circle—that is, the
total distance around the circle. The formula for circumference of a
circle is:
circumference = 2πr
In this formula, r is the radius. Since a
circle’s diameter is always twice its radius, the formula can also be
written c = πd, where d is the diameter. Let’s find the circumference of
the circle below:
Plugging the radius into the formula, c = 2πr =
2π (3) = 6π.
Arc Length
An arc is a part of a circle’s circumference. An arc contains two
endpoints and all the points on the circle between the endpoints. By
picking any two points on a circle, two arcs are created: a major arc,
which is by definition the longer arc, and a minor arc, the shorter one.
Since the degree of an arc is defined by the central or inscribed
angle that intercepts the arc’s endpoints, you can calculate the arc
length as long as you know the circle’s radius and the measure of either
the central or inscribed angle.
The arc length formula is:
arc length =
In this formula, n is the measure of the degree
of the arc, and r is the radius. This makes sense, if
you think about it: There are 360° in a circle, so the degree of an arc
divided by 360 gives us the fraction of the total circumference that arc
represents. Multiplying that by the total circumference
(2pr) gives us the length of the arc.
Here’s the sort of arc length question you might see on the test:
Circle D has radius 9. What is the length
of arc AB?
To figure out the length of arc AB, we need to
know the radius of the circle and the measure of ∠C,
the inscribed angle that intercepts the endpoints of arc
AB. The question provides the radius of the circle, 9,
but it throws us a little curveball by not providing the measure of
∠C. Instead, the question puts ∠C in a triangle and
tells us the measures of the other two angles in the triangle. Like we
said, only a little curveball: You can easily figure out the measure of
∠C because, as you know by now, the three angles of
a triangle add up to 180º:
∠c = 180° – (50° + 70°)
∠c = 180° – 120°
∠c = 60°
Since ∠c is an inscribed angle, arc
AB must be twice its measure, or 120º. Now we can plug
these values into the formula for arc length:
Area of a Circle
If you know the radius of a circle, you can figure out its area.
The formula for area is:
area = πr^{2}
In this formula, r is the radius. So when you
need to find the area of a circle, your real goal is to figure out the
radius. In easier questions the radius will be given. In harder
questions, they’ll give you the diameter or circumference and you’ll
have to use the formulas for those to calculate the radius, which you’ll
then plug into the area formula.
Area of a Sector
A sector of a circle is the area enclosed by a central angle and
the circle itself. It’s shaped like a slice of pizza. The shaded region
in the figure on the left below is a sector. The figure on the right is
a slice of pepperoni pizza. See the resemblance?
The area of a sector is related to the area of a circle just as
the length of an arc is related to the circumference. To find the area
of a sector, find what fraction of 360° the sector makes up and multiply
this fraction by the total area of the circle. In formula form:
area of a sector =
In this formula, n is the measure of the central
angle that forms the boundary of the sector, and r is
the radius. An example will help. Find the area of the sector in the
figure below:
The sector is bounded by a 70° central angle in a circle whose
radius is 6. Using the formula, the area of the sector is:
MishMashes: Figures with Multiple Shapes
The GRE test makers evidently feel it would be no fun if all of these
geometric shapes you’re learning appeared in isolation, so sometimes they
mash them together. The trick in these problems is to understand and be able
to manipulate the rules of each figure individually, while also recognizing
which elements of the mishmashes overlap. For example, the diameter of a
circle may also be the side of a square, so if you use the rules of circles
to calculate that length, you can then use that answer to determine
something about the square, such as its area or perimeter.
Mishmash problems often combine circles with other figures. Here’s an
example:
What is the length of minor arc BE in circle A if the area of rectangle ABCD is 18?
To find the length of minor arc BE, you have to know
two things: the radius of the circle and the measure of the central angle
that intersects the circle at points B and
E. Because ABCD is a rectangle, and
rectangles only have right angles, ∠BAD is 90°. In this
question, they tell you as much by including the rightangle sign. But in a
harder question, they’d leave the rightangle sign out and expect you to
deduce that ∠BAD is 90° on your own. And since that angle
also happens to be the central angle of circle A
intercepting the arc in question, we can determine that arc
BE measures 90°.
Finding the radius requires a bit of creative visualization as well,
but it’s not so hard. The key is to realize that the radius of the circle is
equal to the width of the rectangle. So let’s work backward from the
rectangle to give us what we need to know about the circle. The area of the
rectangle is 18, and its length is 6. Since the area of a rectangle is
simply its length multiplied by its width, we can divide 18 by 6 to get a
width of 3. As we’ve seen, this rectangle width doubles as the circle’s
radius, so we’re in business: radius = 3. All we have to do is plug in the
values we found into the arc length formula, and we’re done.
That covers the twodimensional figures you should know, but there are
also some threedimensional figures you may be asked about as well. We’ll
finish up this geometry section with a look at those.
Rectangular Solids
A rectangular solid is a prism with a rectangular base and edges that
are perpendicular to its base. In English, 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 volume, surface area, and diagonal length.
Volume of a Rectangular Solid
The formula for the volume of a rectangular solid builds on the
formula for the area of a rectangle. As discussed earlier, the area of a
rectangle is equal to its length times its width. The formula for the
volume of a rectangular solid adds the third dimension, height, to get:
volume = lwh
Here’s a good oldfashioned example:
What is the volume of the figure presented below?
The length is 3x, the width is
x, and the height is 2x. Just plug the values
into the volume formula and you’re good to go: v =
(3x)(x)(2x) =
6x3.
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 pairs:
surface area = 2lw + 2lh +
2wh.
Let’s try the formula out on the same solid we saw above. Find the
surface area of this:
Again, the length is 3x, the width is
x, and the height is 2x. Plugging into the
formula, we get:
surface area = 2(3x)(x) +
2(3x)(2x) +
2(x)(2x)
=6x^{2} +
12x^{2} +
4x^{2}
= 22x^{2}
Dividing Rectangular Solids
If you’re doing really well on the Math section, the CAT
program will begin scrounging for the toughest, most esoteric
problems it can find to throw your way. One such problem may
describe a solid, give you all of its measurements, and then tell
you that the box has been cut in half, like so:
A number of possible questions could be created from this
scenario. For example, you may be asked to find the combined surface
area of the two new boxes. Or maybe a Quantitative Comparison
question would ask you to compare the volume of the original solid
with that of the two new ones. Actually, the volume remains
unchanged, but the surface area increases because two new sides
(shaded in the diagram) emerge when the box is cut in half. You may
need to employ a bit of reasoning along with the formulas you’re
learning to answer a difficult question like this, but it helps to
know this general rule: Whenever a solid is cut into smaller pieces,
its surface area increases, but its volume is unchanged.
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 a question will test to see if you can find the
length of a diagonal. Here’s the formula:
Again, l is the length, w is the
width, and h is the height. The formula is like a
pumpedup version of the Pythagorean theorem. Check it out in action:
What 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:
The problem could be made more difficult if it forced you to first
calculate some of the dimensions before plugging them into the
formula.
Cubes
A cube is a threedimensional square. The length, width, and height of
a cube are equal, and each of its six faces is a square. Here’s what it
looks like—pretty basic:
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 solid: We just need
to multiply the length, width, and height. However, since a cube’s
length, width, and height are all equal, the formula for the volume of a
cube is even easier:
volume = s3
In this formula, 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, and
h. This boils down to:
volume =
6s^{2}
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, and h. This yields , which simplifies to:
volume =
Right Circular Cylinders
A right circular cylinder looks like one of those cardboard things
that toilet paper comes on, except it isn’t hollow. It has two congruent
circular bases and looks like this:
The height, h, is the length of the line segment whose endpoints are
the centers of the circular bases. The radius, r, is the
radius of its base. For the GRE, all you need to know about a right circular
cylinder is how to calculate its volume.
Volume of a Right Circular Cylinder
The volume of this kind of solid is the product of the area of its
base and its height. Because a right circular cylinder has a circular
base, its volume is equal to the area of the circular base times the
height or:
volume =
πr^{2}h
Find the volume of the cylinder below:
This cylinder has a radius of 4 and a height of 6. Using the
volume formula, its volume = π(4)^{2}(6) = 96π.
We’ve covered a lot of ground so far in this Math 101 chapter,
working our way through arithmetic, algebra, and now geometry. We’ll
bring it on home with our final subject, data analysis.
