SAT Prep Black Book: The Most Effective SAT Strategies Ever Published (31 page)

The formula for the area of a triangle is given in the front of every real SAT Math section.

In every triangle, the length of each side must be less than the sum of the lengths of the other sides. (Otherwise, the triangle would not be able to “close.”)

A parallelogram is a four-sided figure where both pairs of opposite sides are parallel to each other.

In a parallelogram, opposite angles are equal to each other, and the measures of all the angles added up together equal 360.

Example:

In
ABCD below, all the interior angles taken together equal 360
o
, and opposite angles have equal measurements.

Rectangles are special parallelograms where all the angles measure 90 degrees. In a rectangle, if you know the lengths of the sides then you can always figure out the length from one corner to the opposite corner by using the Pythagorean theorem.

Example:

In the rectangle below, all angles are right angles, and we can use the Pythagorean theorem to determine that the diagonal AC must have a length of 13, since 5
2
+ 12
2
= 13
2
.

Squares are special rectangles where all the sides have equal length.

The area of a two-dimensional figure is the amount of two-dimensional space that the figure covers.

Area is always measured in square units.

All the area formulas you need for the SAT appear in the beginning of each Math section, so there’s no need to memorize them—you just need to know how to use them.

The perimeter of a two-dimensional object is the sum of the lengths of its sides or, for a circle, the distance around the circle.

To find the perimeter of a non-circle, just add up the lengths of the sides.

The perimeter of a circle is called the “circumference.”

The formula for the circumference of a circle appears in the beginning of every real SAT Math section.
It’s C = 2
pi
.

The SAT might give you questions about special polygons, like pentagons, hexagons, octagons, and so on.

The sum of the angle measurements of any polygon can be determined with a simple formula: Where
s
is the number of sides of the polygon, the sum of the angle measurements is (
s
– 2) * 180.

Example:

A triangle has 3 sides, so the sum of its angle measurements is given by (3 – 2) * 180, which is the same thing as (1) * 180, which is the same thing as 180. So the sum of the measurements of the angles in a triangle is 180 degrees. (Remember that we already knew this!)

A hexagon has 6 sides, so the sum of its angle measurements is (6 – 2) * 180, or (4) * 180, which is 720. So all the angles in a hexagon add up to 720 degrees.

To find the perimeter of any polygon, just add up the lengths of the sides.

To find the area of a polygon besides a triangle
, parallelogram, or circle, just divide the polygon into smaller triangles, polygons, and/or circles and find the areas of these pieces. A real SAT math question will always lend itself to this solution nicely.

A circle is the set of points in a particular plane that are all equidistant from a single point, called the center.

Example:

Circle
O
has a center point
O
and consists of all the points in one plane that are 5 units away from the center:

A radius is a line segment drawn from the center point of a circle to the edge of the circle
at point R.

Example:

In the circle above, the line segment OR is a radius because it stretches from the center of the circle (O) to the edge of the circle

All the radii of a circle have the same length, since all the points on the edge of the circle are the same distance from the center point.

A diameter is a line segment drawn from one edge of a circle, through the center of the circle, all the way to the opposite edge.

Example:

LR is a diameter of circle
O
because it starts at one edge of the circle, stretches through the center of the circle, and stops at the opposite edge of the circle.

Because a diameter can be broken into two opposite radii, a diameter always has a length equal to twice the radius of the circle.

A diameter of a circle is the longest line segment that can be drawn through the circle.

A tangent line is a line that intersects a circle at only one point.

A tangent line is perpendicular to the radius of the circle that ends at the one point shared by the tangent and the circle.

Example:

Circle
O
has a tangent line TS that intersects the circle at point R, and is perpendicular to radius OR.

The circumference of a circle is the length around the circle, similar to the perimeter of a polygon.

An arc is a portion of a circle. We can measure an arc by drawing radii to the endpoints of the arc, and then measuring the angle formed by the radii at the center of the circle.

Example:

Circle
O
has a 90
o
arc
PR
, which we can measure by measuring the angle formed by radius
PO
and radius
RO
.

The formulas for area and circumference of a circle appear in the beginning of all real SAT math sections, so t
here’s no need to memorize them if you don’t already know them.

On the SAT, solid geometry may involve cubes, rectangular solids, prisms, cylinders, cones, spheres, or pyramids.

All necessary volume formulas will be given to you, so there’s no need to memorize them.

The surface area of a solid is the sum of the areas of its faces (except for spheres or other “rounded” solids, which you won’t have to worry about on the SAT).

The arithmetic mean of a set of numbers is the result you get when you add all the numbers together and then divide by the number of things that you added.

Example:

The avera
ge of {4, 9, 92} is 35, because
(4 + 9 + 92)/3 = 35.

The median of a set of numbers is the number that appears in the middle of the set when all the numbers in the set are arranged from least to greatest.

Example:

The median of {4, 9, 92} is 9, because when we arrange the three numbers from least t
o greatest, 9 is in the middle.

If there is an even number of elements in the set, then the median of that set is the arithmetic mean of the two numbers in the middle of the set when the elements of the set are arranged from least to greatest.

Example:

The median of {4, 9, 11, 92} is 10, because the number of elements in the set is even, and 10 is the average of the two numbers in the middle of the set.

The mode of a set of numbers is the number that appears most frequently in the set.

Example:

The mode of {7, 7, 23, 44} is 7, because 7 appears more often than any other number in the set.

The probability of an event is a fraction that describes how likely the event is to happen. If the fraction is closer to 1, the event is more likely to happen; if the fraction is closer to zero, the event is less likely to happen.

To determine the fraction, you first calculate the total number of possible outcomes and place this number in the denominator of the fraction; then, you determine the number of outcomes that satisfy the event’s requirements, and place this number in the numerator of the fraction.

Example:

The probability of rolling a 3 on a normal 6-sided die is 1/6. There are 6 possible outcomes, so 6 goes in the denominator of the fraction. Out of those 6 outcomes we only want one, the one where a 3 comes up, so 1 goes in the numerator of the fraction.

The probability of rolling an odd number on a normal 6-sided die is 3/6. Again, there are 6 possible numbers we might roll, so 6 is our denominator. But now, since we want any odd number, the numbers 1, 3, and 5 all satisfy the requirements of our event, so there are 3 possible outcomes that we’ll be happy with—that means 3 goes in the numerator.

Probability fractions can be manipulated just like any other fractions.

To find the probability of two or more events happening in a sequence, we just find the probabilities of each event by itself, and then multiply them by each other.

Example:

The probability of rolling double-sixes on two normal 6-sided dice is 1/36, because the probability of rolling a six on either die is 1/6, and
(1/6)(1/6) = 1/36.

We’ve just covered all the math concepts that the College Board will allow itself to cover on the SAT. As I mentioned at the beginning of the Toolbox, it’s important to keep in mind that simply knowing these concepts is not enough to guarantee a good score on the SAT. It’s much more important to focus on the design of the SAT Math section and learn how to take apart challenging questions.

And that’s exactly what we’ll start talking about on the next page . . .