Example:
{3, 4, 5} is a Pythagorean triple because
32 + 42 = 52.
{1, 1, √2} is a Pythagorean triple because
12 + 12 = √22
{1, √3, 2} is a Pythagorean triple because
12 + √32 = 22
{5, 12, 13} is a Pythagorean triple because
52 + 122 = 132
When we multiply each number in a Pythagorean triple by the same number, we get another Pythagorean triple.
Example:
If we know {3, 4, 5} is a Pythagorean triple, then we also know {6, 8, 10} is a Pythagorean triple, because {6, 8, 10} is what we get when we multiply every number in {3, 4, 5} by 2.
In a {1, 1, √2} right triangle, the angle measurements are 45o, 45o, 90o.
In a {1, √3, 2} right triangle, the angle measurements are 30o, 60o, 90o.
Two triangles are “similar triangles” if they have all the same angle measurements.
Between two similar triangles, the relationship between any two corresponding sides is the same as between any other two corresponding sides.
Example:
Triangles ABC and DEF below are similar. Side AB has length 8, and side DE has length 24, so every side measurement in DEF must be three times the corresponding side in
ABC.
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.”)
Parallelograms
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 360o, and opposite angles have equal measurements.
Rectangles
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 52 + 122 = 132.
Squares
Squares are special rectangles where all the sides have equal length.
Area
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.
Perimeters (squares, rectangles, circles)
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 = 2pi.
Other polygons
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.
Circles (diameter, radius, arc, tangents, circumference, area)
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.