Saturday, February 12, 2011

Perfect Squares

There's a certain amount of memorization that should be done for the math portion of the SAT.  Relying on just the formulas given on the test is not enough (those are just a small, incomplete subset of what you need to know, and you shouldn't need to refer to them while taking the test as you should have them memorized).  It's possible to get away with a lot using your calculator, but it's better not to.

In school, you should have learned perfect squares up to at least 15^2 or 16^2.  These are also the ones you should know for the SAT.  The more you know, the better.  If you don't know these perfect squares now, take the time to memorize the ones you don't know so well.  (At the least, know how to generate a short list of them on your calculator very quickly--but memorization is the better choice.)

Here's a list up to 20^2.

1^2 = 1
2^2 = 4
3^2 = 9
4^2 = 16
5^2 = 25
6^2 = 36
7^2 = 49
8^2 = 64
9^2 = 81
10^2 = 100

11^2 = 121
12^2 = 144
13^2 = 169
14^2 = 196
15^2 = 225
16^2 = 256
17^2 = 289
18^2 = 324
19^2 = 361
20^2 = 400

There are many tricks and patterns with perfect squares, but I'll leave that for another time.

Tuesday, February 8, 2011

Isosceles Triangles - calculating area (given sides)

This is an essential skill for the SAT.  It is commonly needed to directly solve a math problem on the SAT or as an important step in a larger problem.

Definition:  Isosceles triangles are triangles that have at least 2 sides of equal length.

There are many ways to find the area of various types of triangles.  In this post, I will show you how to calculate the area of an isosceles triangle given the lengths of the 3 sides.  The method illustrated is relatively straight-forward and is used by many students.

Given: isosceles triangle with 3 given sides.

To find: area of triangle (height is by-product).


1. Draw an altitude from the vertex between the 2 equal sides to the opposite side.

This opposite side serves as the base corresponding to the altitude just drawn.  The base is now split in half.  The isosceles triangle is split into 2 congruent right triangles (can you prove this?).

2. Next, find the length of this altitude using the Pythagorean theorem.

3. Plug & chug into A = (1/2) b * h.


A triangle is given with sides 5, 5, 8.  Choose the side of 8 to be the base.  Draw the altitude to this base.  Each of the right triangles formed has a hypotenuse of 5 and one leg of 4 (half of the side of 8).  The other leg is the same as the altitude of the original triangle.  It has length 3 by the Pythagorean theorem (3^2 + 4^2 = 5^2).  Thus, the original triangle has base 8, height 3, and area (1/2) * 8 * 3 = 12.

Note:  Equilateral triangles are isosceles.  You can find their areas in the same way.  The 2 right triangles you construct will be 30-60-90 triangles.  The height of the equilateral triangle will then be x * sqrt(3)/2, where x is the length of one of the sides.  The area of the equilateral triangle is then x^2 * sqrt(3) / 4.  These formulas for the height and area of an equilateral triangle are worth memorizing because equilateral triangles are so common on the SAT.


1. Find the height and area of a triangle with sides 13, 13, 10.  (Area is 60.)

2. Find the height and area of a triangle with sides 25, 25, 30.  (Area is 300.)

3. Find the height and area of a triangle with sides 3, 3, 3.  (Area is 9 * sqrt(3) / 4.)

Monday, February 7, 2011

First post


I will use this blog for a variety of things.  Probably, mostly for SAT math tips, test-taking strategies, and study strategies for now...

WARNING on math: I could get very technical about the way I write math, but I try not to in order to make things easier to understand.  For instance, I will write things like "triangle with sides 3, 4, 5" when I really mean the sides have lengths of 3, 4, and 5.  This is also the way I tend to speak in person.

I'm not that careful about grammar, but feel free to correct me.

I also tend to edit things a lot, so you might see things change after you read it the first time.

Please feel free to ask questions or add other comments.