7.1 SAT Math - Basic Geometry

 

Since Geometry usually deals with pictures, there are some unique Rules for Geometry that get special mention here.  Know these rules to help improve your SAT math score!

1) Draw It.  If they describe how something is supposed to look, draw it.  It doesn't have to be perfect, but give yourself a general idea of what it should look like.  Geometry is a visual math.
-SAT Math Hint: Your drawing doesn't have to be perfect, but especially when dealing with coordinate geometry, that is, graphing points, draw it as accurately as you can in the short amount of time you have.  For example, if you were to draw a quick coordinate plane in order to plot some points, don't just slap some tick marks on there any which way like this:

 

Those are horribly inaccurate and will not help you get a clear answer when you plot your points.  Keep the lines evenly spaced from each other and from the axes:

2) Know if the Object Given is Drawn to Scale.  Almost all the drawings given to you on this test are drawn to scale, but those that aren't are usually NOT drawn to scale in order to trick you.  How can you tell if an object is NOT drawn to scale?  It will say "Note: Figure not drawn to scale" underneath the picture.  If there is no note, the figure is drawn to scale.

How can they trick you with a figure not drawn to scale?  In any number of ways.  The points given might be much closer or much farther apart than they look, the angle given could be only a 5° angle, or 170° angle, the lines drawn could be parallel, perpendicular, or neither…  The only thing that you can count on with a figure not drawn to scale is that the points given are in the proper order – so if there are 4 points on a line, A, B, C, and D, they are in that order, though their differences apart might vary greatly.  Or if given a triangle, it will be a triangle, it just might not be acute or right or however else it looks.  If a figure is not drawn to scale, trust only the numbers and facts that you are explicitly given, not the way the picture looks.

3) Fill in Everything You Know.  Too often on geometry questions you'll get a picture with almost no labels on it, and a question that you don't quite know how to solve.  There are two versions of this scenario:
1) You know the general concept they're asking about and the question is clear, you just don't know how to solve it from there.
2) You have no idea what they're talking about at all.

Sometimes you think you're in Scenario #1, and it turns out you're clueless.  Sometimes you think you're lost, but suddenly something clicks and you know exactly how to solve it.  The best way to quickly get to one of those two conclusions is to always, immediately fill in anything you know on the picture.  If you know what an angle equals, write it in.  If you don't know the measurements but you know those two lines or angles are equal, mark them as such.  If the lines are parallel, label them.  If points are listed, plot them.   If the question mentions Area, write down the formula for area.  And then lean back and take a look.  Oftentimes, just seeing the information in front of you will help the lightbulb go off.  Which leads us to a related rule that we'll call 3b.

3b) Jump In.  If you've got all that information written down and you find you're in Scenario 1 above (you get the question, you just don't get how to solve it from here), jump in and start solving stuff.  Find a radius, find a distance, find an angle's measurement.  Once you start seeing what's solvable, it's usually a pretty quick leap to what you need to solve.

On the other hand, if you write down everything you know and you're still stuck in Scenario 2 (you have no idea what the heck they're talking about), skip it!  Come back.  If you don't see it, you don't see it, and you need a change of view or you're just not going to get that one right.

4) Memorize the Formulas.  At the beginning of every math section on the SAT, you are given a box with various formulas listed.  Do NOT make the mistake most students make of not memorizing those formulas because you know you have a cheat sheet!  There are two reasons this is a bad idea.  First, it takes too much time to flip back and forth checking formulas in the middle of a timed test.  Secondly, and more importantly, there are many questions in which the way to solve seems pretty easy if you know the formulas by heart – your brain will quickly search its files to pull out the one you need, even if you're not told that that's the formula you should use.  If you don't know the formulas you will never make that leap and you will never be able to figure out the question, even though the formulas are listed at the front.  Memorize them!  You should be able to spit out the formula without even thinking about it; if you can't, keep studying.

SAT Formulas

The number of degrees of arc in a circle is 360.
The measure in degrees of a straight angle is 180.
The sum of the measure in degrees of the angles of a triangle is 180.

Those are the formulas listed at the beginning of every Math section.  Let's talk briefly about a few of them that tend to lead to errors:

Perimeter vs. Area

Perimeter vs. Area – Know the difference.  Perimeter is the distance around and is found in everything except circles by adding up all the sides; Area covers the whole inside of a 2-Dimensional shape (like a triangle or rectangle) and is found by multiplying the length and width in a quadrilateral, or by various other formulas involving multiplying in other shapes. 
            -SAT Math Tip: Squares with Perimeters that are Perfect Squares.  One of the favorite tricks on the SATs is to give you a square with a perimeter of 64 or the like.  You're in a hurry, you're trying to get through as many questions as possible, you instantly see "Square.  64.  Cool.  Square root of 64 is 8.  Each side is 8.  Easy."  And now you just got it wrong.  To find the side of a square when you know the perimeter, you must divide the perimeter by 4, not find the square root.  When you know the area of a square, you can find the square root to find a side.

Volume

Volume – There are more specific volume formulas that will be discussed in turn, but the general formula is "Area of the base times the height."  This general formula applies to all shapes, including cylinders, and is a quick easy way to remember the volume of any figure on this test.

Pi π

Pi – You need a calculator that has a pi button for the SAT exam.  Pi is shown by the symbol π, and is approximately equal to 3.14.  If you need to quickly estimate pi to eliminate wrong answer choices, use 3 to do the math in your head quickly.  Otherwise, use your calculator to get an accurate answer.  If you are looking to find the Circumference of a circle, and the radius is 3, the answer will be 6π or 18.849…, NOT 18.  Unless the question asks for an approximation, 18 will be incorrect as an answer.
-SAT Math Strategy: Eliminate Obviously Wrong Answers
The nice thing about geometry on the SAT is that a lot of people don't know the rules.  Many people, for whatever reason, forget that pi has a value, or that square roots are greater than 1, or the difference between acute and obtuse angles, or the degrees in a circle.  Because geometry has so many formulas and facts that need to be memorized, and because each answer choice on the SATs is carefully worked out to be one that people who make common errors will choose, the wrong answers in the Geometry section are often pretty obvious if you know your rules.  Get rid of the blatantly wrong answer choices.  Take a look at this one:

7. If x = 127°, what is the measure of ∠ADC?

(A) 37°
(B) 53°
(C) 127°
(D) 143°
(E) 233°

This isn't a particularly difficult question as long as you read carefully, but if you know your rules and formulas, you don't even have to do any math at all.  If you remember that a circle has 360°, and if you notice that they are asking for ∠ADC, which is the wide angle of the circle, you know that answer is going to have to greater than 127°, and most likely much larger than that.  You can already eliminate answer choices A, B, and C, and just by looking you can probably take an educated guess that E is the most likely choice.  Most geometry questions can be narrowed to one to three remaining answer choices without doing any actual work.  You simply have to have the important facts memorized, and get rid of the impossible choices.

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