The Best Geometry Regents Review Guide 2020

If you're currently attending a public high school in the state of New York, then you'll need to pass the New York Regents exams in order to get your diploma. The Geometry Regents exam tests you on a huge array of geometry-related topics, from angles to 3-D shapes. The upcoming Geometry Regents exam is on Wednesday, January 22, 2020, at 9:15 am.

This extensive Geometry Regents review guide will tell you everything you need to know about the format of the exam, what it tests you on, and what questions look like. We'll also give you our best tips for acing it.

What's the Format of the Geometry Regents Exam?

Let's start off our Geometry Regents review by looking at the structure of the math test. Geometry Regents is a four-part exam consisting of 35 questions, and you'll have three hours to complete it.

Here's an overview of the format:

Each multiple-choice question has four answer choices (labeled 1-4) from which you will choose one and then record it on a separate answer sheet.

For the constructed-response questions, you must do the following in order to earn full credit, per the official Geometry Regents instructions:

"Clearly indicate the necessary steps, including appropriate formula substitutions, diagrams, graphs, charts, etc. Utilize the information provided for each question to determine your answer. Note that diagrams are not necessarily drawn to scale."

Essentially, you've got to show your work. Putting down only a correct answer will net you 1 point, but that's it.

While you won't get separate scrap paper for the exam, you may use any blank spaces provided in the test booklet (though it will not be scored). Note that you will be given one sheet of scrap graph paper (should you choose to use it) at the back of the test booklet; anything written on this will not be scored.

The following tools and devices must be available for you to use on the Geometry Regents exam:

• A graphing calculator
• A ruler
• A compass

Finally, you'll get a "High School Math Reference Sheet" with basic formulas and conversions that could come in handy during the exam. This reference sheet can be found at the back of your test booklet; you may tear it out along the perforated edges and use at any time during the Geometry Regents exam.

Here's how the reference sheet will appear:

What Do Geometry Regents Questions Look Like?

Each of the four parts of Geometry Regents contains different types of math questions:

• Multiple choice (Part I)
• Short response (Part II)
• Medium response (Part III)
• Long response (Part IV)

Let's take a look at examples of these below.

All questions come from the August 2019 administration of the Geometry Regents test, and all student responses come from NYSED Regents.

Multiple-Choice Sample Question (Part I)

This question is essentially asking you to do two things:

• Find the volume of a sphere (i.e., the golf ball)
• Using the volume, find the total weight of the golf ball

We know it's asking us to find the volume because it gives us the weight of the golf ball per cubic inch. (Whenever you hear the word "cubic," it's likely going to be a volume-related problem!)

To find the volume of a sphere, you must know the following formula (which is fortunately on the reference sheet above, so you don't need to memorize it):

$V={4/3}π{r^3}$

In this formula, $V$ stands for "volume" and $r$ stands for "radius," which is equal to half the diameter of a circle or sphere. We're told that the diameter of the golf ball is 1.680 inches. Divide this number in half to get the radius:

$1.680/2=0.84$

Now, we can plug this number into our volume formula and solve:

$V={4/3}π{0.84^3}$
$V={4/3}π(0.593)$
$V=(4.189)(0.593)$
$V=2.484$

The volume of the golf ball is roughly 2.484 cubic inches.

But this problem isn't asking for the volume—it's asking for the weight of the ball. We already know that 1 cubic inch weighs 0.6523 ounces, so all we need to do is multiply this by the total number of cubic inches, i.e., the volume we found above:

$2.484*0.6523=1.62$

This gives us 1.62 total ounces, which is the same as answer choice 2.

Short-Response Sample Question (Part II)

There are some key rules about angles you'll need to know in order to solve this problem:

• All angles in a triangle add up to 180°
• Opposite angles are congruent (i.e., the same) in a parallelogram
• The rule of alternate interior angles

We've been asked to find the measure of angle $∠ B$. First things first: you likely noticed that the line from angle $∠ A$ to angle $∠ C$ that divides the parallelogram has created two triangles, the left of which already has two angles filled out for us: $98°$ and $36°$.

To find angle $∠ D$⁠—the last angle in that triangle—we need to subtract the measures of the other two angles ($\bo 98°$ and $\bo 36°$) from $\bo 180°$, since all angles in a triangle must add up to $180°$:

$98+36=134$
$180-134=46$
$∠ D=46°$

Because opposite angles are congruent in parallelograms, we can conclude that $∠ \bi D= ∠ \bi B$, meaning that $∠ \bi B$ must also equal $\bo 46°$.

You would need to show your thinking clearly in order to get full credit for this. Below is a sample student response that approaches the problem a little differently but still gets the correct answer and earns full points:

Medium-Response Sample Question (Part III)

To solve this problem, think of the cargo trailer as two separate shapes:

• A rectangle/rectangular prism on the left
• A triangle/triangular prism on the right

We need to find the total volume of the trailer, so we should start by finding the volume of the rectangular prism. To find the volume of a rectangular prism, you just multiply the length, width, and height:

$V=(10)(6)(6.5)$
$V=390$

We know that the volume of the rectangular part of the cargo trailer is equal to 390 cubic feet. But what about the triangular part? The easiest way to find the volume of a triangular prism is to solve for the area of the triangle first and then multiply it by the height (which we know is 6.5 ft).

To find the area of a triangle, we must use the following formula (which is on the reference sheet):

$A={1/2}bh$

Think of the triangle as if it's been turned on its side. In this formula, $b$ is the "base" of the triangle (the vertical dotted line) and $h$ is the "height" (the horizontal dotted line). Note that the height we are talking about here is not the height of the cargo trailer, which is 6.5 ft!

Because the base of the triangle is also one side of the rectangle, we know that the measurement of this side has to be 6 ft (like its opposite side).

Now, we just have to find the height of the triangle. To do this, divide it into two right triangles using the dotted height line; this causes the base to divide in half as well, giving us one right triangle with a base of 3 and a hypotenuse of 4.

We'll call the unknown measurement (the height of our original triangle) $x$. To find $x$, we can use the Pythagorean theorem (on the reference sheet):

$a^2+b^2=c^2$

Plug the numbers we have for our base (3) and the hypotenuse (4) into the formula to solve for $x$:

$(3^2)+x^2=4^2$
$9+x^2=16$
$x^2=7$
$x=√7$
$x=2.6458$

Since we now have the height, we can go back and solve for the area of our original triangle using the original base (6) and plugging everything into the triangle area formula:

$A={1/2}bh$
$A=(1/2)(6)(2.6458)$
$A=(3)(2.6458)$
$A=7.9374$

Now, multiply the area of the triangle by the height of the trailer (6.5 ft) to get the volume of this section of the trailer:

$V=(6.5)(7.9374)$
$V=51.5931$

Finally, add the two volumes together to get the total volume of the trailer:

$V=51.5931+390$
$V=441.5931$
$V=442$

Our answer, rounded to the nearest cubic foot, is 442 cubic feet.

Here's a sample student response that earned full credit:

Long-Response Sample Question (Part IV)

This three-part question is worth a whopping 6 credits. Let's look at how we'll need to approach it, one section at a time.

Part 1

Given three sets of coordinates for each point on the triangle $ABC$, we must determine whether this triangle is isosceles (in other words, does it have two sides that are equal in length?).

We can figure out the lengths between $A$, $B$, and $C$ using the distance formula (which is not on your reference sheet). To find the distance between points ($x_1$, $y_1$) and ($x_2$, $y_2$), we use this formula:

$√{(x_2-x_1)^2+(y_2-y_1)^2}$

Let's use the points $A$ and $B$ first to find the distance of side $\ov {AB}$. Point $A$ is located at $(1, 2)$ and point $B$ is located at $(-5, 3)$:

$√{(-5-1)^2+(3-2)^2}$
$√{-6^2+1^2}$
$√{36+1}$
$√{37}$

The length of side $\ov {AB}$ is $\bo √{37}$.

Now, let's solve for the length of side $\ov {BC}$. Remember that $B$ is at $(-5, 3)$ and $C$ is at $(-6, -3)$:

$√{(-6--5)^2+(-3-3)^2}$
$√{(-6+5)^2+(-3-3)^2}$
$√{-1^2+-6^2}$
$√{1+36}$
$√{37}$

The length of side $\ov {BC}$ is also $\bo √{37}$, so we can say that triangle $ABC$ is isosceles.

Here's how one student wrote this out to earn full credit:

Part 2

This part isn't too tricky if you use the set of axes to graph out the triangle and square. Here's one student's correctly graphed diagram:

You can see here that the student first started by graphing out the triangle and connecting points $A$, $B$, and $C$. Because opposite sides are parallel in a square, we can find point $\bi D$ by drawing a bottom line that's parallel to $\ov {BA}$ and a vertical line that's parallel to $\ov {BC}$.

In other words, to find point $D$ (and to make side $\ov {CD}$), all you need to do is copy the slope of $\ov {BA}$ by counting to the right 6 units and down 1 unit. (Or, you could find side $\ov {AD}$ first by copying the slope of side $\ov {BC}$ and counting to the left 1 unit and down 6 units.)

Both methods should yield the correct result: D = (0, -4).

Part 3

The final part of this question is to prove that the quadrilateral above with the coordinates $(0, -4)$ for $D$ is, in fact, a square.

We know that a square has four equal sides, so we must show that sides $\ov {AD}$ and $\ov {CD}$ equal $√{37}$ (the same length for sides $\ov {AB}$ and $\ov {BC}$). To do this, simply plug these coordinates into the distance formula and solve. Remember that point $A$ is $(1, 2)$, $C$ is $(-6, -3)$, and $D$ is $(0, -4)$:

$\ov {AD}=√{(0-1)^2+(-4-2)^2}$
$\ov {AD}=√{-1^2+-6^2}$
$\ov {AD}=√{1+36}$
$\ov {AD}=√{37}$

$\ov {CD}=√{(0--6)^2+(-4--3)^2}$
$\ov {CD}=√{(0+6)^2+(-4+3)^2}$
$\ov {CD}=√{6^2+-1^2}$
$\ov {CD}=√{36+1}$
$\ov {CD}=√{37}$

This alone won't get you full credit, though—you must also prove that the slopes of consecutive sides in a square are opposite reciprocals (meaning that a number is flipped upside down into a fraction with the opposite sign). Doing this proves that consecutive sides are perpendicular to each other, thereby forming right angles—an essential feature of a square.

To show that the slopes are opposite reciprocals, find the slope of two perpendicular sides (e.g., $\ov {AB}$ and $\ov {BC}$) using the slope formula (which is not on your reference sheet):

$\Slope={y_2-y_1}/{x_2-x_1}$

Here are the slopes of $\ov {AB}$ and $\ov {BC}$ using their provided coordinates:

$\Slope \of$ $\ov {AB}={3-2}/{-5-1}$
$\Slope \of$ $\ov {AB}={1}/{-6}$
$\Slope \of$ $\ov {AB}=-1/6$

$\Slope \of$ $\ov {BC}={-3-3}/{-6--5}$
$\Slope \of$ $\ov {BC}=-6/{-6+5}$
$\Slope \of$ $\ov {BC}=-6/-1$
$\Slope \of$ $\ov {BC}=6$

${-1/6}$ is the opposite reciprocal of 6, proving that $\ov {AB}$ and $\ov {BC}$ are perpendicular to each other and that $∠ B$ is a right angle.

Here's a student's answer to show you how you could write this out:

What Topics Does Geometry Regents Cover?

• Circles
  • Pi
  • Circumference
• Triangles
  • Types (equilateral, isosceles, scalene, right)
  • Special right triangles
  • Pythagorean theorem
• Quadrilaterals (rectangles, squares, parallelograms, rhombuses, trapezoids)
• Other polygons (pentagons, hexagons)
• Lines and slopes
• Angles
  • Acute angles
  • Obtuse angles
  • Right angles
  • Rules of angles (e.g., opposite interior angles)
• Transformations
  • Rotations
  • Reflections
  • Translations
  • Resizing
• Perimeters of shapes, including circles, squares, triangles, and other polygons
• Areas of shapes, including circles, squares, triangles, and other polygons
• Volumes of 3-D shapes, including spheres, cylinders, cones, pyramids, and rectangular prisms

Now, what percentage of Geometry Regents does each topic take up? Refer to the chart below:

Source: Engage NY via the New York State Education Department

How to Pass Geometry Regents: 5 Essential Tips

If you're taking the Geometry Regents exam to fulfill your high school graduation requirements, it's incredibly important that you pass it. Passing means scoring at least 65, or earning about 29-30 credits/points.

You can use the official Geometry Regents raw score conversion charts to see how points generally translate into final scaled scores on the exam. Each year differs slightly, so it will never be the exact same number of credits you need.

Here are five tips to help you get this passing score on test day.

#1: Take Real Practice Tests

One of the best ways to do some Geometry Regents review is to take official practice tests, which you can find and download for free on the New York State Education Department website. These are real, previously administered exams, so you can rest assured you'll be getting accurate and realistic test-taking practice when you use them.

It's best to take one practice exam at the beginning of your prep, one around the middle, and one close to test day. This way you can see your progress and get a sense for what topics or question types are giving you the most trouble.

Make sure to time yourself as you would on the real exam (three hours) and have out a graphing calculator, ruler, and compass. You should also take the test in a quiet room, away from other people, so you can focus better and get used to real testing conditions.

Once you're finished, score your test with its answer key to see how well you did and whether you passed with a scaled score of 65 or higher. Refer to the model student responses so you can see how certain math problems are supposed to be solved, as well as what types of answers earn (and don't earn) full credit.

#2: Review Major Topics Using Class Materials

Everything that's tested on the Geometry Regents exam is stuff you should have already learned in your high school geometry class.

Therefore, you can use your old homework assignments, graded quizzes and tests, and geometry textbook to review the concepts tested on the exam. Look for patterns in your mistakes to see whether there are any major areas you're still struggling to understand conceptually.

If you're looking for practice questions, see whether your geometry textbook has any you didn't do for in-class practice or homework assignments.

#3: Get Help From Your Math Teacher (If Needed)

Your geometry teacher wants you to pass Geometry Regents (assuming that's the math test you're choosing to take for your high school diploma), so don't be afraid to ask them for help with any concepts or areas you're having problems with.

You could also talk to your teacher about what a full-credit answer should look like for the trickier constructed-response questions.

#4: Don't Spend Too Much Time on Part I

Although Part I has more questions than any other part on Geometry Regents, it's critical that you save your time for the constructed-response questions in Parts II, III, and IV, since these are worth more credits and require you to show your work.

You'll have three hours in total for the exam, but you don't want to waste the majority of that time on the first (and arguably easiest) part, so try to spend no more than an hour on Part I. This will give you a full two hours for Parts II, III, and IV (and ideally some leftover time you can use to check your work at the end).

An hour on Part I comes out to about two and a half minutes per question, making this a solid pace to aim for.

#5: Answer Every Question

There's no guessing penalty on Geometry Regents, so you should put down an answer for every question on the exam, even if you have no idea how to solve it.

For multiple-choice questions, use the process of elimination first to see if you can get rid of one or two obvious wrong answers. This way you'll raise your chances of guessing the correct answer from 25% to 33% or even 50%!

You might also want to come up with a guessing number (1-4), which you will choose whenever you don't know which answer choice to pick. For example, if your guessing number is 2, then you'd always choose answer choice 2 for any multiple-choice questions you didn't know how to solve.

For the constructed-response questions, try to solve as much of the problem as you can. Even if all you know is the first step in a problem, write it down—Parts III and IV can give you partial credit for showing your work, even if it's incomplete, so it's always worth putting down something!

Final Words: The Importance of Geometry Regents Review

The Geometry Regents test is one of three math Regents tests that students at public high schools in New York can choose from as part of their graduation requirements.

The test consists of 35 questions spread across four parts, including a multiple-choice section and three constructed-response sections, which are worth anywhere from 2 to 6 credits per question. A passing score is 65, equal to about 29-30 credits.

Geometry Regents covers a wide range of geometry topics, from polygons, circles, and triangles to angles, reflections, and 3-D shapes.

The easiest way to make sure you pass this exam is to take real Geometry Regents tests from past administrations, review notes and materials from your geometry class, and ask your geometry or math teacher for guidance.

On test day, be sure to answer every question, use the process of elimination as needed for trickier questions, and save most of your time for the hardest parts of the exam: the constructed-response sections.

We hope that this Geometry Regents review guide was helpful. Good luck!

What's Next?

What exactly are the New York State Regents Examinations? Learn all about these graduation requirements and what topics you'll have to take Regents in.

Considering taking a different math Regents exam to fulfill your graduation requirements? Then check out our comprehensive review guides to Algebra 1 Regents and Algebra 2 Regents.

In addition to a math Regents exam, you'll have to take a science Regents exam. Learn about what to expect on Chemistry Regents, Earth Science Regents, and Living Environment Regents.