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Many geometry problems either directly ask about the degree of certain angles or will require you to figure out the degree of an angle in the course of solving some problem. In successive posts, we will cover the basics of what an angle is, how we can use parallel lines to find unknown angles, and how we can use triangles/circles/polygons to find unknown angles.

Angles: Some Preliminaries

To understand angles, we need to begin with the idea of a line:

Line: a straight line that continues in both directions without end.

Now, an angle appears where two lines intersect. More formally,

Angle: a measure of how much you would need to turn one line to make them part of the same line

For example,

And as you can see in the above diagram, there are actually four angles formed when two lines intersect. So there is a convention for naming each of the four angles:

[Practice problems with matching angles: label points on the lines and degree measures and have them match the two]

Now, here are two important facts about angles:

Lines Have 180 Degrees: The angle formed by a single line has 180 degrees.

Circles Have 360 Degrees: The angle formed by a circle is 360 degrees.

Now, we also have two relationships among lines which are very important:

Parallel Lines: Parallel lines go in the same direction and therefore never intersect. We often denote that lines and are parallel by writing , as in the below diagram:

Perpendicular Lines: Perpendicular lines intersect at a 90 degree angle. We denote two lines as perpendicular by writing :

In our next post, we will talk about some of the relationships that exist between angles and lines and shapes.

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Below, we include some additional geometry practice problems.

1. What is the area of ? Of ?

2. In the following diagram, What is the area of ?

3. What is the area of ?

4. The radius of the below circle is 3. Find the area of the shaded region.

5. The circle below has a radius of . Find the ratio of the shaded region's area to the area of the square below.

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The next 3D shape we shall look at is the circular cylinder (also called the “right circular cylinder” by the GRE for its right angles):

Circular Cylinder: A 3D shape that has a circle for both bases and a perpendicular line connecting the center of those bases.

Now, it follows from the definition that a circular cylinder’s top and bottom circles will be the same size. So we use to denote the radius of either circle, and we use to denote the height of the cylinder. Without further ado, we present:

Volume of a Circular Cylinder: The volume of a right circular cylinder with a radius of and a height of is

Surface Area of a Circular Cylinder: The surface area of a circular cylinder with a radius of and a height of is

Practice Problems:

1. The cylinder below has a radius of and a height of 6. Find .

2. The surface area of a right circular cylinder is and its radius is 2. What is the height of the cylinder?

3. The surface area of a cylinder is twice its volume. Its radius equals its height. What is the volume of the cylinder?

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One important kind of 3D shape is the rectangular solid:

which is defined as follows:

Rectangular Solid: A 3D shape with six faces which are all rectangles placed perpendicularly to one another.

The fact that the faces are all perpendicular to each other helps a great deal in calculating the volume, for we simply have:

Volume of a Rectangular Solid: For the following rectangular solid,

its volume is equal to

Example 1
Find the area of a rectangular solid whose dimensions are .

Example 2
A rectangular solid has two sides that are 3 and 8 units long. It has a volume of 48 units. What is the length of its remaining side?

And, to find the surface area of any rectangular solid, we simply need to add up the area of each of the 6 rectangles which surround it.

Surface Area of a Rectangular Solid: For the following rectangular solid,

its surface area is equal to

How did we get this formula? Well, since each of the faces are perpendicular to each other, we know that there are two rectangles with sides of and units long, one in front and one at the back:

and there are two with sides of units long:

and there are two with sides of units long:

Thus, adding up the area of all of those rectangles, we get: which is the same as what we wrote above.

Now, one common 3D shape is the cube, and the cube is just a special kind of rectangular solid:

Cube: a cube is a rectangular solid whose six sides are all squares.

Our formulas become especially simple for a cube. Let be the length of any side of the cube:

Volume of a Cube: The volume of a cube is .

Surface Area of a Cube: The surface area of a cube is .

These formulas follow directly from just applying our above formulas for the volume/surface area of any rectangular solid.

Finally, it is worth noting how we can use the Pythagorean Theorem to find certain distances on a rectangular solid. Consider the following problem:

Example 3
For a rectangular solid with a length of , a width of and a height of , find the length of :

Practice Problems

1. Find the length of

2. Find the length of .

3. Find the area of the shaded plane below:

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Finally, we turn to the kinds of shapes we see in our daily lives: 3D shapes. Now, by contrast to 2D shapes, 3D shapes appear on relatively few GRE problems, so it may help your scores more to really master the different aspects of 2D shapes.

There are two main 3D shapes that the GRE tests on: the "rectangular solid" (also known as the rectangular prism) and the cylinder (also called the "right circular cylinder").

And there are two main properties of such objects that we will care about:

1. Surface Area: this is the total area of all the surfaces of a 3D shape
So, in a rectangular solid, we add up the area of each of the 6 sides. And, in a cylinder, we add up the area of the circle on the top, the circle on the bottom, and the curved portion in between the two.
2. Volume: the space enclosed by a 3D shape
Imagine filling up the 3D shape with water. The amount of space that the water takes up, that’s the volume of the object.
By analogy, you could think of the area of a 2D shape as capturing how much space the lines enclosed. Similarly, volume is a measure of how much space the surfaces of a 3D shape enclose.

In subsequent posts, we will talk about the rectangular solid and cylinder in greater detail and show how to calculate the surface area and volume for each.

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The perimeter is just the distance it would take if you were to walk the outside edges of a given shape. So, for the following triangle:

the perimeter is just

Finding the perimeter of a square or a rectangle is similarly straightforward:

Perimeter of a Square or Rectangle: In the following diagrams, the perimeter of the square is and the perimeter of the rectangle is

Perimeter of a Rhombus: Since a rhombus is defined by having four sides of equal length, the perimeter of the below rhombus is just

Perimeter of a Trapezoid:

The perimeter of a trapezoid is a little trickier, but recall that we can break a trapezoid up into two right triangles and a rectangle:

And then, if we have the bases of the right triangles, we can use the Pythagorean Theorem to find the length of the diagonal bits:

Thus, we get that the perimeter is:

Perimeter of a Parallelogram
Remember that the parallel sides of a parallelogram have the same length. Thus, for the below parallelogram:

the perimeter is just .

include some simpler problems here for finding the perimeter of when they give you basically all the right information

Sometimes, the perimeter figures in a word problem. So, for example, you might have:

Example 1

A farmer has a square field whose perimeter is twice its area. What is the area of the field?

Practice Problems

1. The outer rectangle below is units apart from the smaller rectangle on the top and bottom sides, and units apart on the left and right sides. The outer rectangle has a base of 8 units and a height of 12 units. What is the perimeter of the inner rectangle?

2. The perimeter of a regular -sided shape is Find the length of a side.

3. The length of a side of a regular hexagon is What is the perimeter?

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Many geometry problems ask you to find the area of a shape or to compare the area of two shapes or to use the area of a shape to deduce some other value. In previous posts, we have already given the standard formulas for finding the area of such shapes (see our appendix for a full list of such formulas). But the GRE will often give you the relevant information in subtle ways, or ask you to do surprising things with the information they’ve given. So in what follows, we give some practice problems intended to help you become used to using these standard formulas in less standard ways.

But first, here are some general steps to follow in trying to solve a geometry problem:

1. Record what the problem tells you.

- The problem may give you certain lengths or angle measurements or areas or perimeters; keep track of this. It is best to write it down somewhere, either on a diagram or just in a list.

2. If you’re stuck, write down some formulas you think may be relevant.

- Thus, if the problem asks you to find the area of some shape, write down the area formula for that shape! Or if the problem gives you a 45-45-90 right triangle, write down the ratio of the side lengths. Sometimes writing such stuff down can help spark connections in your mind.

3. If you’re stuck, look back carefully at the problem and see if there’s any information you missed.

- Generally, these problems are quite parsimonious: they give you exactly what is needed to solve the problem, and no more. So if you find yourself with some unused fact, try and fit it into the problem somewhere. It’s unlikely they would’ve included something totally irrelevant to your problem.

4. If you don’t think you can get the solution quickly, just move on!

- Remember that the math section gives you 35 minutes for 20 questions, so you can only afford to spend an average of one minute and 45 seconds per question. There’s no shame in flagging a question that looks tough so that you can come back to it if time permits. Look at it this way: if you don’t get through the test, there may be easy questions down the road that you’re effectively giving up on. By flagging it and coming back later, you help ensure that you get all the low-hanging fruit in the test.

Practice Problems

1. In the following diagram, and are both squares with side lengths of 2 and area of area of area of Find the area of the shaded rhombus.

2. In the following diagram, , and are equally spaced. How many scalene triangles can be formed with vertices at those points?

3. The following points are equally spaced. How many equilateral triangles can be formed? If three distinct points are randomly chosen, what is the likelihood that they form an equilateral triangle?

4. The outer rectangle below is units apart from the smaller rectangle on the top and bottom sides, and unit apart on the left and right sides. The inner rectangle has a height of 4 units and a base of 2 units. What is the area of the outer rectangle?

5. The radius of both circles below is 1. What is the area of triangle ?

6. What is the area of ?

7. In the following diagram, What is the area of ?

Challenge Problem: You are not likely to see anything as convoluted as the following on the GRE. Still, it might be a good way to push your understanding of the above material:

8. The radius of the following circle is 5. Find the shaded area.

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Quadrilaterals are just the four-sided shapes (“quad” meaning four); they include:

Rectangles: four sides connected by four 90-degree angles

Rectangle Area Formula:

Squares: four sides of equal length connected by four 90-degree angles.

Square Area Formula:

Parallelograms: four-sided figure where opposite sides are parallel

Parallelogram Area Formula:

Parallelograms also have the nice property that opposite angles are of the same degree, and opposite sides are the same length.

Rhombus: a parallelogram with equal sides

Rhombus Area Formula:

Trapezoids: four-sided figure with one pair of parallel opposing sides

Trapezoid Area Formula:

Now a lot of these definitions overlap with others, so here is a diagram that explains how all of these shapes relate to one another:

Finally, one fact that is sometimes helpful in solving problems with quadrilaterals is:

Quadrilaterals Have 360 Degrees: The sum of the angles in a quadrilateral is 360.

This fact will come up again when we talk about finding the values of unknown angles.

Practice Problems:

1. Find the area of the entire figure below.

2. Find the area of the entire figure below.

3. The area of the below parallelogram is 30, is perpendicular to , and the ratio of the area of to the area of is 1 to 3. What is the value of ?

4. The dotted lines below meet at a right angle. Find

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Not all triangles are equally loved. Right triangles, for example, were adored by early mathematicians, and even now, are cherished by standardized-test makers. These are the triangles that have a ninety-degree angle, like so

It is tradition to assign to the shortest side, to the second shortest, and to the longest side. We also call the side opposite from the right angle (in the above diagram, ) the hypotenuse of the right triangle, and we call the other two sides the legs of the right triangle.

These figures are so well-beloved partly because they have many interesting properties. The most famous property is:

Pythagorean Theorem: If and are the legs of a right triangle and is the hypotenuse of that right triangle, then

And even among the right triangles, there are two kinds that are especially cherished:

30-60-90 Right Triangle: Any right triangle with angles of 30, 60, and 90 degrees will have the following side lengths (where is some fixed number):

45-45-90 Right Triangle: Any right triangle with angles of 45, 45 and 90 degrees will have the following side lengths (where is some fixed number):

Knowing the ratios of the side lengths of such triangles can help (and may sometimes be crucial) to solving a geometry problem on the GRE. For example:

Example 1
Find the area of the following right triangle:

Example 2

In the following diagram, the area of triangle A is 18. Find . Also, find the area of triangle B.

Now, for the 30-60-90 right triangle, one just has to memorize the side ratios. But you can easily derive the side ratios of the 45-45-90 right triangle:

Finally, some problems may give you the side lengths of a triangle and you will have to infer that the triangle in question is actually a right triangle. In other words, we have:

Reverse Pythagorean Theorem: If a triangle has side lengths such that , then the triangle is a right triangle.

In applying the Reverse Pythagorean Theorem, here are some common side-ratios to look out for:

3-4-5

5-12-13

So if you see a triangle like:

You should note that its side lengths have the ratio 3:4:5 (simply divide all the side lengths by 3) and thus it is a right triangle.

Practice Problems

1. Find the value of

2. Find the value of

3. Find the length of AC.

4. In the following diagram, What is the ratio of the area of to the area of ?

5. What is the length of ?

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The name “triangle” means, sensibly enough, “three angles” and indeed every triangle has three angles and three sides:

This much is hopefully familiar. But just as with circles, there are many special terms people have devised for triangles. When it comes to the sides of a triangle, we can have:

Equilateral Triangles: All sides have the same length.

Isosceles Triangles: Two sides have the same length as each other; the third has a different length.

Note that in an isosceles triangle:

Equal Sides Have Equal Angles: In an isosceles triangle, the angles opposite from the sides of equal length must have equal degree.

Thus, in the below diagram:

angles and are equal.

Scalene Triangles: All of the sides are of different lengths.

We can also classify triangles according to their angles:

Acute Triangles: All angles are less than 90 degrees.

Right Triangles: One angle is exactly 90 degrees.

We use a square to mark a 90 degree angle.

Right triangles have lots of special further features which we will talk about here.

Obtuse Triangles: One angle is more than 90 degrees.

Now, we can also compare two triangles to each other. We can say that they are:

Similar: Two triangles are similar if their angles have the same values.

So in the above diagram, even though one of the triangles is obviously bigger than the other, we can say that they are “similar” (in the technical, mathematical sense defined above) because their angles have the same values. And the definition makes sense since, after all, those two triangles do look pretty similar (in our ordinary, day-to-day sense)!

These triangles are also similar to one another; it doesn’t matter if you rotate or flip the second triangle. All that matters is whether the angle values for the two figures are the same.

The reason why we care about the similarity of triangles is:

Similar Triangles Share Proportions: If two triangles are similar, then there is some constant ratio you can multiply the sides of one triangle by in order to get the sides of the other triangle.

For example, if the following two triangles are similar:

then there is some constant, call it , such that:

This can be very helpful in trying to find certain side lengths or the area of a triangle as we shall see.

Congruent: Two triangles are congruent if their angles are the same and their sides are the same length.

You can think of congruence as signaling that those two triangles are effectively the same triangle. The lines are the same length, the angles are the same, the area is the same, and so on.

And again, it doesn’t matter if you rotate or flip one of the triangles. It is still true that their angles and side lengths are the same, since rotating or flipping a shape won’t change any of that.

Now, when it comes to finding the area of a triangle, you need to know the base of the triangle and its height. The base can be any side of the triangle, but the height needs to extend perpendicular (at a right angle to) the base, and reach the highest point of the triangle, away from the base, like so:

Traditionally, we use to denote the base and to denote the height. Thus, we get:

Triangle Area Formula:

Finally, we note two fundamental facts about triangles:

Angles of a Triangle Sum to 180: For any triangle, the sum of its interior angles is 180 degrees.

Triangle Inequality: The sum of the length of any two sides of a triangle is greater than the length of its third side.

Practice Problems:

1. Find the value of

2. Find the value of

3. A triangle has legs that are and units long. What integer values are possible for the third side?

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