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Textbook
1. Welcome
2. Vocabulary approach
3. Quantitative reasoning
3.1 Quant intro
3.2 Arithmetic & algebra
3.3 Statistics and data interpretation
3.4 Geometry
3.4.1 Angles
3.4.2 Triangle basics
3.4.3 Sum of interior angles
3.4.4 Pythagorean theorem
3.4.5 Right triangles (45-45-90)
3.4.6 Right triangles (30-60-90)
3.4.7 Triangle inequality theorem
3.4.8 Coordinate plane
3.4.9 Equation for a line
3.4.10 Graphing inequalities
3.4.11 Graphing parabolas
3.4.12 Graphing circles
3.4.13 Parallel and perpendicular lines
3.4.14 Quadrilaterals
3.4.15 Circles
3.4.16 3D shapes
3.4.17 Polygons
3.4.18 Regular polygons
3.4.19 Shaded region problems
3.5 Strategies
4. Verbal reasoning
5. Analytical writing
6. Wrapping up
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3.4.18 Regular polygons
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3. Quantitative reasoning
3.4. Geometry

Regular polygons

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Regular polygons have special characteristics that make them easier to work with.

Definitions
Regular polygon
Any shape with equal angles and side lengths

You already know some of these. For example, a regular triangle is an equilateral triangle, and a regular quadrilateral is a square. When reading geometry questions, always be on the lookout for the word regular, since this piece of information will likely make the question much easier to solve.

Here are some of the most common regular polygons:

Sides Name Interior angle Figure
3 Triangle 60° Regular triangle with 3 sides, equilateral triangle
4 Square 90° Regular quadrilateral with 4 sides, square
5 Pentagon 108° Regular pentagon with 5 sides
6 Hexagon 120° Regular hexagon with 6 sides
8 Octagon 135° Regular octagon with 8 sides
10 Decagon 144° Regular decagon with 10 sides

All regular polygons can be perfectly circumscribed, i.e., they can have a perfect circle drawn around them with all corners precisely on the circumference. Some irregular polygons can also be circumscribed, but not always.

Regular quadrilateral a.k.a. square circumscribed by a circle The figure directly above shows a circumscribed regular quadrilateral (a square).

Irregular quadrilateral not circumscribed by a circle The figure directly above shows an irregular quadrilateral that is not circumscribed.

Let’s see how these shapes might show up in an example question.

The triangle and quadrilateral shown below are regular polygons. The area of the quadrilateral is twice as large as the area of the triangle.

Quantity A: b
Quantity B: z

Regular triangle and quadrilateral with highlighted areas

Ready for the answer?

(spoiler)

Answer: Quantity A is greater

First, note that the question stated these are regular polygons. This means each shape has sides with equal lengths.

a=b=cw=x=y=z​

Remember the equations for the area of a polygon?

  • A△​=bh/2
  • A□​=s2

The question states that the area of our square is twice as large as the area of the triangle, and we can write this as an equation.

A□​s2s2​=2∗A△​=2∗bh/2=bh​

The question asks us to compare b with z. This gets a little easier when we understand what we’re really being asked:

  • Quantity A: b (any side of the triangle)
  • Quantity B: z (and side of the square)

Because this is a regular triangle, i.e., an equilateral triangle, the base of the triangle is greater than the height. Since the side lengths are all equal, it follows that the length of side b is also greater than the height.

Regular triangle with ascender

We’re almost done. Let’s revisit that equation we set up earlier: s2=bh

With our knowledge that b>h, we can infer that b>s and h<s, and therefore, Quantity A is greater.

GRE figures are not drawn to scale, but now we have enough information to make a sketch of how this would look if drawn to scale:

Regular triangle with half the area of a square

For the triangle to have half the area of the square, the base needs to be a little big bigger than the height. It checks out!

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