GRE Exam: A Simple Guide To Solving Geometry Problems - Polygons

A polygon is any two-dimensional shape. Familiar examples of polygons are squares, triangles, and rectangles.

Here are some definitions that you will find useful in working with polygons:
- The Sum of the Angles: The sum of the angles of any polygon can be determined by using the following formula: sum of the angles = (n – 2)180, where n = the number of sides.
- The Perimeter of a Polygon: The perimeter of any polygon will be the sum of its side lengths.
- Quadrilaterals: A quadrilateral is any four-sided polygon. The sum of the angles of any quadrilateral is 360. The quadrilaterals most commonly tested on the GRE are parallelograms.
- Parallelograms: A parallelogram is a quadrilateral in which opposite sides are parallel.


A parallelogram has the following properties:
1. Opposite sides are equal.
2. Opposite angles are equal.
3. The area = base × height.
4. The diagonal of a parallelogram creates two equivalent triangles.
5. Adjacent angles add up to 180.


In the parallelogram above, a = 2b. What is c?
SOLUTION: From Property 4, you know that a + b = 180. Substitute 2b for a:

From Property 2, you know that a = c. Thus c = 120.

Rectangles and Squares
A rectangle is a special type of parallelogram in which all angles equal 90 degrees. The area of a rectangle is l × w. The perimeter of a rectangle is 2(l + w).

A square is a special type of rectangle in which all sides are equal. The area of a square = s2 (where s = the length of a side). The perimeter of a square is 4s (where s = the length of a side).


What is the area of a square that has a perimeter of 20?
SOLUTION: Let s = the length of a side of the square.

Thus

The area of the square is 52 = 25.

In a certain rectangle, the length is double the width. If the area of the rectangle is 72, what is the length of the rectangle?
SOLUTION: Let w = the width of the rectangle. The length therefore equals 2w. The area of the rectangle is lw = 2w × w = 2w2 = 72. Thus


Maximizing the Area of a Polygon
Some tougher GRE questions will give you the perimeter of a parallelogram or triangle and ask you for the maximum area. Given a fixed perimeter, the maximum area of any polygon will be reached when all sides are equal.

Maximum Area of a Rectangle
To maximize the area of a rectangle, you should make the side lengths equal. Notice that when you do so, you create a square!

What is the maximum area of a rectangle with a perimeter of 28?
SOLUTION: To maximize the area of the rectangle, make all the sides equal. Let x = the length of one side. he perimeter is thus

If x = 7, the area is 72 = 49.

Maximizing Area with Given Side Lengths
In the preceding examples, you used the perimeter of the shapes to determine the side lengths that would maximize the area. Sometimes you will be given the lengths of the two sides of a parallelogram or triangle and asked to determine the maximum area from this information.

In these situations, use the following rule: Given the lengths of two sides of a triangle or parallelogram, you can maximize the area by making the two sides perpendicular.

In the case of a parallelogram, this means creating a rectangle. In the case of a triangle, this means creating a right triangle.


SOLUTION: To maximize the area of the triangle, make it a right triangle. Assume that the legs are AB and BC. In this case, the area of the triangle is . Quantity B is greater.