# Other Shapes

### Rules

A polygon's angles add up to 180º plus an additional 180º for every side over 3.

The angles of a pentagon add up to 540º.

The angles of a enneakaidecagon add up to 3060º.

(An enneakaidecagon is a figure with 19 sides). Polygons other than triangles and quadrilaterals are rare on the GRE; when they do appear, they are typically shapes which employ a combination of triangles and quadrilaterals.

### Practice

1. In the figure to the left, angles a, d, h, and e are each equal to half of the measures of each of angles b, c, g, and f. What is the measure of angle a?

2. How many degrees do the angles of a 102-sided figure add up to?

3. The angles of a particular polygon add up to 2340º. How many sides does the polygon have?

### Answers to Practice Questions

1. 90º

The angles of an octagon add up to 1080º. Since a, d, h, and e are equal and are each half of the b, c, g, or f, then it can be inferred that 4(a) + 4(2a) = 1080º. This means that 12a = 1080º and that a is equal to 90º.

2. 18,000º

A polygon has 180º plus an additional 180º for every side over 3. Thus, the angles of a 102-sided figure equal (102-2)180º or 18,000º.

3. 15

A polygon has 180º plus an additional 180º for every side over 3. Thus, the question can be solved using algebra. Let n equal the number of sides of the polygon.

(n -2)180º = 2340º Take the initial equation.

180ºn - 360º = 2340º Expand the parentheses.

180ºn = 2700º Add 360º to both sides.

n = 15 Divide both sides by 180º.

The polygon has 15 sides.