The 4 angles of a quadrilateral add to 360

The 5 angles of a pentagon add to 540

The 6 angles of a hexagon add to 720

The easiest way to work this is arguably to examine the exterior angles, each of which forms a linear pair with an interior angle. If an interior angle measures , then each exterior angle, which is supplementary to an interior angle, measures

The measures of the exterior angles of a polygon, one per vertex, total ; in a regular polygon, they are congruent, so if there are  such angles, each measures . Since the number of vertices is equal to the number of sides, if we set this equal to  and solve for , we will find the number of sides.

Multiply both sides by :

The polygon has 72 vertices and, thus, 72 sides.

To determine the measure of the angles of a regular polygon use:

Thus, 

While lesser-used than most area formulas, the area of a trapezoid is fair game for the GMAT to test so you should be sure to know it. A trapezoid has two parallel sides, which for the purposes of calculating area will serve as your bases. Here those sides are AB and EC. The formula, then, is 

, whereas always the height must be perpendicular to the bases.

Here you're given the top base as , and need to derive the full bottom base. Since AB must equal ED, you can add ED, which is , to DC, which is , to find the second base as .

To find the height, recognize that triangle BCD is a right triangle with a hypotenuse of 5 and a shorter side of 3. This then means that you're looking at a  Pythagorean triplet, and line segment BD must then measure .

Now you can plug into the trapezoid area formula: 

Therefore  is the correct answer choice.

The area of a parallelogram is calculated as Base × Height, where the height must be perpendicular to the base (much like in a triangle).

Here you can use your knowledge of Pythagorean Theorem (and common Pythagorean triplets) to find the height. Since the right triangle on the left-hand side of the shape has a side of  and a hypotenuse of , it fits the 3-4-5 side ratio, meaning that the height must be 8. (or you could use Pythagorean Theorem and call the height , so , meaning that  so again the height is ).

You know that the area, Base × Height, is , so you can now solve for the base: , so .

You now need the perimeter, which is the sum of all four sides. That will be .

Therefore the correct answer is .

The perimeter of a rectangle can be calculated as  (where  = length and  = width). Here you're told that . This allows you to solve for the ratio of .

First, subtract  from both sides to get .

Next, divide both sides by  to get .

Finally, divide both sides by  to express the ratio as a fraction: , meaning that .

The correct answer is .

The area of any square can be calculated as . Here, you're given the area and need to work backward to find the length of a side. That means that , so the length of a side is .

What is the greatest distance between any two points on a square? It's the square's diagonal, which has a length of . That's a rule you should memorize, but of course, it derives from the hypotenuse of an isosceles right triangle.

So the calculation you're looking for is  times  (the diagonal ratio multiplied by the length of a side). That becomes .

The key to solving this problem is in dividing the given figure into two: a rectangle on the left and a right triangle on the right:

If you do so, you should recognize something familiar with the right triangle: the hypotenuse has a length of  and one side has a length of 30, meaning that this triangle will fit the  side ratio. You then know that the bottom side of the triangle must measure .

With that, you can fill in figures for the bottom of the quadrilateral. The bottom of the rectangle will measure , symmetrical to the top, and the bottom of the triangle will also measure , meaning that the entire bottom side of the quadrilateral measures . Therefore the perimeter is .

As you are given the perimeter of each square as , you can then divide  by  to determine that the side of each square equals .

From there, recognize how the dimensions of the squares fit to the dimensions of the rectangle. The rectangle's height is exactly the same as the diagonal of the square. You can calculate that diagonal using the rule for isosceles right triangles: the diagonal forms the hypotenuse of an isosceles right triangle, where the hypotenuse measures the same as one side times . So the diagonal equals .

That's the height of your rectangle, and the width is equal to  times that amount, since the width spans exactly three squares laid diagonal-to-diagonal. So you can calculate the area of the triangle as height times width:

As you are given that the total area is , and you know that that area is the sum of the areas of five identical squares, you can solve for the side length of a square using , where  represents the side length. That allows you to divide both sides by  to get . And then take the square root to get .

So you now know that the side length of a square must be 9, and the question becomes how many of those sides you add up. Recognize that the middle square is not at all part of the perimeter, and that each of the four outer squares shares one side with that middle square. So the perimeter will be composed of the other three sides of each of the outer four squares. That gives you  squares times  sides times the side length of :