Types of Polygon
Regular A polygon with all sides and interior angles the same. Regular polygons are always convex.
Convex All interior angles less than 180°, and all vertices 'point outwards' away from the interior. The opposite of concave. Regular polygons are always convex.

Definitions, Properties and Tips
• Sum of Interior Angles \(180(n-2)\) where \(n\) is the number of sides
• For a regular polygon, the total described above is spread evenly among all the interior angles, since they all have the same values. So for example the interior angles of a pentagon always add up to 540°, so in a regular pentagon (5 sides), each one is one fifth of that, or 108°. Or, as a formula, each interior angle of a regular polygon is given by: \(\frac{180(n-2)}{n}\), where \(n\) is the number of sides.

• The apothem of a polygon is a line from the center to the midpoint of a side. This is also the inradius - the radius of the encircle.



• The radius of a regular polygon is a line from the center to any vertex. It is also the radius of the circumcircle of the polygon.



GRE is dealing mainly with the following polygons:

Quadrilateral A polygon with four 'sides' or edges and four vertices or corners.



Types of quadrilateral:
Square All sides equal, all angles 90°. See Definition of a square.
Rectangle Opposite sides equal, all angles 90°. See Definition of a rectangle.
Parallelogram Opposite sides parallel. See Definition of a parallelogram.
Trapezoid Two sides parallel. See Definition of a trapezoid.
Rhombus Opposite sides parallel and equal. See Definition of a rhombus.


Parallelogram A quadrilateral with two pairs of parallel sides.



Properties and Tips
• Opposite sides of a parallelogram are equal in length.
• Opposite angles of a parallelogram are equal in measure.
• Opposite sides of a parallelogram will never intersect.
• The diagonals of a parallelogram bisect each other.
• Consecutive angles are supplementary, add to 180°.
• The area, \(A\), of a parallelogram is \(A = bh\), where \(b\) is the base of the parallelogram and \(h\) is its height.
• The area of a parallelogram is twice the area of a triangle created by one of its diagonals.

A parallelogram is a quadrilateral with opposite sides parallel and congruent. It is the "parent" of some other quadrilaterals, which are obtained by adding restrictions of various kinds:
• A rectangle is a parallelogram but with all angles fixed at 90°
• A rhombus is a parallelogram but with all sides equal in length
• A square is a parallelogram but with all sides equal in length and all angles fixed at 90°


Rectangle A 4-sided polygon where all interior angles are 90°



Properties and Tips
• Opposite sides are parallel and congruent
• The diagonals bisect each other
• The diagonals are congruent
• A square is a special case of a rectangle where all four sides are the same length.
• It is also a special case of a parallelogram but with extra limitation that the angles are fixed at 90°.
• The two diagonals are congruent (same length).
• Each diagonal bisects the other. In other words, the point where the diagonals intersect (cross), divides each diagonal into two equal parts.
• Each diagonal divides the rectangle into two congruent right triangles. Because the triangles are congruent, they have the same area, and each triangle has half the area of the rectangle.
• \(Diagonal=\sqrt{w^2+h^2}\) where: \(w\) is the width of the rectangle, h is the height of the rectangle.
• The area of a rectangle is given by the formula \(Width*Height\).

A rectangle can be thought about in other ways:
• A square is a special case of a rectangle where all four sides are the same length. Adjust the rectangle above to create a square.
• It is also a special case of a parallelogram but with extra limitation that the angles are fixed at 90°.


Squares A 4-sided regular polygon with all sides equal and all internal angles 90°



Properties and Tips
• If the diagonals of a rhombus are equal, then that rhombus must be a square. The diagonals of a square are (about 1.414) times the length of a side of the square.
• A square can also be defined as a rectangle with all sides equal, or a rhombus with all angles equal, or a parallelogram with equal diagonals that bisect the angles.
• If a figure is both a rectangle (right angles) and a rhombus (equal edge lengths), then it is a square. (Rectangle (four equal angles) + Rhombus (four equal sides) = Square)
• If a circle is circumscribed around a square, the area of the circle is \(\frac{\pi}{2}\) (about 1.57) times the area of the square.
• If a circle is inscribed in the square, the area of the circle is \(\frac{\pi}{4}\) (about 0.79) times the area of the square.
• A square has a larger area than any other quadrilateral with the same perimeter.
• Like most quadrilaterals, the area is the length of one side times the perpendicular height. So in a square this is simply: \(area=s^2\), where \(s\) is the length of one side.
• The "diagonals" method. If you know the lengths of the diagonals, the area is half the product of the diagonals. Since both diagonals are congruent (same length), this simplifies to: \(area=\frac{d^2}{2}\), where \(d\) is the length of either diagonal
• Each diagonal of a square is the perpendicular bisector of the other. That is, each cuts the other into two equal parts, and they cross and right angles (90°).
• The length of each diagonal is \(s\sqrt{2}\) where \(s\) is the length of any one side.

A square is both a rhombus (equal sides) and a rectangle (equal angles) and therefore has all the properties of both these shapes, namely:
The diagonals of a square bisect each other.
• The diagonals of a square bisect its angles.
• The diagonals of a square are perpendicular.
• Opposite sides of a square are both parallel and equal.
• All four angles of a square are equal. (Each is 360/4 = 90 degrees, so every angle of a square is a right angle.)
• The diagonals of a square are equal.

A square can be thought of as a special case of other quadrilaterals, for example
• a rectangle but with adjacent sides equal
• a parallelogram but with adjacent sides equal and the angles all 90°
• a rhombus but with angles all 90°


Rhombus A quadrilateral with all four sides equal in length.



Properties and Tips
• A rhombus is actually just a special type of parallelogram. Recall that in a parallelogram each pair of opposite sides are equal in length. With a rhombus, all four sides are the same length. It therefore has all the properties of a parallelogram.
• The diagonals of a rhombus always bisect each other at 90°.
• There are several ways to find the area of a rhombus. The most common is: \(base*altitude\).
• The "diagonals" method. Another simple formula for the area of a rhombus when you know the lengths of the diagonals. The area is half the product of the diagonals. As a formula: \(\frac{d1*d2}{2}\), where \(d1\) is the length of a diagonal \(d2\) is the length of the other diagonal.


Trapezoid A quadrilateral which has at least one pair of parallel sides.



Properties and Tips
• Base - One of the parallel sides. Every trapezoid has two bases.
• Leg - The non-parallel sides are legs. Every trapezoid has two legs.
• Altitude - The altitude of a trapezoid is the perpendicular distance from one base to the other. (One base may need to be extended).
• If both legs are the same length, this is called an isosceles trapezoid, and both base angles are the same.
• If the legs are parallel, it now has two pairs of parallel sides, and is a parallelogram.
• Median - The median of a trapezoid is a line joining the midpoints of the two legs.
• The median line is always parallel to the bases.
• The length of the median is the average length of the bases, or using the formula: \(\frac{AB+DC}{2}\)
• The median line is halfway between the bases.
• The median divides the trapezoid into two smaller trapezoids each with half the altitude of the original.
• Area - The usual way to calculate the area is the average base length times altitude. The area of a trapezoid is given by the formula \(h*\frac{a+b}{2}\) where
\(a\), \(b\) are the lengths of the two bases \(h\) is the altitude of the trapezoid
• The area of a trapezoid is the \(altitude*median\)