Quadrilaterals and polygons

This chapter applies to all AMC 8/10/12 test takers.

This chapter focuses on shapes with four or more sides.

• A quadrilateral is a polygon with exactly four sides.
• A polygon can have any number of sides (including triangles).

Since triangles are covered in a separate chapter, the focus here is on quadrilaterals and other polygons.

You’ll want to be comfortable with:

• How interior and exterior angles relate
• How to find area, perimeter, and diagonals for common shapes

Most of this chapter is about definitions and classification. A more detailed discussion of area formulas appears in the area and perimeter chapter. Near the end, we’ll also point out a few AMC-style strategies that come up often with these shapes.

Quadrilaterals

Every quadrilateral has four sides. The diagram below shows how common quadrilaterals are categorized. If one shape appears inside another category, it satisfies the larger category’s definition too.

For example, a square is both a rectangle and a rhombus. That also makes it a parallelogram, and (of course) a quadrilateral.

We’ll define each shape below, starting with parallelograms.

A parallelogram is a quadrilateral with both pairs of opposite sides parallel. This gives you several useful facts:

• Opposite angles are equal.
• Adjacent angles are supplementary (they add to $180$).
• Since all quadrilaterals have interior angles summing to $360$, knowing one angle lets you find the other three.
• The diagonals intersect at the center of the shape (they bisect each other).
• The area is base times height.

A rhombus is a parallelogram with all sides the same length. In addition to the parallelogram properties:

• Its diagonals intersect at the center.
• Its diagonals are perpendicular.

A rectangle is a parallelogram with all angles equal. That forces every angle to be $90$ degrees.

A square is both a rectangle and a rhombus: all sides are equal and all angles are $90$ degrees. The diagonal of a square is always the side length times $sqrt2$.

A trapezoid has exactly one pair of parallel sides. To find its area, multiply the height by the average of the two parallel bases.

An isosceles trapezoid is a trapezoid where the two non-parallel (tilted) sides have the same length. The same area formula applies, but the symmetry often makes angle and length relationships easier to use. This shape has two pairs of equal angles, and the equal angles are adjacent (not opposite).

The most flexible of these common categories is the kite. A kite has two pairs of adjacent equal sides, but it doesn’t need any parallel sides. Its diagonals are perpendicular. You can find the area by multiplying the diagonal lengths and dividing by two.

There is also the irregular quadrilateral, which is any quadrilateral that doesn’t fit the categories above. These usually don’t have equal sides or equal angles. You can often find the area by splitting the shape into simpler pieces, or by using Brahmagupta’s formula (found in this chapter) if it is a cyclic quadrilateral.

The problem below involves an irregular quadrilateral. One way to approach it is to sketch possibilities from the answer choices and use the idea that a side can’t be so long that the remaining sides can’t “reach” to close the quadrilateral.

Example: The question below is from 2023 AMC 10A

A quadrilateral has all integer sides lengths, a perimeter of $26$, and one side of length $4$. What is the greatest possible length of one side of this quadrilateral?
A. $9$
B. $10$
C. $11$
D. $12$
E. $13$

Polygons

This section focuses on regular polygons with five or more sides.

A regular polygon has:

• All sides the same length
• All interior angles the same measure

For example, a seven-sided regular polygon has seven equal sides and seven equal angles. As the number of sides increases, a regular polygon looks more and more like a circle, and its interior angles get larger.

Sides	Name	Interior angle	Figure
5	Pentagon	108°
6	Hexagon	120°
8	Octagon	135°
10	Decagon	144°

Example: The question below is from 2014 AMC 10B

Six regular hexagons surround a regular hexagon of side length $1$ as shown. What is the area of $△ABC$?

A. $2\sqrt{3}$
B. $3\sqrt{3}$
C. $1+3\sqrt{2}$
D. $2+2\sqrt{3}$
E. $3+2\sqrt{3}$

Sum of interior angles for an n-sided shape

The sum of the interior angles of any polygon is $(n-2)\times 180$, where $n$ is the number of sides.

For instance, a pentagon ($5$ sides) has interior angles with a sum of $(n-2)\times 180=540$.

This is especially useful for irregular polygons. One quick observation is that the sum of interior angles must always be a multiple of $180$.

Example: The question below is from 2017 AMC 12A

Claire adds the degree measures of the interior angles of a convex polygon and arrives at a sum of $2017$. She then discovers that she forgot to include one angle. What is the degree measure of the forgotten angle?
A. $37$
B. $63$
C. $117$
D. $143$
E. $163$

Common themes

• All regular shapes are symmetrical along multiple diagonals.
• The average interior angle increases as the number of sides increases.
• Always draw out diagrams to help you reason through a question.
• A regular hexagon is made up of six equilateral triangles.
• You can deconstruct parallelograms into four triangles that meet in the center by drawing both diagonals.
• Projecting a diagram onto a coordinate plane may be helpful with certain problems.
• The number of diagonals in a polygon is found by taking the number of sides and plugging it into this formula: $\frac{n(n-3)}{2}$.