Quadrilaterals & Polygons

A polygon is a closed, flat shape made of straight line segments. Each segment is a side, each corner where two sides meet is a vertex, and a segment that connects two non-adjacent vertices is a diagonal.

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• Convex: no vertex is “pushed in”; every diagonal stays inside the shape.
• Concave: at least one vertex is “pushed in” (it has a dent), and at least one diagonal falls outside.
• Regular: all sides congruent and all angles congruent.
• Irregular: not regular (sides or angles differ).

A regular hexagon.

From a single vertex you can draw $\mathbf{n-3}$ diagonals (you cannot connect a vertex to itself or to its two neighbors). Counting every diagonal of the whole polygon:

Every polygon can be split into triangles by drawing all diagonals from one vertex. A polygon with $n$ sides splits into $n-2$ triangles, and each triangle holds $180^\circ$. So:

A regular hexagon.

In a regular polygon all $n$ interior angles are equal, so each one is the total divided by $n$:

A regular hexagon.

At each vertex, extend one side; the angle between that extension and the next side is an exterior angle. No matter how many sides a convex polygon has,

For a regular polygon each exterior angle is the same:

An interior angle and its exterior angle form a linear pair, so they add to $180^\circ$.

A regular hexagon.

A parallelogram is a quadrilateral with both pairs of opposite sides parallel. From that one fact, four properties always follow:

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• Opposite sides are congruent.
• Opposite angles are congruent.
• Consecutive (next-door) angles are supplementary (add to $180^\circ$).
• The diagonals bisect each other (cut each other in half).

Height is perpendicular to the base.

Each special parallelogram keeps all parallelogram properties and adds more:

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• Rectangle: four right angles; diagonals are congruent.
• Rhombus: four congruent sides; diagonals are perpendicular and bisect the angles.
• Square: a rectangle and a rhombus --- four right angles and four equal sides; diagonals are congruent, perpendicular, and bisect the angles.

Height is perpendicular to the base.

A trapezoid has exactly one pair of parallel sides, called the bases. The other two sides are the legs. The midsegment connects the midpoints of the legs; it is parallel to the bases and its length is the average of the two bases:

An isosceles trapezoid has congruent legs; its base angles are congruent and its diagonals are congruent.

A trapezoid has two parallel bases.

Every square is a rectangle and a rhombus; every rectangle and every rhombus is a parallelogram; every parallelogram is a quadrilateral. As you climb down the tree, shapes gain more properties; as you climb up, they become more general.

A cyclic quadrilateral is one whose four vertices all lie on a single circle. Its defining property is beautiful and useful:

So if $ABCD$ is cyclic, then $m\angle A + m\angle C = 180^\circ$ and $m\angle B + m\angle D = 180^\circ$. (The reason: an inscribed angle is half its intercepted arc, and the two opposite angles together intercept the whole circle, or $360^\circ$; half of that is $180^\circ$.) Every rectangle and every isosceles trapezoid is cyclic; a general parallelogram is cyclic only when it is a rectangle.

A regular hexagon.

Take any quadrilateral and mark the midpoint of each of its four sides. Connect those midpoints in order. The result --- called the Varignon parallelogram --- is always a parallelogram, no matter how lopsided the original shape is.

The key idea is the triangle midsegment: in triangle $ABC$, the segment joining the midpoints of $AB$ and $BC$ is parallel to diagonal $AC$ and half its length. The same holds on the other side, so both pairs of opposite midpoint-sides are parallel --- the definition of a parallelogram.

The apothem $a$ of a regular polygon is the perpendicular distance from the center to the middle of any side. Slice the polygon into $n$ congruent triangles, each with base equal to a side $s$ and height equal to the apothem. Adding the triangle areas gives

where $P = n s$ is the perimeter. In words: half the apothem times the perimeter.

When you know the corner coordinates of a polygon, you can find its area directly --- no base or height needed. List the vertices in order (all the way around) as $(x_1,y_1), (x_2,y_2), \dots, (x_n,y_n)$, then

It is called the shoelace formula because the cross-multiplications lace diagonally like shoelaces. Go around once in a consistent direction and take the absolute value at the end.

Angle formulas run both directions: given a single interior or exterior angle of a regular polygon, you can solve for the number of sides $n$.

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• From an exterior angle $E$:   $n = \dfrac{360^\circ}{E}$.
• From an interior angle $I$: first get the exterior angle $E = 180^\circ - I$, then $n = \dfrac{360^\circ}{E}$; or solve $\dfrac{(n-2)180^\circ}{n} = I$ directly.

A valid answer must be a whole number $n \ge 3$ --- if the division is not exact, no regular polygon has that angle.

The family tree turns every classification claim into a logic question. Compare the definitions of the two shapes:

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• Always --- the first shape's definition forces the second's properties. Every rhombus is a parallelogram: having four equal sides guarantees both pairs of opposite sides are parallel.
• Sometimes --- the property can hold but is not required. A rhombus is sometimes a square: only when its angles are also right angles.
• Never --- the definitions contradict. A square is never a (strict) trapezoid: a square has two pairs of parallel sides, but the strict trapezoid definition demands exactly one.

The habit that never fails: state each definition, then ask whether the first must, may, or cannot satisfy the second.