Quadrilaterals & Polygons

Everything for this chapter, in one place. $a$ is a side, $h$ a height, $d_1$ and $d_2$ the diagonals.

Rectangle and square


Rectangle, area	$lb$
Rectangle, perimeter	$2(l+b)$
Rectangle, diagonal	$\sqrt{l^2+b^2}$ — the square root is not optional
Square, area	$a^2$
Square, diagonal	$a\sqrt{2}$
Rectangle in a circle	the diagonal is the diameter, so $l^2+b^2 = 4R^2$

Parallelogram


Area	$\text{base} \times h = ab\sin\theta$
Height versus side	$h \le$ the slant side, always — this is a hidden inequality
Diagonals	they bisect each other
Diagonal identity	$d_1^2 + d_2^2 = 2(a^2+b^2)$

Rhombus — four equal sides


Diagonals	perpendicular bisectors of each other
Area	$\dfrac{1}{2}d_1d_2$
Side and diagonals	$d_1^2 + d_2^2 = 4a^2$
Area, from the angle	$a^2\sin\theta$
Inradius	$r = \dfrac{\text{area}}{2a}$

Trapezium


Area	$\dfrac{1}{2}(\text{sum of the parallel sides}) \times h$
A leg perpendicular to both parallels	that leg is the height
Legs extended to a point	the two triangles are similar, in the ratio of the parallel sides

Circles in and around


A circle fits inside a quadrilateral exactly when	$AB + CD = BC + AD$
Then	$\text{area} = r \times \text{semiperimeter}$
A circle fits around it exactly when	opposite angles add to $180^\circ$

Any regular polygon, $n$ sides

Regular hexagon, side $a$ — six equilateral triangles


Area	$\dfrac{3\sqrt{3}}{2}a^2$
Short diagonal, spanning two sides	$a\sqrt{3}$
Long diagonal, straight across	$2a$
Distance from centre to a vertex	$a$

Regular octagon, side $a$


Area	$2(1+\sqrt{2})a^2$
Diagonal spanning two sides	$a\sqrt{2+\sqrt{2}}$

Limits


Each side of a quadrilateral	is less than the sum of the other three
Fixed perimeter	the square has the largest area
Fixed area	the square has the smallest perimeter

Formula sheet