⬛ Rectangle, Square & Quadrilaterals

• Rectangle (length $L$, breadth $B$): area $= L\times B$, perimeter $= 2(L+B)$, and the diagonal joins opposite corners so by Pythagoras it is $\sqrt{L^2+B^2}$.

• Square (side $a$): area $= a^2$, perimeter $= 4a$, diagonal $= a\sqrt{2}$. Turn that around and the area straight from a diagonal $d$ is $\dfrac{1}{2}d^2$ — a huge time-saver.

• Rhombus (diagonals $d_1$, $d_2$): area $= \dfrac{1}{2}\,d_1 d_2$. Because the diagonals cross at a right angle and cut each other in half, a side is $\sqrt{(d_1/2)^2+(d_2/2)^2}$.

• Trapezium (parallel sides $a$ and $b$, height $h$): area $= \dfrac{1}{2}(a+b)h$ — average the two parallel sides, then multiply by the height.

• Path / border: the area of a border is always outer area minus inner area. For a path of width $w$ outside an $L\times B$ rectangle, every side grows by $2w$, so the path $= (L+2w)(B+2w)-LB$; for a path inside, every side shrinks by $2w$.

One trap shows up constantly: “a side changes by $x\%$ — by how much does the area change?” Area is a product of two lengths, so the change is not double. For a side that changes by $a\%$ and another by $b\%$, the area changes by $\left(a+b+\dfrac{ab}{100}\right)\%$. For a square both equal $x$, so the area changes by $\left(2x+\dfrac{x^2}{100}\right)\%$. Remember: ratio of areas $= (\text{ratio of sides})^2$.