Rectangles, Squares & Other Quadrilaterals

• Rectangle $l \times b$: area $lb$, perimeter $2(l + b)$, diagonal $\sqrt{l^2 + b^2}$.
• Square of side $s$: area $s^2$, diagonal $s\sqrt2$, so area $= \dfrac{d^2}{2}$.
• Given $l + b$ and $lb$: $l$ and $b$ are the roots of $t^2 - (l + b)t + lb = 0$.
• The square drawn on a square's diagonal has twice its area.
• Areas of squares in ratio $p : q$ means sides (and perimeters) in ratio $\sqrt p : \sqrt q$.

• Path of width $w$ outside an $l \times b$ plot: area $(l + 2w)(b + 2w) - lb$.
• Border of width $w$ inside: area $lb - (l - 2w)(b - 2w)$.
• Covering with tiles: number of tiles $= \dfrac{\text{area to cover}}{\text{area of one tile}}$ (convert units first).
• Length and breadth up by $x\%$ and $y\%$: area factor $\left(1 + \dfrac{x}{100}\right)\left(1 + \dfrac{y}{100}\right)$.

For diagonals $d_1, d_2$ and side $a$:

• Area $= \dfrac12 d_1 d_2$.
• $a^2 = \left(\dfrac{d_1}{2}\right)^2 + \left(\dfrac{d_2}{2}\right)^2$, so $4a^2 = d_1^2 + d_2^2$.
• Given the sum of the diagonals and the area: $d_1^2 + d_2^2 = (d_1 + d_2)^2 - 4\times\text{Area}$.

• Parallelogram with sides $a, b$ and angle $\theta$: area $ab\sin\theta$, which is less than the rectangle $ab$ unless $\theta = 90^\circ$.
• A triangle and a parallelogram on the same base with equal areas: the triangle's height is twice the parallelogram's.
• Trapezium: area $= \dfrac12(\text{sum of parallel sides})\times h$.
• Isosceles trapezium with parallel sides $p > q$ and legs $l$: each leg overhangs by $\dfrac{p - q}{2}$, so $h = \sqrt{l^2 - \left(\dfrac{p - q}{2}\right)^2}$.

• Quadrilateral $ABCD$ with $\angle B = 90^\circ$: $AC = \sqrt{AB^2 + BC^2}$, then $\text{Area} = [ABC] + [ACD]$, the second by Heron.
• A triangle whose vertices sit on the sides of a square: square minus the three right triangles at the corners.
• A right angle at a point on a side of a square makes the two corner triangles similar; use the ratio of their hypotenuses.