Mensuration: Areas & Perimeters

At a glance 12 formulas

1. 1Rectangle and square$\text{area} = lb,\ \text{perimeter} = 2(l + b)$$l, b = \text{sides};\ \text{diagonal} = \sqrt{l^2 + b^2}$
2. 2Rhombus$\text{area} = \tfrac{1}{2} d_1 d_2$$d_1, d_2 = \text{the diagonals, which cross at } 90^\circ$
3. 3Equilateral triangle, hexagon$\text{area} = \tfrac{\sqrt{3}}{4}a^2,\ \text{inradius} = \dfrac{a}{2\sqrt{3}}$$a = \text{side};\ \text{hexagon} = 6 \text{ triangles} = \tfrac{3\sqrt{3}}{2}a^2$
4. 4Circle, radius r$\text{area} = \pi r^2,\ \text{circumference} = 2\pi r$$\text{semicircle boundary} = \pi r + 2r$
5. 5Sector of angle θ$\text{area} = \dfrac{\theta}{360^\circ} \times \pi r^2$$r = \text{radius};\ \text{arc} = \dfrac{\theta}{360^\circ} \times 2\pi r$
6. 6Area of a ring$\pi(R^2 - r^2) = \pi(R + r)(R - r)$$R, r = \text{outer and inner radii}$
7. 7Path of width w$\text{outside } l \times b:\ 2w(l + b + 2w)$$\text{inside: } 2w(l + b - 2w)$
8. 8Two cross roads$\text{area} = w(l + b - w)$$w = \text{road width},\ \text{plot } l \times b$
9. 9Wheel turns$\text{distance} = \text{turns} \times 2\pi r$$r = \text{radius of the wheel}$
10. 10Scaling an area$\text{lengths} \times k \Rightarrow \text{area} \times k^2$$\text{lengths } +x\% \Rightarrow \text{area } +\left(2x + \tfrac{x^2}{100}\right)\%$
11. 11Round at a distance$\text{path length} = \text{perimeter} + 2\pi d$$d = \text{gap};\ \text{the corner arcs make one circle}$
12. 12Rope round a cornerA rope tied at a corner of a building sweeps $\tfrac{3}{4}$ of a circle; past each corner it carries on, shorter by that side, sweeping a quarter circle.