⭕ Circle, Sector & Semicircle

• Area of a circle $= \pi r^2$ (a region, so the unit is cm²).

• Circumference (boundary length) $= 2\pi r = \pi d$, where the diameter $d = 2r$.

• Ring / annulus (the gap between two concentric circles of radii $R$ and $r$) $= \pi(R^2-r^2)$. Square each radius first, then subtract — never subtract the radii and square afterwards.

The exam habit to build: take $\pi=\dfrac{22}{7}$ whenever the radius is a multiple of 7, otherwise use $3.14$. The 7 in the denominator then cancels with the $r^2$ instantly — for example $\dfrac{22}{7}\times 7^2 = \dfrac{22}{7}\times 49 = 22\times 7 = 154$. Cancel the 7 first, multiply last.

• Semicircle (half a circle): area $= \dfrac12\pi r^2$, but the perimeter $= \pi r + 2r$ — the curved arc plus the straight diameter. The number-one trap is forgetting the diameter $2r$.

• Sector (a slice of angle $\theta$): it is just the fraction $\dfrac{\theta}{360}$ of the whole circle. So area $= \dfrac{\theta}{360}\times\pi r^2$, arc length $= \dfrac{\theta}{360}\times 2\pi r$, and the perimeter of the slice $= \text{arc} + 2r$ (add the two straight radii).

• Ratios: area scales as the square of length, so if two circles have radii in the ratio $a:b$ their areas are in the ratio $a^2:b^2$ — and conversely, equal area ratio $a:b$ means the radii (and circumferences) are in ratio $\sqrt{a}:\sqrt{b}$.