Summary and Quick Revision

Chapter at a Glance

Every question in this chapter is built from a handful of formulas applied to circles and their parts. Memorise these and read each problem carefully to see which one is needed.

Key Formulae (Must Memorise)

• Circumference: $C = 2\pi r = \pi d$.
• Area of a circle: $A = \pi r^2$.
• Area of a ring (annulus): $\pi(R^2 - r^2)$.
• Area of a sector (angle $\theta$): $\dfrac{\theta}{360}\pi r^2 = \tfrac12\ell r$.
• Length of an arc: $\ell = \dfrac{\theta}{360}\times 2\pi r$.
• Perimeter of a sector: $\ell + 2r$.
• Area of a segment: area of sector $-$ area of triangle $= \dfrac{\theta}{360}\pi r^2 - \tfrac12 r^2\sin\theta$.
• Major part: whole circle $-$ minor part.

Standard Angles and Facts

• Minute hand: $6^\circ$ per minute. Hour hand: $30^\circ$ per hour ($0.5^\circ$ per minute).
• Quadrant $=\tfrac14$ circle ($90^\circ$); semicircle $=\tfrac12$ circle ($180^\circ$).
• Corner of a square field $\Rightarrow$ quarter circle ($90^\circ$); equilateral triangle corner $\Rightarrow 60^\circ$; regular hexagon corner $\Rightarrow 120^\circ$.
• Four equal quarter circles $=$ one circle; two equal semicircles $=$ one circle.
• Inscribed circle in a square: diameter $=$ side. Inscribed square in a circle: diagonal $=$ diameter, area $=2r^2$.
• Segment triangle areas: $90^\circ\to\tfrac12 r^2$; $60^\circ$ and $120^\circ\to\tfrac{\sqrt3}{4}r^2$.

Common Traps to Avoid

• Area vs circumference: area uses $r^2$, circumference uses $r$. Do not mix them.
• Arc length vs perimeter of a sector: the perimeter adds the two radii ($+2r$) to the arc.
• Segment vs sector: a segment is sector $-$ triangle; never forget to subtract the triangle.
• Minor vs major: get the major part by subtracting the minor part from the whole circle.
• Units: convert to a single unit before computing (cm, m, km), and give area in squared units.
• Use the value of $\pi$ the question specifies ($\tfrac{22}{7}$ or $3.14$), and $\sqrt3=1.73$ (or 1.7) only when told.

Final Tip: For shaded regions, write the plan in words first — "square $-$ circle", "rectangle $+$ two semicircles" — then substitute numbers.