Areas Related to Circles — Cheat Sheet

Class X · Mathematics Ch.11 · 8 Panels · Quick Reference

⭕ Circle Basics & π
Radius = r  |  Diameter d = 2r
π = circumference ÷ diameter
Which value of π?
UseWhen
22/7r is a multiple of 7
3.14otherwise / decimals
π is irrational ≈ 3.14159… — always keep it as a factor until the last step.
📏 Circumference & Arc
Circumference C = 2πr = πd
Arc length ℓ = (θ/360) × 2πr
Perimeter of semicircle = πr + 2r
Perimeter of quadrant = (πr/2) + 2r
Wheel problems: distance in n turns = n × 2πr. Turns = distance ÷ (2πr).
Perimeter ≠ area. Perimeter includes the straight edges (radii/diameter), not just the arc.
🟣 Area of a Circle
Circle A = πr²
Semicircle = ½πr²   Quadrant = ¼πr²
Circular Ring (Annulus)
A = π(R² − r²) = π(R+r)(R−r)
(R = outer radius, r = inner radius)
If diameter given: find r = d/2 BEFORE squaring. A common exam slip.
🍕 Sector of a Circle
Area of sector = (θ/360) × πr²
Arc length = (θ/360) × 2πr
Also: Area = ½ × ℓ × r
Perimeter of a sector = arc length + 2r (two radii).
Major sector = πr² − minor sector. Use 360° − θ for the major angle.
Quadrant is a sector with θ = 90°.
🌙 Segment of a Circle
Minor segment = (sector) − (triangle)
= (θ/360)πr² − ½r²sinθ
Triangle in the sector
θTriangle area
60°(√3/4)r²
90°½r²
120°(√3/4)r²
Major segment = πr² − minor segment.
⭐ All Formulas at a Glance
QuantityFormula
Circumference2πr
Area of circleπr²
Arc length(θ/360)·2πr
Sector area(θ/360)·πr²
Segment areasector − triangle
Ring areaπ(R²−r²)
🧩 Combinations of Figures
  1. Break the shaded region into standard shapes
  2. Write the area of each part
  3. ADD the parts that build it up
  4. SUBTRACT the parts that are cut out
Circle in Squarecorners = a² − π(a/2)²
Square in Circlediagonal = diameter = 2r
Designs: flower beds, windows and tiles combine sectors, segments and polygons.
🧠 Exam Quick Recall
Semicircle perimeter = πr + 2r (don't forget the diameter!).
Sector perimeter = arc + 2r (don't forget the two radii!).
Common mistakes: using d instead of r; forgetting straight edges in perimeter; mixing π = 22/7 with a non-multiple of 7.
Units: length in cm, area in cm². Keep units consistent throughout.