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📐 Mathematics  ·  Circles & Mensuration  ·  JEE

Isoperimetric inequality states: among all closed curves of fixed perimeter, the one enclosing maximum area is:

Answer: Circle.

  • A Square
  • B Equilateral triangle
  • C Circle
  • D Regular hexagon

Correct answer: C. Circle

Explanation: Isoperimetric inequality: 4πA ≤ P², with equality iff the curve is a circle. For circle: P=2πr, A=πr², giving 4π(πr²)=4π²r²=P². No other shape achieves this equality. Ans: Circle.

Cylinder, Cone, SphereV=πr²hCSA=2πrhV=⅓πr²hCSA=πrlV=4/3πr³SA=4πr²

Three standard 3D solids and their key formulas: a cylinder's volume scales with the full height, a cone's volume is exactly one-third of the cylinder with the same base and height, and a sphere's volume and surface area both follow distinct r-based formulas worth memorising separately.

Concept context

Perimeters, areas, and volumes of 2D and 3D shapes

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