Question

a square OABC is inscribed in a quadrant OPBQ of a circle . if OA=21 cm .find the area of the shaded region.

Answer 1

First we find the area of the square OABC
OA=21cm
∴AB =21cm
∴ ar(OABC) = OA×BC
 ∴ar(OABC) = 21²
∴arOABC = 441cm²

By Pythagoras's theorem,
Hypotenuse²=side1²+side2²
∴OB² = OA²+AB²
∴OB²=21²+21²
∴OB² = 882
∴OB=√882

∵area of circle = πr²
And r=√882
∴ area of quadrant = 1/4(πr²)
                               =1/4 × 3.14 × 882 
                                =692.37 cm²

Area of shaded portion = area of quadrant-area of square
                                      = (692.37-441)cm²
                                      =251.37cm²

Answer 2

Draw the diagonals oc of square oabc

By Pythagoras theorem

O b square is equal to o a square +

a b square

Therefore OB is equal 20°

Area of square oabc is 400 CM square