r ≈ 4
Step-by-step explanation:
circle = 360 degrees
360 / 90 = 4
let's call the area of triangle XYZ + the shaded area portion A
because ∠XYZ = 90°, and 360 / 90 = 4, A = 1/4th of the area
total area of circle = πr²
XY = YZ = radius
thus, as ∠XYZ = 90° and two sides are the same, this forms a 45-45-90 triangle, with base YZ and height XY (base/height are interchangeable here)
thus, area of triangle XYZ = r*r/2 = r²/2
area A = r²/2 + 4.64
4 * A = total area of circle
4(r²/2+4.64) = πr²
2r²+18.56 = πr²
subtract 2r² from both sides to put all the variables on one side
(π-2)r² = 18.56
divide both sides by (π-2) to isolate r²
18.56/(π-2) = r² ≈ 16.258
r ≈ 4.032 ≈ 4