KOL is a sector of radius 5cm with KN perpendicular to OL at N such that ON = 4cm.

how to find the perimeter of the shaded region KNL. i had already found the theta which is 0.64rad (3 sig fig). i need full solution for this! thanks

Perimeter KNL = Arclength KL + LN + KN
= OK*arctan(KN/ON) + (OL - ON) + sqrt(OK^2 - ON^2)
= 5*arctan(3/4) + 1 + 3
= 5*arctan(3/4) + 4
= 7.22cm to 3 significant digits (or 2 decimal places)

The perimeter is the distance from point K to point L along the circumference, then from L to N, and from N back to K.
Getting other measurements in place: ON= 4cm, OK=OL= 5cm.
LN=OL-ON=5-4= 1 cm
OKN is a right triangle, so OK^2 = ON^2 + KN^2 which is 5^2 = 4^2 = KN^2, so KN = 3...to be continued

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using pythagorus, NL equals sqrt(5^2-4^2) ie 3cm
arc length KL equals rxtheta ie 5x0.64 equals 3.2 cm
perimeter equals 3+3.2 ie 6.2cm