# how to find the perimeter of the shaded region in terms of π (pi) ? The diagram shows a quadrant with centre O and of radius, 9 cm. Given that the arc AM is 6 cm and the angleAOM is 2/3 rad , how...

how to find the perimeter of the shaded region in terms of π (pi) ?

The diagram shows a quadrant with centre O and of radius, 9 cm. Given that the arc AM is 6 cm and the angleAOM is 2/3 rad , how to find the perimeter of the shaded region in terms of π (pi) ?

http://i49.tinypic.com/2ptp1sl.jpg

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### 2 Answers

You need to write the formula of perimeter of shaded region to check what the problem provides and what you need to find, such that:

`P = MO+OB+MB`

Notice that `MO = OB = R = 9` and you need to find the length of the arc MB.

You should remember that you may find the length of arc using the following formula such that:

`MB = pi*R*(hat(MOB))/180`

`hat(MOB) = pi/2 - 2pi/3 =gt hat(MOB) = (3pi-4pi)/6=gt hat(MOB) = -pi/6 =gt MB = -9pi/6`

Substituting -`9pi/6` for MB in equation of perimeter yields:

`P = 9 + 9- 9pi/6 `

`P = 9(2- pi/6) =gt P = (9/6)(12 - pi) =gt P = (3/2)(12 - pi)`

**Hence,evaluating the perimeter in terms of `pi` , using the given information, yields `P = (3/2)(12 - pi).` **

**Sources:**

Required perimeter P= MO+OB+BM

MO=OB=Radius=9cm

angle AOB = pi/2 radian being angle of the quadrant

angle AOM = 2/3 radian

therefore angle MOB = pi/2-2/3

length of arc MB = angle*radius = (pi/2-2/3)*9 = 9pi/2-6 cm

P = 9+9+9pi/2-6 = 12+9pi/2 = 26.1cm appoximately

**The perimeter of the shaded portion is 12+9pi/2 cm or 26.1cm approximately**