Question

Please help me( ´人` )

The following diagram shows a quadrant OPQ in the sector ORST with centre O and a radius of 21 cm.

Given Q is the midpoint of OR, calculate the area, in cm², of the shaded region.

A 285.025
B 308.045
C 365.075
D 410.025​

Answer 1

D

Explanation:

First let's list down the formulas we will be using.

Area of Quadrant = 1/4 x Area of Circle

=
`(1)/(4) \pi r^(2)`

Area of Part of a circle = (θ/360)
`\pi r^(2)`

First, we will find Angle QOT.

Angle QOT = 360 - 231 = 129

Area of ORST =
`(129)/(360) \pi (21)^(2) \\=(6321)/(40) \pi cm^(2)`

Next we know that, Q is midpoint of OR, therefore OQ = QR

and OQ = 0.5 * OR

OQ = 0.5 * 21 = 10.5cm = radius of Quadrant QOP

Area of Quadrant QOP =
`(1)/(4) \pi (10.5)^(2) \\=(441)/(16) \pi cm^(2)`

Lastly,

Area of Shaded Region = Area of ORST - Area of Quadrant QOP

=
`(6321)/(40) \pi -(441)/(16) \pi \\= 409.86cm^(2)`

Similar to the other question you asked, choose the closes answer as there might be some rounding differences. I kept it in fractions as it will ensure maximum precision.