A Nice geometry math question| find the area of the shaded portion

We are asked to find the area of this shaded region. Okay, now this is a quadrant of a circle so that the area of the shaded portion will be equal to the area of this quadrant so the area of the quadrant, okay, minus the area of this circle, the area of the circle, okay, this full circle So let's write it all out. How do we know about it? We need to find the area of this quadrant. then subtract the area of the full circle from it to obtain the area of the shaded region So please, this is not drawn to scale, okay? Now let's get to work. If the area or the radius Let's call this quadrant r, so this is r, and let's call this the center of the small circle. and let the radius of the small circle be small r, okay? So remember, here will be 90 degrees. and that is because in a circle with center at zero, a tangent line AB is drawn across it, okay if you draw a straight line from the center to the point of contact between the circle and the Then a tangent forms a 90-degree angle, which is why we have 90 degrees here, and if this is the center of the circle, then from here to here is also the radius of the circle, okay, and this is 90 degrees, this is a quadrant of a circle, this is 90, so here is 90, therefore you can see that what We have here a rectangle, so the opposite sides are equal. So if we call this one a and this one b. This is center O, and here C. You can see that BO is equal to r plus r, right? Which is 2. Okay, now let us draw a straight line from the center of this circle to this point, okay? So let us join here to here if we do that you can see that what we have here op let's call this p you can see that OP is also the radius of the quadrant, okay, so OP is also equal to capital R. Now let us Okay, triangle BPO. So if we do that, we have this, this, and this. So this is B, this is O, this is P. Now PB is 20, given that this is 90 degrees and PO is capital R, then BO is 2 small r. Alright, so I wouldn't even write it here right now using the Pythagorean theorem. Using the Pythagorean theorem, we can say that r squared equals 20 squared. plus 2r squared, okay, so we have r squared is equal to 400 plus 2 squared is 4r, so if we solve Furthermore, we can say that r squared minus 4 r squared equals 400. Now let's call this Equation one, so back to the main equation, the question says you find the area of the quadrant, so the area of the quadrant, sorry, the area of the shaded portion, right, is equal to the area of the quadrant minus the area of the full circle area of the full circle, okay, so area is equal to, now area of the quadrant is one over four pi times r, where r is the radius of the quadrant, is capital R squared minus the area of the full circle is equal to pi times r squared, where r is the radius of the smaller circle, okay Now, so we can say the area is equal to, they have pi in common, right? So if we factor that out, this becomes when this is divided by pi we are left with r squared over four okay minus When this is divided by π, we are left with small r if we simplify this further. So we have area equal to pi; if we simplify this further, if we bring out four outside then we are going to form something like this, right, because r squared over four becomes r square it, all right, and this becomes minus four r squared, right? Yeah, okay, so but remember that r squared minus four r squared equals 400, so it simply means that the area of the the portion which is a equals pi over four multiplied by r squared minus four r squared is equal to 400, so we have area is equal to this; we divide this to give us 100. Okay, is equal to. pi times area is equal to 100 multiplied by now pi is 3.142, okay? Now remember that this is an approximated value of pi, so our area is going to be equal to 314.2 units squared, so that should be the area of the shaded region, okay? Thanks so much for watching, bye.