128 square cm
Explanation:
We'll apply the arc length formula
L = theta*r
Theta is the central angle and r is the radius. This formula only works when the angle is in radian mode. In this case, theta = 4/3 radians.
Arc length AB is part of a circle that has radius r = 8, so,
L = theta*r
L = (4/3)*8
L = 32/3
That represents the length of arc AB. This length is in cm.
Let x be the length of segment BD. This is also the length of AC as well due to rotational symmetry.
Arc CD is part of the larger circle that has radius r = 8+x
L = theta*r
L = (4/3)(8+x)
We found the following lengths so far
arc AB = 32/3segment BD = xarc CD = (4/3)(8+x)segment AC = x
These four pieces form the sides of the blue region. Therefore, they form the perimeter of the blue region. Add up these four sides and set the sum equal to the given perimeter 48. Then solve for x.
(arcAB)+(segmentBD)+(arcCD)+(segment AC) = perimeter of blue region
(32/3) + (x) + (4/3)(8+x) + (x) = 48
32/3 + x + (4/3)*8+(4/3)*x + x = 48
32/3 + 2x + 32/3+(4/3)*x = 48
64/3 + 2x +(4/3)*x = 48
64/3 + 6x/3 +(4/3)*x = 48
64/3 + (10/3)*x = 48
(10/3)*x = 48-64/3
(10/3)*x = 144/3-64/3
(10/3)*x = 80/3
10x = 80
x = 80/10
x = 8
Since x = 8, we know that AC and BD are each 8 cm long.
This means that the larger circle has radius 8+8 = 16 cm
Now we'll apply the sector area formula. You can think of a sector as a pie slice or pizza slice. The formula shown below only works if theta is in radian mode.
Let's find the area of sector AOB. This is the smaller pie slice.
A = area of sector
A = (1/2)*theta*r^2
A = (1/2)*(4/3)*8^2
A = (1/2)*(4/3)*64
A = (1*4*64)/(2*3)
A = (2*64)/3
A = 128/3
Sector AOB is 128/3 square cm. Let's make m = 128/3 so we can use it later.
Now let's find the area of sector COD. We use the same theta value, but the radius is now 16 cm.
A = (1/2)*theta*r^2
A = (1/2)*(4/3)*16^2
A = (2/3)*256
A = 512/3
The area of sector COD is 512/3 square cm. Let n = 512/3.
The last thing to do is subtract the values of m and n to get the area we want.
blue area = (area of sector COD) - (area of sector AOB)
blue area = n - m
blue area = 512/3 - 128/3
blue area = (512-128)/3
blue area = 384/3
blue area = 128 square cm
