Find the area of a segment AYB if the radius of a circle is 21cm and angle AOB is equal to 180degree.
To find the area of segment AYB, we need to know the length of the chord AB and the height of the segment (distance from the midpoint of AB to the center of the circle).
Since angle AOB is 180 degrees, we can deduce that AB is a diameter of the circle. Therefore, the length of AB is equal to twice the radius, which is 2 * 21 cm = 42 cm.
Now, let's find the height of the segment. This is the distance from the midpoint of AB to the center of the circle. Since AB is a diameter, its midpoint coincides with the center of the circle. So, the height of the segment is simply the radius, which is 21 cm.
Now, we have the length of the chord AB = 42 cm and the height of the segment = 21 cm.
To find the area of the segment AYB, we can use the formula:
Area = (θ/360) * π * r^2 - (1/2) * b * h
where θ is the central angle in degrees, π is a mathematical constant (approximately equal to 3.14159), r is the radius of the circle, b is the length of the chord, and h is the height of the segment.
Plugging in the values, we get:
Area = (180/360) * π * 21^2 - (1/2) * 42 * 21
Simplifying further:
Area = (1/2) * π * 21^2 - (1/2) * 42 * 21
Area = (1/2) * 3.14159 * 441 - (1/2) * 882
Area = 1.5708 * 441 - 441
Area = 693.00168 - 441
Area ≈ 252.00168 cm^2
Therefore, the area of segment AYB is approximately 252.00168 cm^2.