Final answer:
The area of a circle can be calculated using πr², while the area of a sector is found with ½ r²θ, the area of a right-angle triangle within the circle is ½ ab, and the area of the shaded segment is the area of the sector minus the area of the triangle OAB.
Step-by-step explanation:
Calculating Areas in a Circle and a Sector
To calculate the area of a circle, the formula A = πr² is used, where r is the radius. If O is the centre of the circle, and the radius is given, simply substitute the value of r into the formula to find the area. For the area of the sector OAB, you need the angle of the sector (in radians or degrees) and the radius. The formula is A = ½ r²θ, with θ being the angle in radians. To find the area of the triangle OAB, if the triangle is right-angled at O, which commonly occurs in circle problems, use the formula A = ½ ab, where a and b are the lengths of the two sides that meet at the right angle. For the area of the shaded segment, first calculate the area of the sector and then subtract the area of the triangle OAB from it.
Remember that unit consistency is crucial, so ensure that the radius and angles are in compatible units when substituting into formulas. For segments and sectors, it can be helpful to visualize the circle as part of a coordinate system, with the center at the origin to determine the relevant angles and lengths.