Ohio Assessments for Educators (OAE) Mathematics Practice Exam

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Which formula represents the area of a sector in radians?

A = θπr²/360
A = θr²/2
A = πr²θ
A = πr²/2

The correct answer is: A = θr²/2

The area of a sector of a circle is calculated using the formula A = (θ/2) * r², which can also be expressed as A = θr²/2 when θ is measured in radians. Here, r represents the radius of the circle, and θ is the angle in radians that subtends the sector. This formula derives from the fact that a full circle, which measures 2π radians, has an area of πr². Therefore, when you take a fraction of the circle (the sector), the area of that sector is proportional to the fraction of the angle θ compared to the full angle of 2π. By representing the sector's area directly as a function of the angle in radians, the formula correctly relates the size of the sector to both its angle and radius. Other choices may involve measurements or formulas that correspond to different contexts or angles (such as degrees) but do not effectively convey the relationship needed for calculating the area of a sector when using radians.