Group Activity Area of a Sector A sector of a circle (shaded...

Group Activity Area of a Sector A sector of $a$ circle (shaded in the figure) is a region bounded by a central angle of a circle and its intercepted arc. Use the fact that the areas of sectors are proportional to their central angles to prove that $$ A=\frac{1}{2} r^{2} \theta $$ where $r$ is the radius and $\theta$ is in radians.

Group Activity Area of a Sector A sector of $a$ circle (shaded in the figure) is a region bounded by a central angle of a circle and its intercepted arc. Use the fact that the areas of sectors are proportional to their central angles to prove that
$$
A=\frac{1}{2} r^{2} \theta
$$
where $r$ is the radius and $\theta$ is in radians.

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Key Concepts

Sector of a Circle

A sector is a region of a circle bounded by two radii and the intercepted arc. It represents a portion of the entire circle, enabling the analysis of a part-to-whole relationship in circular geometry.

Central Angle (in Radians)

The central angle is the angle formed at the center of the circle by the two radii that define the sector. When measured in radians, it directly relates to the arc length and simplifies the proportion between the sector’s area and the circle’s total area.

Proportionality Principle

This principle states that the area of a sector is directly proportional to its central angle. Since the entire circle corresponds to an angle of 2? radians, a sector with a central angle ? is ?/(2?) of the total area, which is key to deriving the sector area formula.

Area of a Circle

Understanding the full area of a circle, given by A = ?r², is fundamental. It provides a basis for calculating the area of a sector by relating the fraction of the circle determined by the central angle to the total area.

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