Find the measure of the central angle on a circle of radius r that forms a sector with the given area.
Radius, r: 10 feet Area of the Sector, A: 25 square feet

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Recall the formula for the area of a sector of a circle: \(A = \frac{1}{2} r^{2} \theta\), where \(r\) is the radius and \(\theta\) is the central angle in radians.

Substitute the given values into the formula: \(25 = \frac{1}{2} \times 10^{2} \times \theta\).

Simplify the expression on the right side: \(25 = \frac{1}{2} \times 100 \times \theta\).

Solve for \(\theta\) by isolating it on one side: \(\theta = \frac{25}{\frac{1}{2} \times 100}\).

Calculate \(\theta\) to find the measure of the central angle in radians, then convert to degrees if needed using \(\theta_{degrees} = \theta_{radians} \times \frac{180}{\pi}\).

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주요 개념

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Area of a Sector

The area of a sector of a circle is a portion of the circle's total area, determined by the central angle. It is calculated using the formula A = (θ/360) × πr², where θ is the central angle in degrees and r is the radius. Understanding this formula allows you to relate the sector's area to the angle.

Central Angle Measurement

The central angle is the angle subtended at the center of the circle by the sector. It is usually measured in degrees or radians. Knowing how to isolate and solve for this angle from the sector area formula is essential for finding the angle given the radius and area.

Circle Geometry and Radius

The radius is the distance from the center of the circle to any point on its circumference. It is a fixed length that helps define the size of the circle and its sectors. Recognizing the role of the radius in formulas involving sectors is crucial for solving related problems.

Find the measure of the central angle on a circle of radius r that forms a sector with the given area. Radius, r: 10 feet Area of the Sector, A: 25 square feet