Work each problem. See Example 5. Angle Measure Find the measure (in radians) of a central angle of a sector of area 16 in² a circle of radius 3.0 in.

검증된 단계별 안내

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

Identify the given values: the area \(A = 16\) in² and the radius \(r = 3.0\) in.

Substitute the known values into the formula: \(16 = \frac{1}{2} \times (3.0)^{2} \times \theta\).

Simplify the expression on the right side: calculate \(\frac{1}{2} \times 9 = 4.5\), so the equation becomes \(16 = 4.5 \times \theta\).

Solve for \(\theta\) by dividing both sides of the equation by 4.5: \(\theta = \frac{16}{4.5}\).

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Area of a Sector

The area of a sector of a circle is given by the formula A = (1/2) * r² * θ, where r is the radius and θ is the central angle in radians. This formula relates the sector's area directly to the angle, allowing calculation of one when the other is known.

Radian Measure of Angles

Radians measure angles based on the radius of a circle, where one radian is the angle subtended by an arc equal in length to the radius. Using radians simplifies many trigonometric formulas, especially those involving arc length and sector area.

Solving for the Central Angle

To find the central angle θ when the sector area and radius are known, rearrange the sector area formula to θ = (2 * A) / r². This step involves algebraic manipulation and understanding the relationship between the variables.