(a) Find the length of the arc that subtends the given central angle θon a circle of diameter d

step 1 So in the given question we have been given the angle subtended by an arc theta as 50 degrees an

(a) Find the length of the arc that subtends the given...

(a) Find the length of the arc that subtends the given central angle $\theta$ on a circle of diameter $d$ (b) Find the area of the sector determined by $\boldsymbol{\theta}$.
$$\theta=50^{\circ}, \quad d=16 \mathrm{m}$$

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Key Concepts

Circle Properties

Understanding the properties of a circle, such as its diameter and radius, is fundamental in solving problems related to circles. The radius, which is half the diameter, is a key measurement used in formulas for arc length and sector area, linking the geometry of the circle to its algebraic representations.

Sector Area

The area of a sector is the portion of the circle enclosed by two radii and the arc connecting them. It is determined by the formula A = 0.5 × r² × ?, with the angle in radians. This relationship shows how the fraction of the circle’s area occupied by the sector depends on the central angle.

Radian Measure

Radians are the standard unit of angular measure used in many formulas of circle geometry, including those for arc length and sector area. Converting degrees to radians (by multiplying by ?/180) is essential because the formulas for arc length (L = r?) and sector area (A = 0.5 × r² × ?) require the angle to be in radians.

Arc Length

The arc length is the distance along a portion of the circumference of a circle. It is calculated by multiplying the circle’s radius by the central angle in radians (L = r?). This concept is critical in understanding how changes in the central angle or the circle's radius affect the distance along the circle.

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