(a) What angle in radians is subtended by an arc 1.50 m long on the circumference of a circle o

(a) We know that the angle in radians is given by the formula: angle u003D arc length / radius. In t

(a) What angle in radians is subtended by an arc 1.50 m...

(a) What angle in radians is subtended by an arc $1.50 \mathrm{~m}$ long on the circumference of a circle of radius $2.50 \mathrm{~m} ?$ What is this angle in degrees? (b) An arc $14.0 \mathrm{~cm}$ long on the circumference of a circle subtends an angle of $128^{\circ} .$ What is the radius of the circle? (c) The angle between two radii of a circle with radius $1.50 \mathrm{~m}$ is 0.700 rad. What length of arc is intercepted on the circumference of the circle by the two radii?

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Key Concepts

Circle Geometry

A solid understanding of circle geometry, including the relationships between radius, arc length, and central angles, is critical for solving a range of problems involving circular sectors and arcs, as well as for applications in trigonometry and calculus.

Degree Conversion

Converting between radians and degrees is essential since many problems and real-world applications use degrees. The conversion is based on the equivalence of 2? radians to 360 degrees, leading to the formulas: degrees = radians × (180/?) and radians = degrees × (?/180).

Arc Length Formula

The arc length formula, s = r?, establishes that the length of an arc (s) on a circle is the product of the radius (r) and the central angle (?) in radians. This relationship is central to solving problems involving segments of a circle.

Radian Measure

Radian measure is a natural way to quantify angles based on the arc length relative to the radius; one radian corresponds to the angle created when the arc length equals the radius. This measure is preferred in many areas of mathematics and physics for its direct relation to circle geometry.

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