Radians and Circular Measurement

Develop radian measure, arc length and sector area.

Lesson 1 of 32 parts Practice Flashcards

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Radian Measure

Objective: Explain radian measure, apply the rule, and justify relevant restrictions.

One radian is the central angle subtending an arc equal in length to the radius. Since a full circle has circumference 2πr, one turn is 2π radians or 360°. Degree-to-radian conversion multiplies by π/180; conversion in the reverse direction multiplies by 180/π. A radian is a ratio of lengths, but angle units must still be identified in calculations.

Formulas

$θ(rad)=θ(deg)·π/180$

Worked examples

Try itConvert 105° to radians. Give the coefficient of π.Show answer

Step 1: Multiply 105 by π/180. Step 2: Reduce 105/180 to 7/12. Step 3: The angle is 7/12π radians.

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Radian Measure

Objective: Explain radian measure, apply the rule, and justify relevant restrictions.

Formulas

$θ(rad)=θ(deg)·π/180$

Worked examples

Try itConvert 105° to radians. Give the coefficient of π.Show answer

Step 1: Multiply 105 by π/180. Step 2: Reduce 105/180 to 7/12. Step 3: The angle is 7/12π radians.

Arc Length and Sector Area

Objective: Explain arc length and sector area, apply the rule, and justify relevant restrictions.

For a circle of radius r and central angle θ in radians, arc length is s=rθ and sector area is A=(1/2)r²θ. Both follow by taking the fraction θ/(2π) of a complete circumference or disk. These formulas assume a nonnegative sector angle for ordinary length and area. If an angle is supplied in degrees, convert before applying the formulas.

Formulas

$s=rθ; A=(1/2)r²θ$

Worked examples

Try itA circle has radius 3 cm. A central angle measures 7/5 radians. Find the arc length in cm.Show answer

Step 1: Use s=rθ with θ in radians. Step 2: Substitute s=3·(7/5). Step 3: The arc length is 21/5 cm.

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