Arc Length and Sector Area

15.3: Arc Length and Sector Area. Original editorial SAT preparation with three worked examples, ten flashcards, and thirty practice questions. Educator-review draft.

Lesson 3 of 63 parts Practice Flashcards

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Arc length in degrees

Objective: Take a central-angle fraction of circumference.

Prerequisite skills: circumference, circle area, and fractions. An arc with central angle θ degrees is θ/360 of the complete circumference. Multiply that fraction by 2πr. The arc measure in degrees is not itself a distance. Common error: using area instead of circumference for arc length. Restrictions and interpretation: Use degrees over 360 or radians over 2π consistently; arc length is linear and sector area is quadratic in radius.

Formulas

$s=(θ/360)·2πr$

Worked examples

Try itA circle has radius 69. A 60° central angle intercepts an arc of length Kπ. What is K?Show answer

The arc is 60/360=1/6 of the circumference. Length=(1/6)(2π69)=23π, so K=23. Therefore the requested answer is 23.

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Arc length in degrees

Objective: Take a central-angle fraction of circumference.

Formulas

$s=(θ/360)·2πr$

Worked examples

Try itA circle has radius 69. A 60° central angle intercepts an arc of length Kπ. What is K?Show answer

The arc is 60/360=1/6 of the circumference. Length=(1/6)(2π69)=23π, so K=23. Therefore the requested answer is 23.

Sector area in degrees

Objective: Take a central-angle fraction of circle area.

A sector is the region swept by two radii and their included arc. Its area is the central-angle fraction of πr². Chord-bounded segments are different regions and need a triangle-area correction. Common error: taking the fraction of circumference when asked for area. Restrictions and interpretation: Use degrees over 360 or radians over 2π consistently; arc length is linear and sector area is quadratic in radius.

Formulas

sector area=(θ/360)πr²

Worked examples

Try itA circle has radius 48. The area of a 90° sector is Kπ. What is K?Show answer

A 90° sector is 1/4 of the disk. Area=(1/4)π(48)²=576π, hence K=576. Therefore the requested answer is 576.

Radian arc and sector formulas

Objective: Use radian measure directly in arc-length and sector-area formulas.

A radian is defined by arc length divided by radius, so s=rθ. Combining that with the fraction of a full disk gives A=r²θ/2. These direct forms require θ in radians, not degrees. Common error: using a degree value directly in s=rθ. Restrictions and interpretation: Use degrees over 360 or radians over 2π consistently; arc length is linear and sector area is quadratic in radius. Desmos and method choice: Use Desmos for arithmetic or a graph/table check when that reduces work. Enter parentheses explicitly and verify the requested quantity and units. Use the algebraic steps shown here for an exact result; a rounded graph display is not proof of exact equality.

Formulas

$s=rθ; A=r²θ/2 when θ is in radians$

Worked examples

Try itA sector has radius 26 and central angle 2 radians. What is its area in square units?Show answer

Use A=r²θ/2 with radians: A=(26)²(2)/2=676. The unit is area, unlike arc length 2r. Therefore the requested answer is 676.

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