Arc Length, Sector Area and Radians

The key idea

An arc or sector is a fraction of the whole circle:

\frac{θ}{360°}

in degrees or

\frac{θ}{2 π}

in radians. In radians, arc length is

r θ

and sector area is

\frac{1}{2} r^{2} θ

. An inscribed angle is half the central angle on the same arc.

Using the inscribed angle as the arc's measure, or mixing degrees and radians.

These are formula questions, not graphing questions. The calculator only does the arithmetic.

College Board skill: Circles · Geometry and Trigonometry

One question from each level. The drill has 12 at every level, and moves you up after four right in a row.

Level 1 · Warm-up

A circle has a radius of 4. An arc of the circle has a central angle of

45°

. The length of the arc is

k π

. What is the value of

k

Show the answer

Answer: 1The arc is

\frac{45}{360}

of the circumference

2 π (4) = 8 π

\frac{45}{360} \times 8 π = 1 π

. So

k = 1

Level 2 · Test level

A sector of a circle with radius 3 has a central angle of

\frac{π}{6}

radians. The area of the sector is

k π

. What is the value of

k

Show the answer

Answer: 3/4In radians, sector area is

\frac{1}{2} r^{2} θ = \frac{1}{2} (3)^{2} \cdot \frac{π}{6} = 0.75 π

. So

k = 0.75

Level 3 · Beyond the test

In a circle, an inscribed angle measuring

35°

intercepts an arc of length

\frac{35}{9} π

. What is the radius of the circle?

Show the answer

Answer: 10The central angle on the same arc is twice the inscribed angle,

70°

. The arc is

\frac{70}{360} \cdot 2 π r = \frac{7}{18} π r

. Setting this equal to

\frac{35}{9} π

gives

r = 10

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