Arc Length and Sector Area, SAT math lesson

Cut a round pizza into six equal slices. Each slice is one sixth of the pizza, so its curved crust is one sixth of the whole crust and its surface is one sixth of the whole top. That single idea is behind nearly every arc and sector question on the SAT. Find what fraction of the full circle the angle covers, then take that fraction of the circumference for a length or of the area for a region.

See also Arc Length and Area of a Sector in Precalculus.

Key terms

Circumference

Definition The circumference of a circle is the length of the circle itself, the distance once around it. For a circle with radius $r$ and diameter $d = 2r$ ,

C = 2\pi r = \pi d.

The area of the disk inside the circle is $A = \pi r^2$ . Both formulas are on the SAT reference sheet.

In plain words Circumference is the length of the crust. Area is how much cheese covers the top. One is measured in units like centimeters, the other in square units like square centimeters.

Example A circle with radius $9$ has circumference $2\pi \cdot 9 = 18\pi$ and area $\pi \cdot 9^2 = 81\pi$ . If the radius is in inches, the first is $18\pi$ inches and the second is $81\pi$ square inches.

Central angle

Definition A central angle of a circle is an angle whose vertex is the center of the circle and whose sides are two radii. The part of the circle between the two radii is the intercepted arc, and the region bounded by the two radii and that arc is a sector. For a minor arc, the arc’s degree measure equals the measure of its central angle.

In plain words Stand at the center and point at two spots on the circle. The angle between your arms is the central angle, and the piece of circle between the two spots is its arc.

Example If radii $OA$ and $OB$ form a $60^\circ$ angle at the center $O$ , then the minor arc $AB$ measures $60^\circ$ . The rest of the circle, the major arc, measures $360^\circ - 60^\circ = 300^\circ$ .

Fraction of a circle

Definition A full turn around the center is $360^\circ$ , which is also $2\pi$ radians. Both facts are on the reference sheet. A central angle of $\theta^\circ$ , or of $\theta$ radians, covers this fraction of the full circle,

\frac{\theta^\circ}{360^\circ} \qquad \text{or} \qquad \frac{\theta}{2\pi}.

Because the arc and the sector grow in direct proportion to the angle, they are the same fraction of the circumference and of the area.

In plain words Ask what part of a full turn the angle is. Every other number in the problem then comes from scaling the whole circle by that part.

Example Divide a circle of radius $6$ into six equal sectors. Each central angle is $360^\circ \div 6 = 60^\circ$ , and the fraction is $\frac{60}{360} = \frac{1}{6}$ . So each arc is $\frac{1}{6} \cdot 12\pi = 2\pi$ long, and each sector has area $\frac{1}{6} \cdot 36\pi = 6\pi$ .

Arc length

Definition The arc length $s$ of an arc is the length measured along the circle. For a central angle of $\theta^\circ$ in a circle with radius $r$ ,

s = \frac{\theta}{360} \cdot 2\pi r.

This formula is not printed on the reference sheet. It comes from the circumference $2\pi r$ and the fraction $\frac{\theta}{360}$ , so you can rebuild it any time.

In plain words Arc length is a slice of the circumference. It is a length, so its units are plain units such as meters.

Example In a circle with radius $9$ , a central angle of $80^\circ$ intercepts an arc of length

s = \frac{80}{360} \cdot 18\pi = 4\pi.

That is about $12.57$ units, a bit more than one fifth of the circumference.

Sector area

Definition A sector is the region bounded by two radii and the arc between them. For a central angle of $\theta^\circ$ in a circle with radius $r$ , the sector area is

A = \frac{\theta}{360} \cdot \pi r^2.

Like arc length, this is derived from a reference sheet formula, here $A = \pi r^2$ , and is not printed there. Arc length and sector area have different dimensions. Arc length is a length, while sector area is measured in square units, so the two can never be compared or added.

In plain words Sector area is a slice of the whole area, the same fraction as the arc but taken of $\pi r^2$ instead of $2\pi r$ .

Example Same circle and angle as before, radius $9$ and $80^\circ$ .

A = \frac{80}{360} \cdot 81\pi = 18\pi.

If the radius is $9$ centimeters, the arc is $4\pi$ centimeters and the sector is $18\pi$ square centimeters. The same fraction $\frac{2}{9}$ produced both.

Radians

Definition The radian measure of a central angle is the length of its intercepted arc divided by the radius, $\theta = \frac{s}{r}$ . A full turn has arc $2\pi r$ , so it measures $2\pi$ radians, and $180^\circ = \pi$ radians. When $\theta$ is in radians, the arc and sector formulas simplify to

s = r\theta \qquad \text{and} \qquad A = \frac{1}{2} r^2 \theta.

These two short formulas are true only for $\theta$ in radians. With degrees, use the fraction $\frac{\theta}{360}$ instead. Combining them also gives $A = \frac{1}{2} r s$ .

In plain words A radian is the angle whose arc is exactly one radius long. Measured that way, the arc is simply the radius times the angle.

Example A circle has radius $6$ and a central angle of $\frac{2\pi}{3}$ radians. The fraction of the circle is $\frac{2\pi/3}{2\pi} = \frac{1}{3}$ , and the formulas agree with it.

\begin{aligned} s &= 6 \cdot \frac{2\pi}{3} = 4\pi \\[0.4em] A &= \frac{1}{2} \cdot 36 \cdot \frac{2\pi}{3} = 12\pi \end{aligned}

A common slip is to plug $120$ , the degree measure, into $s = r\theta$ . That gives $720$ , which is far longer than the whole circumference $12\pi \approx 37.7$ .

Perimeter of a sector

Definition The perimeter of a sector is the length of its whole boundary, which is the arc plus the two radii,

P = 2r + s.

In plain words Walk around the edge of a pizza slice. You cover the crust and both straight cuts, so the answer is more than the arc length.

Example A sector has area $15\pi$ and central angle $150^\circ$ . Recover the radius first.

\frac{150}{360} \cdot \pi r^2 = 15\pi \quad \text{so} \quad r^2 = 36.

A radius is positive, so $r = 6$ . The arc is $\frac{150}{360} \cdot 12\pi = 5\pi$ , and the perimeter is $12 + 5\pi$ , about $27.7$ . If a question asks for the arc, the answer is $5\pi$ . If it asks for the perimeter, the two radii must be added.

Check yourself

A circle has radius $12$ . A central angle of $135^\circ$ intercepts an arc of the circle. What is the length of this arc? (A) $9\pi$ , (B) $18\pi$ , (C) $24 + 9\pi$ , (D) $54\pi$

Show the answer

The answer is (A). The angle is $\frac{135}{360} = \frac{3}{8}$ of the circle, and the circumference is $24\pi$ , so the arc is $\frac{3}{8} \cdot 24\pi = 9\pi$ . Choice (B) uses the diameter in place of the radius, choice (C) is the perimeter of the sector, and choice (D) is the sector area $\frac{3}{8} \cdot 144\pi$ .

In a circle with radius $5$ , a central angle of $\theta$ radians intercepts an arc of length $4$ . What is the area of the sector formed by this angle?

Show the answer

The answer is $10$ . From $s = r\theta$ the angle is $\theta = \frac{4}{5}$ radians, so

A = \frac{1}{2} \cdot 5^2 \cdot \frac{4}{5} = 10.

The shortcut $A = \frac{1}{2} r s = \frac{1}{2} \cdot 5 \cdot 4$ gives the same value.

A sector of a circle with radius $8$ has a perimeter of $28$ . What is the measure of its central angle, in radians? (A) $\frac{3}{4}$ , (B) $\frac{3}{2}$ , (C) $\frac{7}{4}$ , (D) $\frac{7}{2}$

Show the answer

The answer is (B). The perimeter includes two radii, so the arc is $28 - 16 = 12$ , and the angle is $\theta = \frac{s}{r} = \frac{12}{8} = \frac{3}{2}$ . Choice (D) treats all of $28$ as arc length, choice (C) divides $28$ by the diameter, and choice (A) divides the correct arc by the diameter instead of the radius.

Arc Length and Sector Area

Theory, part 105

Practice, part 106

Key terms

Circumference

Central angle

Fraction of a circle

Arc length

Sector area

Radians

Perimeter of a sector

Check yourself

How this lesson connects

Builds on

Used later in