6.3 Arc Length & Sector Area

Arc Length and Sector Area by Proportion

Standard GEO-G.C.5 asks you to find one of the central angle, arc length, radius, or sector area given the others. The unifying idea is proportionality: an arc is the same fraction of the circumference that its central angle is of 360 degrees, and a sector is the same fraction of the circle's area.

• Arc length = (theta / 360)(2)(pi)(r)
• Sector area = (theta / 360)(pi)(r squared)

Here theta is the central angle in degrees. Both formulas are just "the fraction of the circle" (theta / 360) times the whole circumference or the whole area. Because the same fraction controls both quantities, you can set up arc / circumference = sector / area = theta / 360 and solve for whatever is missing.

Worked Examples in Degrees

Arc length. A circle has radius 10 and a 90-degree central angle. Since 90/360 = 1/4 and the circumference is 2(pi)(10) = 20(pi), the arc length is one fourth of 20(pi), or 5(pi).

Sector area (simple fraction). A sector has radius 6 and central angle 60 degrees. Because 60/360 = 1/6 and the full area is (pi)(6 squared) = 36(pi), the sector area is 36(pi)/6 = 6(pi).

Sector area (larger angle). A sector has radius 8 and central angle 135 degrees. Here 135/360 = 3/8 and the full area is (pi)(8 squared) = 64(pi), so the sector area is (3/8)(64(pi)) = 24(pi).

Working backward for the angle. If an arc length is 10(pi) on a circle of radius 12, then 10(pi) = (theta/360)(24(pi)). Dividing, theta/360 = 10/24 = 5/12, so theta = 150 degrees.

Working backward for the radius. Suppose a 60-degree sector has area 6(pi). Then 6(pi) = (60/360)(pi)(r squared) = (1/6)(pi)(r squared). Multiplying both sides by 6 and dividing by (pi) gives r squared = 36, so the radius is 6. The single proportion theta / 360 lets you solve for angle, arc, radius, or area interchangeably.

Real-world modeling. These formulas power the modeling questions (8-15% of the exam). Arc length measures a curved edge such as the crust of a pizza slice or a lane on a curved track; sector area measures a wedge such as a slice of pie, a windshield-wiper sweep, or the ground watered by a rotating sprinkler. If a sprinkler waters a 90-degree wedge of radius 20 feet, the watered area is (90/360)(pi)(20 squared) = 100(pi) square feet.

Finding the angle from two measurements. A pie of radius 9 inches is cut so that one slice has an arc-length (crust) of 3(pi) inches. Then 3(pi) = (theta/360)(2)(pi)(9) = (theta/360)(18(pi)). Dividing gives theta/360 = 3/18 = 1/6, so the slice spans 60 degrees. Notice you never needed a decimal for (pi); it cancels because both sides carry the same factor.

Radian Measure

The Next Generation standard also defines radian measure. The length of the arc intercepted by a central angle is proportional to the radius, and the radian measure is that constant of proportionality: radians = arc length / radius. A full circle is 2(pi) radians = 360 degrees, so a half circle is (pi) radians and a quarter circle is (pi)/2 radians.

Convert between the two systems with these rules:

• Degrees to radians: multiply by (pi)/180. Example: 60 degrees = 60 * (pi)/180 = (pi)/3 radians.
• Radians to degrees: multiply by 180/(pi). Example: (pi)/4 radians = (pi)/4 * 180/(pi) = 45 degrees.

In radians the formulas streamline to arc length = r(theta) and sector area = (1/2)r squared (theta), with theta in radians. For instance, a central angle of (pi)/2 radians on a circle of radius 10 gives arc length = 10 * (pi)/2 = 5(pi), the same 5(pi) you get from the degree formula with 90 degrees. The two systems always agree because a radian is defined precisely so that arc length equals radius times angle.

A New York Scope Note

NYSED clarifies that on the Geometry Regents the emphasis is proportional reasoning in degrees, with the whole circle as 360 degrees; radians are defined as the constant of proportionality here, and the deeper radian work is developed in Algebra II. So for the Geometry exam, expect the degree-based proportion (theta/360) to do most of the lifting, while still knowing what a radian is and how to convert. In practice, if a problem gives you an angle in degrees, work in degrees; only convert to radians if the problem itself uses radian notation.

Common Traps

• Arc length vs. sector area: arc length uses the circumference (2(pi)r); sector area uses the area ((pi)r squared). Don't swap them.
• Exact vs. approximate: leave answers in terms of (pi) unless a problem explicitly says to round, and use the calculator's (pi) key when a decimal is required.
• Degree/radian mix-up: r(theta) and (1/2)r squared (theta) require theta in radians, while the theta/360 forms require degrees.