SAT® · Math: Geometry and Trigonometry
Arc Length and Sector Area
Arc length and sector area are both just fractions of the whole circle: the central angle tells you what fraction of 360 degrees you have, and that same fraction applies to both the circumference and the area.
A circle's rim and its interior both get cut by a central angle in the same proportion. A 60 degree angle is 60/360, or one sixth, of the way around the circle. That one sixth applies equally to the circumference and to the area, because the angle is slicing the circle like a pizza cutter — the crust and the filling shrink by the same fraction. That's the intuition behind both formulas: arc length = (angle/360) x 2 π r, and sector area = (angle/360) x π r2.
In radians the relationship gets even simpler, because a radian is defined so that arc length = r x theta directly, no fraction needed. If theta is 2 radians and the arc is 14 units, then r = 14/2 = 7, straight from that formula. Sector area in radians works the same way: area = (1/2) r2 theta. Degrees and radians are just two different rulers for the same angle, so always check which one a problem is using before picking a formula.
This topic sits right alongside circle equations and right-triangle trig in the geometry portion, and it leans on the same core fact every circle problem does: everything scales off the radius and the angle. Once arc length and sector area feel like 'fraction of the whole circle,' they stop being separate formulas to memorize and become one idea applied twice — once to the rim, once to the interior.
Key terms
- Central angle
- An angle whose vertex sits at the center of the circle and whose sides are two radii.
- Arc length
- The distance along the curved rim of the circle between the two endpoints of a central angle.
- Sector
- The pie-slice region bounded by a central angle and the arc it cuts off.
- Radian
- A unit of angle defined so that an angle of 1 radian cuts an arc equal in length to the radius.
Watch it explained
- How it worksHow does a fraction of 360 degrees turn circumference into arc length and area into sector area?
- Worked exampleFind the arc cut off by a 60 degree central angle in a circle of radius 9.
- Worked exampleFind the area of a sector with a 45 degree central angle in a circle of radius 8.
- Worked exampleFind the radius of a circle in which a central angle of 2 radians cuts an arc of length 14.
- Common mistakeWhy doesn't the SAT print arc length or sector area anywhere on the formula sheet it gives you?
Check yourself
End of topic.