Find the Central Angle

Part of the Circles section of Coddy's Geometry journey. Lesson 18 of 62.

The arc length rule also runs backwards. If you know the arc and the radius, the rule gives an equation for the central angle. The board writes the line with both sides already divided by π: cancel π over π and the line has only numbers and x.

$Circle O with radius 3, an arc of 2 pi and central angle x$

Here the arc is 2π and the radius is 3. The rule says 2π = x/360 × 2 × π × 3. Divided by π and cancelled, 2 = x/360 × 6. Clear the 360: 720 = 6x, so x = 120. Check it: 120° is a third of the turn, and a third of 6π is 2π.

Tap the central angle, pick Arc length, and solve the line the board writes.

Challenge

OA and OB are radii of circle O, and the arc from A to B has the length given. Tap the central angle, pick the rule that links it to the arc, and solve the equation the board writes for x.

Then press Check.

Try it yourself

$Use the figure to find x$