The Missing Central Angle

Part of the Circles section of Coddy's Geometry journey. Lesson 19 of 62.

Sometimes the central angle of an arc or a sector is not marked. Then find it first with the angle rules you already know, and use the circle rule second. Radii meet at the center like rays at a point, so angles on a straight line add to 180° and angles all the way round add to 360°.

$Diameter AC and radius OB of 3: angle AOB is 120 degrees, so arc BC is pi$

Here AC is a diameter, so the angles AOB and BOC sit on a straight line: ∠BOC = 180° - 120° = 60°. The arc from B to C is then 60/360 of 2 × π × 3 = 6π, which is π. Two steps: Angles on a straight line, then Arc length.

Challenge

Three radii of circle O split the angle around the center, and two of the angles are marked. Find the area of the sector OAC, exact in terms of π, then press Check.

Try it yourself

$Find the marked part of the figure$

Find the marked part of the figure

Test yourself

This lesson includes a short quiz. Start the lesson to answer it and track your progress.

All lessons in Circles

1Parts of a Circle

Center and Radius Chords and Diameters All Radii Are Equal Diameter Is Two Radii Recap: Parts of a Circle

4Arcs and Sectors

Arc Length Area of a Sector Find the Central AngleThe Missing Central AngleRecap: Arcs and Sectors

7Angles in a Semicircle

Thales' Theorem The Right Angle Inside Is That Enough?Semicircle With Letters Recap: Semicircles

10Tangents

What a Tangent Is Tangent and Radius Tangent Lengths Two Tangents From a Point Tangents With Letters Recap: Tangents