EQN-04-04

Circles — Arcs, Sectors & Angles

Take fractions of a circle for arc length and sector area, and use the central-vs-inscribed angle rule.

~6 min read•Difficulty 3/5

circumference = 2 π r area = π r^{2} full angle = 360 °

A central angle of $θ$ degrees cuts off the fraction $θ /360$ of the circle.

arc length = (θ /360) \times 2 π r

sector area = (θ /360) \times π r^{2}

A 90° sector is one quarter; a 60° sector is one sixth.

An inscribed angle is half the central angle that subtends the same arc.

central angle = 2 \times inscribed angle (same arc)

A special case: an angle inscribed in a semicircle (diameter as one side) is always 90°.

A tangent touches the circle at one point and is perpendicular to the radius drawn to that point — a frequent right-angle source.

Diagram

A 60° sector is $60/360 = 1/6$ of the whole circle.

Worked examples

Example 1 (difficulty 2/5)

A circle has radius 6. What is the area of a 60° sector? (Leave in terms of

π .)

◎ Predict the answer first:

Example 2 (difficulty 2/5)

A central angle measures 80°. What is the inscribed angle subtending the same arc?

◎ Predict the answer first:

Turn the central angle into the fraction $θ /360$ first, then multiply by circumference or area.
Inscribed angle $=$ half the central angle; an angle in a semicircle is 90°.
A radius meeting a tangent makes a right angle — look for hidden right triangles.

Using

θ /180

instead of

θ /360

for a sector

Why: Confusing the 180° of a straight line with the 360° of a full turn.

Fix: A full circle is 360°, so the fraction is

θ /360.

Setting the inscribed angle equal to the central angle

Why: They share the same arc, so they feel equal.

Fix: The inscribed angle is half the central angle.