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Calculate theta

Start with arc length and radius

arc length = 0.7π, r = 3.5

Some questions give the arc length and radius, then ask for the central angle.

Detailed notes

Worked Example

Sometimes the question gives the arc length and radius, then asks for the angle. First find the minor angle, then decide whether the question wants the minor angle or the major angle.

Step 1: Identify the radius and the requested angle

The radius is 3.5 cm and the minor arc length is 0.7π. The question asks for the major angle y, so we first find the minor angle x.

r = 3.5 cm

minor arc length = 0.7π

y = 360° - x

Step 2: Substitute into the arc length formula

When the angle is in degrees, use the angle as a fraction of the full circumference.

arc length = 2πr × θ360

0.7π = 2π(3.5) × x360

0.7π = 7π × x360

Step 3: Solve for the minor angle

Cancel π from both sides, then undo the multiplication by 7 over 360.

0.7 = 7x360

0.7 × 360 = 7x

x = 0.7 × 3607 = 36°

Step 4: Convert to the major angle

The question asks for the major angle y, so subtract the minor angle from a full turn.

y = 360° - x

y = 360° - 36° = 324°