# Video: US-SAT03S4-Q34-579170942169

The given figure shows a circle centered at the origin. If the arc 𝐵𝐶𝐷 has a length of (27𝜋 − 4)/3, find the measure of ∠𝐴𝑂𝐵 in radians.

04:29

### Video Transcript

The given figure shows a circle centered at the origin. If the arc 𝐵𝐶𝐷 has a length of 27𝜋 minus four over three, find the measure of angle 𝐴𝑂𝐵 in radians.

This is the arc 𝐵𝐶𝐷. And this is the angle we’re interested in, 𝐴𝑂𝐵. We know that the arc length is 27𝜋 minus four over three. But the information we’ll need to solve this problem is the angle measure that creates this arc. We know that an arc length 𝑠 is equal to the angle that creates that arc times the radius. And so if we wanted to calculate this angle 𝜃, we would take the arc length and divide it by the radius. And because we know that the circle is centered at the origin and that 𝐶 is located at six, zero, we can say that the radius is six units. And that means 𝜃 will be equal to the arc length 27𝜋 minus four divided by three, divided by the radius of six. We can rewrite that to say 27𝜋 minus four over three times one-sixth. Dividing by six is the same thing as multiplying by one over six. And so we’ll multiply those denominators together and the numerators together. 27𝜋 minus four over 18 is the angle measure that creates the arc 𝐵𝐶𝐷.