### Video Transcript

Circle π is divided into three sectors. Points π΄, π΅, and πΆ are on the circumference of the circle. Sector π΄ππΆ has an area of 12π, and sector πΆππ΅ has an area of 18π. If the radius of the circle is six, what is the measure of the central angle π΅ππ΄ in degrees?

It is sensible to start this question by sketching the circle π. Points π΄, π΅, and πΆ lie on the circumference such that the circle is split into three sectors. Weβre told that the area of sector π΄ππΆ is 12π. And weβre also told that the area of sector πΆππ΅ is 18π. The radius of the circle is equal to six. Therefore, the lengths ππ΄, ππ΅, and ππΆ are all equal to six. Weβve been asked to calculate the central angle π΅ππ΄ in degrees, labelled π on the diagram.

The area of any circle can be calculated using the formula ππ squared, where π is the radius of the circle. In our question, we need to multiply π by six squared. Six squared is equal to 36. Therefore, we need to multiply π by 36. This can be written as 36π.

As we already know the areas of sectors π΄ππΆ and πΆππ΅, we can calculate the area of sector π΅ππ΄ by subtracting these values from 36π. The area of sector π΅ππ΄ is equal to 36π minus 12π plus 18π. 12π plus 18π is equal to 30π. Therefore, we need to subtract 30π from 36π. This gives us six π. The area of sector π΅ππ΄ equals six π.

The area of any sector can be calculated using the formula π divided by 360 multiplied by ππ squared. ππ squared is the area of the whole circle, and π is the angle of the sector. Weβve already worked out that the area of the sector π΅ππ΄ is equal to six π. The area of the whole circle, ππ squared, is equal to 36π. This means that six π is equal to π divided by 360 multiplied by 36π.

We can divide both sides of this equation by π. This leaves us with six is equal to π divided by 360 multiplied by 36. The numerator and denominator of the right-hand side of this equation can be divided by 36. 36 divided by 36 is equal to one, and 360 divided by 36 is equal to 10. This means that six is equal to π divided by 10. Six multiplied by 10 is equal to 60. Therefore, the angle π equals 60 degrees.

We can therefore conclude that if sector π΄ππΆ has an area of 12π, sector πΆππ΅ has an area of 18π, and the radius of the circle is six, then the central angle π΅ππ΄ will be equal to 60 degrees. We could also use this method to calculate the central angle of sectors π΄ππΆ and πΆππ΅.