Video Transcript
Circle 𝑂 is divided into three sectors. Points 𝐴, 𝐵, and 𝐶 are on the circumference of the circle. Sector 𝐴𝑂𝐶 has an area of 12𝜋, and sector 𝐶𝑂𝐵 has an area of 18𝜋. If the radius of the circle is six, what is the measure of the central angle 𝐵𝑂𝐴 in degrees?
It is sensible to start this question by sketching the circle 𝑂. Points 𝐴, 𝐵, and 𝐶 lie on the circumference such that the circle is split into three sectors. We’re told that the area of sector 𝐴𝑂𝐶 is 12𝜋. And we’re also told that the area of sector 𝐶𝑂𝐵 is 18𝜋. The radius of the circle is equal to six. Therefore, the lengths 𝑂𝐴, 𝑂𝐵, and 𝑂𝐶 are all equal to six. We’ve been asked to calculate the central angle 𝐵𝑂𝐴 in degrees, labelled 𝜃 on the diagram.
The area of any circle can be calculated using the formula 𝜋𝑟 squared, where 𝑟 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 𝐴𝑂𝐶 and 𝐶𝑂𝐵, we can calculate the area of sector 𝐵𝑂𝐴 by subtracting these values from 36𝜋. The area of sector 𝐵𝑂𝐴 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 𝐵𝑂𝐴 equals six 𝜋.
The area of any sector can be calculated using the formula 𝜃 divided by 360 multiplied by 𝜋𝑟 squared. 𝜋𝑟 squared is the area of the whole circle, and 𝜃 is the angle of the sector. We’ve already worked out that the area of the sector 𝐵𝑂𝐴 is equal to six 𝜋. The area of the whole circle, 𝜋𝑟 squared, is equal to 36𝜋. This means that six 𝜋 is equal to 𝜃 divided by 360 multiplied by 36𝜋.
We can divide both sides of this equation by 𝜋. This leaves us with six is equal to 𝜃 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 𝜃 divided by 10. Six multiplied by 10 is equal to 60. Therefore, the angle 𝜃 equals 60 degrees.
We can therefore conclude that if sector 𝐴𝑂𝐶 has an area of 12𝜋, sector 𝐶𝑂𝐵 has an area of 18𝜋, and the radius of the circle is six, then the central angle 𝐵𝑂𝐴 will be equal to 60 degrees. We could also use this method to calculate the central angle of sectors 𝐴𝑂𝐶 and 𝐶𝑂𝐵.
