If the measure of the angle 𝐴 is 76 degrees and the radius of the circle equals three centimeters, find the length of the major arc 𝐵𝐶.
We should always start by identifying exactly what it is we already know about the diagram. We are given that the measure of the angle 𝐴 is 76 degrees. We are also told that the radius has a length of three centimeters. It is useful to include both of these radii. The reason for which will become clear in a moment.
Here, we need to recall an important circle theorem — that is the radius and the tangent meet at 90 degrees. This is useful as it will allow us to calculate some more angles. Angles in a quadrilateral add to 360 degrees. We can therefore subtract the known angles from 360 to find the measure of the angle at the center of the circle 𝑂. 360 minus 90 add 90 add 76 is 104 degrees.
The question however wants us to find the length of the major arc 𝐵𝐶. That is to say, it wants us to find the measure of the longest part of the circumference of the circle between 𝐵 and 𝐶. In order to do this, we need to find the side of the angle marked 𝜃. We know the angles around a point add up to 360 degrees. So 360 minus 104 is 256. 𝜃 is 256 degrees.
The formula for the arc length of a sector with an angle of 𝜃 radians is 𝑟 times 𝜃. We currently only know the size of 𝜃 in degrees. Let’s recall. To change degrees into radians, we multiply by 𝜋 over 180. Therefore, 𝜃 is 256 times 𝜋 over 180. In its simplest form, 𝜃 is 64 out of 65𝜋.
We now have a radius of three centimeters and an angle of 64 out of 65𝜋. The arc length is three times 64 out of 65𝜋. This is equivalent to 13.4 centimeters correct to three significant figures.