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
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 size 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
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.