From my research, the image below belongs to the question. From what is seen in the image, there are 2 arcs already drawn with respect to the new vertex. The third arc corresponds to the point where the line DM is made. This line is similar to line BC. So at the point where the second and third arcs would meet, it would be a point similar to a point on line AB.
In a full rotation, 360°, a point on the edge of a gear would travel a distance equal to its circumference. The circumference of a circle is:
C=2πr, and since we are only traveling 150°, we need to set up an appropriate ratio for circumference to the distance the point travels:
d/C=150/360
d=5C/12 and since C=2πr
d=10πr/12
d=5πr/12, and since r=4in
d=5*4π/12 in
d=20π/12 in
d=5π/3 in
d≈5.24 in (to nearest hundredth of an inch)
Area is the space inside a shape
