video explanation

text explanation

there are two ways to solve this problem. the first involves recognizing that triangle oac is an equilateral triangle. since side oa and side oc are both radii of the circle, that means their lengths are 6. equal sides in a triangle have equal angles opposite those sides. since the sum of all the angles in a triangle must equal 180, we know that the other two angles are also 60 degrees:

x + x + 60 = 180
2x + 60 = 180
2x = 120
x = 60

since all three angles are equal, all three side lengths must be equal as well, making side ac also 6.

the shortest distance between two points is a straight line. thus, the shortest distance between point a and point c is the straight line of distance 6 between them. arc abc, on the other hand, is curved. this means that it travels a longer distance. whatever that distance is, it must be greater than 6. therefore, column a is larger and (a) is the correct answer.

the other way to solve this problem is to recognize that arc abc represents a fraction of the circle's total circumference. a circle's circumference is defined as:

the ratio of an arc of a circle and that circle's circumference is proportional to the ratio of the central angle that corresponds to the arc and the total degrees (360) that make up that circle. in other words, we can find the length of the arc by using this equation:

in this problem, since the central angle is 60 degrees, this means that the arc represents of the circumference:

since the radius is 6, arc abc is equal to 2pi.  pi is roughly 3.14, so the length of arc abc would be roughly 6.28, which is greater than column b's 6.