Problem Solving

In the figure above, equilateral triangle ABC is inscribed in the circle. If the length of arc ABC is 24, what is the approximate diameter of the circle?

A. 5
B. 8
C. 11
D. 15
E. 19

Equilateral triangle ABC is inscribed in a circle (points ABC are on the circle). IF the length of arc ABC is 24, what is the approximate diameter of the circle

A) 5
B) 8
C) 11
D) 15
E) 19

A central angle is formed by two radii.
An inscribed angle is formed by two chords.
When an inscribed angle and a central angle intercept the same arc on the circle, the degree measurement of the inscribed angle is 1/2 the degree measurement of the central angle:



Circles display the following proportionality:

(Central Angle)/360 = (intercepted arc length)/circumference = (sector area)/(circle area)

Since 120/360 = 1/3, the intercepted arc in the circle above is 1/3 the circumference of the circle. The sector enclosed by the two radii is 1/3 the area of the entire circle.

Now here's a drawing of the problem above:



Let c = circumference.
Since angle A is 60 degrees, the corresponding central angle is 120 degrees. Since 120/360 = 1/3, arc BC = (1/3)c.
Using similar logic, arc AB = (1/3)c.
Thus, arc ABC = (2/3)c.
Since arc ABC = 24:
24 = 2/3c
c = 36.

Thus, �d = 36.
d ≈ 11.

The correct answer is C.