IMO D

Since it is an equilateral triangle, the three chords/sides are equal. So, the angles of the arcs (AB, BC, CA) are all equal. So, its 360 degrees/3 arcs = 120 degrees/arc

We know that the length of ABC = 24. The total angle of arc ABC is 240 (Because there are 2 arcs out of 3).

Apply the formula (arc length)

Arc length = (Angle of Sector/Angle of circle) * 2 * pi * radius

24 = 240/360 * 2 * pi * radius [We need to find the radius]

radius = 7.63 (approx.)
Therefore, diameter = 7.63 * 2 = 15.27 (approx) = 15.

Please post the OA

OA is D. But I could not get it how did you get 240 degree angle.

papgust wrote:IMO D

Since it is an equilateral triangle, the three chords/sides are equal. So, the angles of the arcs (AB, BC, CA) are all equal. So, its 360 degrees/3 arcs = 120 degrees/arc

We know that the length of ABC = 24. The total angle of arc ABC is 240 (Because there are 2 arcs out of 3).

Apply the formula (arc length)

Arc length = (Angle of Sector/Angle of circle) * 2 * pi * radius

24 = 240/360 * 2 * pi * radius [We need to find the radius]

radius = 7.63 (approx.)
Therefore, diameter = 7.63 * 2 = 15.27 (approx) = 15.

Please post the OA

Please re-check your calculation, it should be 11 not 15...

Thanks Giorgio. You are right. It should be 11.

hmboy, can you check the OA again? I double-checked my answer and looks like it must be 11.

Here's the diagram to explain why it is 240? Arc AB + Arc BC = 240 (Because, the question gives perimeter only for Arc ABC and NOT Arc AC)

Can you please tell me how you identify the angle of Arc ABC , iam still not clear even after this diagram

I hope now u'll b little clear.
its an equilateral triangle. angle subtended @ O(i.e. center of the circle) will b double of the angle subtended @ b. its a property in geometry( inscribed angle). Now since angle aoc is 120 degree. we get major angle O as 240 (360-120). hence answer.

acc to me it should b 11.

i thnk image is nt clear

There are two approaches to take.
If you know the central angle of any circle, and know its arc length then u can use

Arc length = radius x central angle ( measured in RADIANS)

12= R x 120 x pi/180 ( to change angles to radians) ( Figuring how to get the arc length 12 is half the problem)
You get it b/c Angle C is 60 degree so angle of arc AB must be 2 x 60 -=120. If ABC is 24 and if Arc aB = ARC BC ( you know from equalateral triangle) then Arc AB has a measure of 12. So with Arc and central angle known just use the above formula and solve for R.

R = 5. something, and when u multiply by 2 you get approx 11.

If you don't like to convert to RADIANS then use


Central Angle/360= Arc Length/ Circumference ( 2PiR). You will still have to figure the arc length of the CENTRAL ANGLE. You you can see R in the circumference formula which u can solve for.
You get the same 11.
120/360 =12/2piR

R = 5. somthing.

The real meat in this question is to FIGURE OUT the CENTRAL ANGLE and its ARC LENGth. Once you get that you should be done.
Choose C.