Sun of Reuleaux

Geometry Level 5

A Reuleaux triangle is defined in the following way:

  • The 3 apex points A, B, and C are one unit from one another
  • Each pair of apex points are connected by the arc of a circle of radius 1 whose center is the third apex
  • We end up with the yellow region as shown above

What is the radius of the Reuleaux triangle's circumcircle as defined by apex points A, B, and C?

Give your answer to 5 decimal places.


The answer is 0.57735.

W Rose by W Rose , 63, USA

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3 solutions

Let the circumcentre of △ A B C \triangle ABC be O O and the circumradius be r r . Connect O A OA and O B OB . Notice that O A = O B = r OA=OB=r . Also notice that ∠ A O B = 12 0 ∘ \angle AOB=120^\circ by symmetry. Applying the cosine rule to △ A O B \triangle AOB , we find that 2 r 2 − 2 r 2 cos ⁡ ( 12 0 ∘ ) = 1 2r^2-2r^2\cos(120^\circ)=1 Solving, we find r r to be 0.57735.

3 Helpful 0 Interesting 1 Brilliant 0 Confused

+1 Your solution is a great read.

Note: There is a slight typo in your last equation. It should be 12 0 ∘ 120 ^ \circ .

Thanks for contributing and helping other members aspire to be like you!

Calvin Lin Staff - 4 years, 7 months ago

Sir,the incircle data is superfluous,the triangle is equilateral with side length 1,so circumradius is 1/(2*(sin60°)) and hence the ans,it would have been better to ask for the radius of the bigger circle or the triangle side length which circumscribe any of this

3 Helpful 0 Interesting 0 Brilliant 0 Confused

Altitude of equilateral triangle side s, =s * Sin60.=3/2 R.
So R=2/3
3 \sqrt3 \2 s=1/ 3 \sqrt3 s=.57735*s.
But s=1, so R=0.57735.

2 Helpful 0 Interesting 0 Brilliant 0 Confused

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