Circumcentre of an equilateral triangle

In summary, the distance from one corner of an equilateral triangle of sides a to the circumcentre can be found by dropping a perpendicular from the circumcenter to a side, creating a 30-60-90 triangle. The radius of the circumcircle is then r = a/sqrt3, which can also be found using the Pythagorean theorem by subtracting the shorter leg of the 30-60-90 triangle from the distance computed. Trigonometry is not necessary for this solution.
  • #1
Chronos000
80
0
1. Homework Statement [

what is the distance from one corner of an equilateral triangle of sides a to the circumcentre?


I can figure out the length from one corner to the opposite side to be sqrt3*a/2 but that's about it. I just can't see how to do this.

thanks
 
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  • #2
Drop a perpendicular from the circumcenter to a side. That gives you a 30-60-90 triangle whose hypotenuse is the radius you are looking for.
 
  • #3
thanks, the answer is a/sqrt3, I thought there was some way to do this without cos or sin but perhaps not
 
  • #4
Chronos000 said:
thanks, the answer is a/sqrt3, I thought there was some way to do this without cos or sin but perhaps not

You could just use the pythagorean theorem if you really want to. If you call the circumcircle radius r, then the a*sqrt(3)/2 distance you computed minus r is the shorter leg of your 30-60-90 triangle. Now use the pythagorean theorem on it to solve for r. You don't really NEED the trig.
 

FAQ: Circumcentre of an equilateral triangle

What is the definition of the circumcentre of an equilateral triangle?

The circumcentre of an equilateral triangle is the point where the perpendicular bisectors of all three sides intersect. It is equidistant from all three vertices of the triangle.

How is the circumcentre of an equilateral triangle calculated?

The circumcentre can be calculated by finding the point where the perpendicular bisectors of the sides intersect. This can be done using various geometric constructions or by using the coordinates of the vertices of the triangle.

What is the significance of the circumcentre in an equilateral triangle?

The circumcentre is important because it is the centre of the circumcircle, which is a circle that passes through all three vertices of the triangle. It is also the point where the three angle bisectors of the triangle intersect.

How does the circumcentre relate to the centroid of an equilateral triangle?

The centroid of an equilateral triangle is the point where the three medians of the triangle intersect. The medians and the perpendicular bisectors are concurrent at the circumcentre of the triangle. Additionally, the distance between the circumcentre and the centroid is equal to two-thirds of the distance between the circumcentre and any vertex of the triangle.

Can the circumcentre of an equilateral triangle be located outside of the triangle?

No, the circumcentre will always be located inside the triangle or on one of its sides. This is because the perpendicular bisectors of the sides can only intersect inside or on the boundary of the triangle. If the triangle is obtuse, the circumcentre will be located on the side opposite the obtuse angle.

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