How Do You Calculate the Circumcircle Radius with Given Triangle Sides?

[primarygun]

For a triangle, 3 sides are given.
What's the radius of its circumcircle?
Are we able to get it without using cosine law or sine law or heron formula?

 

[CollectiveRocker]

What is a circumcircle?

 

[primarygun]

An outern circle .

 

[CollectiveRocker]

I'm really sorry, but I have no idea what you are talking about.

 

[primarygun]

A circle surrounds a triangle, the three vertexs of the triangle touches the circumference

 

[CollectiveRocker]

Oh, I apologize. The only way which I know of to get that is with the law of cosines.

 

[primarygun]

My teachers haven't taught us the cosine law, but this question is in the part of properties of circle. Moreover, he hasn't taught heron formula.

 

[Astronuc]

Try - http://mathworld.wolfram.com/Circumcircle.html

It gives the solution and explanation for a circumcircle and 3 points (vertices of a triangle).

 

[VietDao29]

Hi,
First, use the law of cosine:
$$a^{2} = b^{2} + c^{2} - 2bc\cos{A}$$
$$b^{2} = a^{2} + c^{2} - 2ac\cos{B}$$
$$c^{2} = a^{2} + b^{2} - 2ab\cos{C}$$
to find out cos of one of the angle in the triangle, then use $sin^{2}\theta + cos^{2}\theta = 1$ (In a triangle, $sin\theta$ is always positive) to find its sine and then use the law of sine to find out the radius of the circumcircle.
The law of sine is:
$$\frac{a}{\sin{A}} = \frac{b}{\sin{B}} = \frac{c}{\sin{C}} = 2 \times R$$
Where R is the radius of the circumcircle.
Hope you get it.
Viet Dao,

 

[primarygun]

No need sin law.
I know many methods, but currently I want the most simpliest one.
Anyway, thank you. :P