Circumscribed Circle Calculator

Our circumscribed circle calculator will become your best friend if you deal a lot with circumcircles, i.e., circles circumscribed about a triangle. Read on to learn more about this important geometric topic!

In what follows, we'll cover the following topics:

• What is the circumscribed circle? Does it always exist?
• How to circumscribe a circle about a triangle?
• How to calculate the circumradius of a triangle?

And a few more! Let's go!

The circumscribed circle (a.k.a. the circumcircle) of a given triangle is a circle that passes through all three vertices of this triangle.

The radius of the circumcircle is called the circumradius of the triangle and the center of the circumcircle is called the circumcenter. The circumcenter coincides with the point where the perpendicular bisectors of the triangle's sides intersect. Discover more with Omni's circumcenter calculator.

The formula for the radius of the circumcircle of a triangle with sides $a$, $b$, and $c$ reads:

where $A$ is the area of the triangle. We can compute $A$ using Heron's formula:

where $S$ is a half of the triangle's perimeter: $S = \frac{1}{2}(a+b+c)$. If you're not familiar with it, visit our Heron's formula calculator for a quick introduction. Alternatively, plug the triangle sides into our circumradius calculator and find $R$ in no time without struggling to find the area first.

When we know how to calculate circumradius of a triangle, many paths open! We can easily determine several values related to the circumscribed circle:

• The area of the circumcircle:

• The diameter of the circumcircle:

• The circumference of the circumcircle:

as well as the ratio of the areas of the circumcircle and of the triangle:

Operating this circumscribed circle calculator is easy as pie. Just plug the lengths of the sides of your triangle into the fields labeled as a, b, and c, and let the magic happen! All the other fields get filled in automatically, which means the radius, circumference, diameter, and area of the circumcircle of your triangle are already there!

If you need extra info, including the area of the triangle and the ratio of the two areas, there's several bonus fields in the Additional results box. Enjoy!

The construction of the circumcircle boils down to the construction of its center, i.e., of the circumcenter of our triangle. (Once you have the center, you just draw the circle through all vertices of the triangle and that's it.) To find the circumcenter, draw any two perpendicular bisectors of the triangle sides. The point where they intersect is the circumcenter you need.

To determine the circumradius of an arbitrary triangle:

1. Compute the area of the triangle. If you only know its sides, use Heron's formula.
2. Multiply the area by 4.
3. Multiply together all the side lengths: a × b × c.
4. Divide the result from Step 3 by that of Step 2.
5. Well done! You've just found the radius of the circumscribed circle! You can verify it using an online circumscribed circle calculator.
6. To summarize, the circumradius formula reads: R = a × b × c / (4 × Area).

The circumradius of an equilateral triangle with side a is equal to a / √3. The circumcenter coincides with the orthocenter, which is the point where the three altitudes of the triangle intersect.

For right triangles, the circumradius is exceptionally easy to determine: it's equal to half of the length of the hypotenuse (which is the triangle's longest side). The circumcenter is the mid-point of the hypotenuse.