# Circumscribed Circle Calculator

Table of contents

What is a circumscribed circle?Formula for the circumradius of a triangleOther formulas related to circumcirclesHow to use this circumscribed circle calculator?How do I circumscribe a circle about a triangle?FAQsOur 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!

## What is a circumscribed circle?

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.

💡 **Every triangle has a circumcircle.** However, this is not true for all other polygons! For instance, among quadrilaterals, all **rectangles** (including squares, of course) have circumscribed circles, but **no non-square rhombus** has the circumscribed circle.

## Formula for the circumradius of a triangle

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.

## How to use this circumscribed circle calculator?

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!

## How do I circumscribe a circle about a triangle?

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.

### How do I calculate the radius of the circumscribed circle?

To determine the circumradius of an arbitrary triangle:

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

### What is the circumradius of an equilateral triangle?

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.

### What is the circumradius of a right triangle?

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**.