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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!

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.

Circumscribed circle on a triangle.
Wmheric, CC BY-SA 3.0, via Wikimedia Commons

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 aa, bb, and cc reads:

R=abc4A\small R = \frac{abc}{4A}

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

A=S(Sa)(Sb)(Sc)\small A = \sqrt{S(S-a)(S-b)(S-c)}

where SS is a half of the triangle's perimeter: S=12(a+b+c)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 RR 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:
Ac=πR2\small \qquad A_c = \pi R^2
  • The diameter of the circumcircle:
d=12R\small \qquad d = \tfrac 12 R
C=2πR\small \qquad C = 2\pi R

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

AcAt=πR2S(Sa)(Sb)(Sc)\small \frac {A_c}{A_t} = \frac{\pi R^2}{ \sqrt{S(S-a)(S-b)(S-c)}}

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.

FAQ

How do I calculate the radius of the circumscribed circle?

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).

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.

Anna Szczepanek, PhD
Triangle's sides
a
in
b
in
c
in
Circumcircle
Radius (R)
in
Diameter
in
Circumference
in
Area
in²
Trigger
Additional results
Triangle perimeter
in
Triangle area
in²
Ratio of areas
Inradius (r)
in
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