How to Find the Radius of the Circumscribed Circle

Geometry Lessons

The short answer

R = abc / 4A

Multiply the three sides, then divide by four times the area. No coordinates, no construction and no angles required.

The circumradius is the distance from the circumcenter out to any corner, and it is the radius of the one circle passing through all three. What makes it worth its own page is that you can get the length without ever locating the centre.

Two routes, and when to use each

If you have the three side lengths, use R equals abc over 4A. Get the area with Heron from those same sides and the whole calculation runs off three numbers.

If you have coordinates instead, you can find the circumcenter first and then measure its distance to any one corner. That works, but it is more effort, and the side length route is usually faster even when coordinates are what you started with.

There is a third route worth knowing. The sine rule says a over sin A equals 2R, so any side divided by twice the sine of its opposite angle also gives the circumradius. That is the one to reach for when an angle is what you have.

1

Multiply the three sides

Order does not matter. This is the abc in the numerator.

2

Find the area and quadruple it

Heron works from the same three sides if you have nothing else.

3

Divide

The result is the circumradius, and it must exceed every half side length.

A worked example

Take the 13, 14, 15 triangle again. Multiplying the sides gives 13 times 14 times 15, which is 2730. The area from Heron is 84, so four times the area is 336. Dividing, R is 2730 over 336, which is 8.125.

Sanity check that against the shape. The longest side is 15, so the circle has to be wide enough to span it, meaning R must exceed 7.5. It comes out at 8.125, comfortably above and not wildly so, which is what you would expect for a triangle this close to equilateral.

A 13, 14, 15 triangle with a circle drawn through all three corners, its radius of 8.125 marked from the centre to a vertex, beside the working multiplying the three sides and dividing by four times the area.
Every corner sits exactly on the circle, which is the defining property. Note how much of the circle lies outside the triangle, since the two only meet at three points.

Four times the area, not the area

The 4 in the denominator gets dropped more often than any other part of this formula, and the result is four times too large. Checking against half the longest side catches it instantly.

abc / Agives2730 / 8432.5

four times too large, and far wider than the triangle needs

abc / 4Agives2730 / 3368.125

comfortably above half the longest side, as it should be

The right triangle shortcut

On a right triangle the circumradius is exactly half the hypotenuse, every time. Check it on the 3, 4, 5: the formula gives 60 over 24, which is 2.5, and half of 5 is also 2.5.

The reason is that the circumcenter of a right triangle sits on the hypotenuse midpoint, so the hypotenuse is a diameter of the circle. If you know a triangle is right angled, halve the hypotenuse and skip the formula entirely.

How it compares with the inradius

The circumradius is always larger, and by a defined margin. Euler showed that R is at least twice r for every triangle, with the two only reaching that limit on an equilateral one. For the 13, 14, 15 triangle, R is 8.125 and r is 4, which is just past twice, matching how close to equilateral it sits.

That relationship is a quick plausibility check. If you calculate both and find R smaller than twice r, at least one of them is wrong.

Mistakes that catch people out

  • Forgetting the 4 in the denominator. The answer comes out four times too large.
  • Getting a radius smaller than half the longest side. Impossible, since the circle has to span that side.
  • Confusing the circumradius with the circumcenter. One is a length, the other a point.
  • Using the sine rule without doubling. The rule gives a over sin A equal to 2R, not R.
  • Reaching for the full formula on a right triangle. Half the hypotenuse is the same answer with no arithmetic.

Practice questions

  1. Sides of 3, 4 and 5. Find the circumradius.
  2. Sides of 5, 12 and 13. Find the circumradius.
  3. Sides of 6, 6 and 6. Find the circumradius.
  4. Sides of 7, 8 and 9. Find the circumradius.
  5. A right triangle has a hypotenuse of 26. Find the circumradius.
Show the answers
  1. A = 6, so R = 60 / 24 = 2.5, which is half the hypotenuse
  2. A = 30, so R = 780 / 120 = 6.5, again half the hypotenuse
  3. A = 15.588, so R = 216 / 62.35 = 3.464
  4. A = 26.833, so R = 504 / 107.33 = 4.696
  5. half of 26, so 13, with no further calculation needed

Common questions

How do I find the radius of the circumscribed circle?

Multiply the three sides together and divide by four times the area. That gives R directly from the side lengths.

What is the circumscribed circle?

The one circle passing through all three corners of the triangle. It is also called the circumcircle, and its centre is the circumcenter.

Do I need the coordinates?

No. The formula works from the three side lengths alone. Coordinates are only needed if you also want the location of the centre.

What is the circumradius of a right triangle?

Exactly half the hypotenuse. On a 3, 4, 5 triangle it is 2.5, and the centre sits on the hypotenuse midpoint.

Is the circumradius always bigger than the inradius?

Yes, on every triangle. Euler proved the circumradius is at least twice the inradius, with equality only for an equilateral triangle.

What is the difference between the circumcenter and the circumradius?

The circumcenter is a point, given as coordinates. The circumradius is a length, measured from that point out to any corner.

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