Circumscribed Circle Calculator
Enter a triangle's sides or angles, or a regular polygon's side count and length, to find the circumscribed circle instantly.
⭕ What is a Circumscribed Circle?
A circumscribed circle (also called a circumcircle) is a circle that passes through all vertices of a polygon. The center of this circle is called the circumcenter, and the distance from the circumcenter to any vertex is the circumradius (R). Every triangle has a unique circumscribed circle, as does every regular polygon. The circumcircle is the smallest circle that can completely contain the polygon with all vertices lying exactly on its boundary.
Circumscribed circles appear throughout engineering and design. In mechanical engineering, the circumscribed circle of a hexagonal bolt head tells you the minimum wrench socket size or the clearance radius needed for the bolt. In structural engineering, the circumcircle of a triangular truss cross-section defines the minimum circular pipe or column that can house the truss. In land surveying, the circumcircle of a triangular plot defines the smallest circular boundary that encloses all three corners. Geodesic domes, crystal lattices, and antenna arrays all use circumscribed circle calculations to determine spacing and coverage radii.
There are three common ways to compute the circumradius of a triangle. The first uses all three side lengths with Heron's formula for the area: R = abc divided by (4K). The second uses the Law of Sines directly: R = a divided by (2 sin A), where A is the angle opposite side a. These two are mathematically equivalent but the second is faster when you already know a side-angle pair. For regular polygons with n equal sides of length s, the circumradius is R = s divided by (2 sin(pi/n)).
A key theorem (Thales' theorem) states that for any right triangle, the hypotenuse is a diameter of the circumscribed circle, making the circumradius exactly half the hypotenuse. For obtuse triangles, the circumcenter lies outside the triangle. For acute triangles, it lies inside. This calculator handles all triangle types and regular polygons from a triangle (3 sides) to a dodecagon (12 sides), returning the circumradius, circumcircle area, circumference, and either the triangle area or polygon area.
📐 Formulas
Triangle (3 sides) R = abc ÷ (4K)
a, b, c = the three side lengths of the triangle
K = triangle area via Heron's formula: K = √(s(s−a)(s−b)(s−c)), where s = (a+b+c)/2
Example: 3-4-5 triangle: K = 6; R = (3×4×5) / (4×6) = 60/24 = 2.5 units
Triangle (Law of Sines) R = a ÷ (2 sin A)
a = any side of the triangle
A = the angle directly opposite side a (in degrees)
Note: For a right triangle (A = 90°), sin A = 1, so R = a/2 = hypotenuse/2
Example: side a = 10, angle A = 30°; R = 10 / (2 × 0.5) = 10 units
Regular Polygon R = s ÷ (2 sin(π/n))
n = number of sides (integer, 3 or more)
s = side length (all sides equal for a regular polygon)
Polygon area: K = (n × s²) / (4 tan(π/n))
Example: Regular hexagon n=6, s=5; R = 5 / (2 sin(30°)) = 5 / 1 = 5 units
📖 How to Use This Calculator
Steps
1
Select the shape type - Click Triangle by Sides, Triangle by Angle, or Regular Polygon. Each tab shows the correct inputs for that calculation method.
2
Enter the dimensions - For Triangle by Sides, enter all three side lengths. For Triangle by Angle, enter one side and the angle directly opposite it (in degrees). For Regular Polygon, enter the number of sides (3 to 12) and the side length.
3
Click Calculate - Press Calculate to get the circumradius, the circumcircle area (pi times R squared), the circumference (2 pi times R), and the triangle or polygon area as a secondary metric.
4
Read the results - Circumradius R is the primary result. The circumcircle area and circumference describe the enclosing circle. Use the share buttons to copy or link the result for later use.
💡 Example Calculations
Example 1 - Classic 3-4-5 Right Triangle
Find the circumscribed circle of a right triangle with legs 3 and 4 and hypotenuse 5
1
Sides a = 3, b = 4, c = 5. Compute semiperimeter: s = (3+4+5)/2 = 6.
2
Heron's area: K = √(6 × 3 × 2 × 1) = √36 = 6 sq units.
3
Circumradius: R = (3 × 4 × 5) / (4 × 6) = 60 / 24 = 2.5 units. Verify: R = hypotenuse / 2 = 5 / 2 = 2.5 (Thales' theorem).
Circumradius = 2.5 units | Circumcircle Area = 19.635 sq units
Try this example →Example 2 - Equilateral Triangle (Law of Sines method)
An equilateral triangle has side 8 m. Use the Law of Sines to find the circumradius.
1
All angles = 60 degrees. Side a = 8, angle A = 60 degrees.
2
R = a / (2 sin A) = 8 / (2 × sin 60°) = 8 / (2 × 0.866) = 8 / 1.732 = 4.619 m.
3
Circumference = 2 π × 4.619 = 29.02 m. Circumcircle area = π × 4.619² = 67.02 sq m.
Circumradius = 4.619 m | Circumcircle Area = 67.02 sq m
Try this example →Example 3 - Regular Hexagon Bolt Head
A hexagonal bolt head has a 24 mm across-flats measurement. Each side of the hexagon is 24/sqrt(3) = 13.856 mm. Find the circumradius (minimum socket clearance radius).
1
Regular hexagon: n = 6, side s = 13.856 mm.
2
R = s / (2 sin(pi/6)) = 13.856 / (2 × 0.5) = 13.856 / 1 = 13.856 mm.
3
A socket must have inner radius at least 13.856 mm (a 27.7 mm socket) to fit over this bolt. Circumcircle area = π × 13.856² = 602.9 sq mm.
Circumradius = 13.856 mm | Circumference = 87.07 mm
Try this example →Example 4 - Obtuse Triangle by Three Sides
A triangular land survey plot has sides 7 m, 9 m, and 14 m. Find the circumscribed circle.
1
Sides: a = 7, b = 9, c = 14. Check triangle inequality: 7 + 9 = 16 > 14. Valid (obtuse triangle since 7² + 9² = 49 + 81 = 130 < 196 = 14²).
2
s = (7+9+14)/2 = 15. Area K = √(15 × 8 × 6 × 1) = √720 = 26.83 sq m.
3
R = (7 × 9 × 14) / (4 × 26.83) = 882 / 107.33 = 8.218 m. Note: circumcenter lies outside the triangle because it is obtuse.
Circumradius = 8.218 m | Triangle Area = 26.833 sq m
Try this example →❓ Frequently Asked Questions
What is the formula for the circumscribed circle of a triangle?
The circumradius R = abc / (4K), where a, b, c are side lengths and K is the area found by Heron's formula. Equivalently, by the Law of Sines: R = a / (2 sin A), where A is the angle opposite side a. For a 3-4-5 triangle: K = 6, R = (3 times 4 times 5) / (4 times 6) = 2.5 units. This formula holds for any triangle: acute, right, or obtuse.
What is the circumradius of a right triangle?
For a right triangle, the circumradius equals exactly half the hypotenuse: R = c / 2. This follows from Thales' theorem: a triangle inscribed in a semicircle with the diameter as the hypotenuse always has a right angle at the opposite vertex. Using the Law of Sines: R = c / (2 sin 90 degrees) = c / 2. For a 5-12-13 right triangle: R = 13 / 2 = 6.5 units.
How do you find the circumscribed circle of a regular polygon?
For a regular n-gon with side length s: R = s / (2 sin(pi/n)). For a square (n = 4): R = s / (2 sin 45 degrees) = s / sqrt(2) = s times sqrt(2) / 2. For a regular pentagon (n = 5): R = s / (2 sin 36 degrees) = s / 1.176 = 0.851s. For a hexagon (n = 6): R = s / (2 sin 30 degrees) = s. Enter n and s in the Regular Polygon mode for instant results.
What is the circumscribed circle of an equilateral triangle?
For an equilateral triangle with side s: R = s / sqrt(3) = s times sqrt(3) / 3. This equals s / (2 sin 60 degrees) = s / sqrt(3). For side 6: R = 6 / 1.732 = 3.464 units. The circumcenter coincides with the centroid and the incenter for an equilateral triangle. The circumradius is always twice the inradius for equilateral triangles: R = 2r.
What is the difference between a circumscribed circle and an inscribed circle?
The circumscribed circle (circumcircle) passes through all vertices of the polygon and is the smallest circle enclosing the polygon. The inscribed circle (incircle) is the largest circle fitting inside the polygon, tangent to all sides. For a triangle with sides a, b, c, area K, and semiperimeter s: inradius r = K / s, circumradius R = abc / (4K). The two circles share the same center only for equilateral triangles and other regular polygons.
Where is the circumcenter located for acute, right, and obtuse triangles?
For an acute triangle (all angles less than 90 degrees), the circumcenter lies inside the triangle. For a right triangle, the circumcenter lies exactly at the midpoint of the hypotenuse. For an obtuse triangle (one angle greater than 90 degrees), the circumcenter lies outside the triangle, beyond the side opposite the obtuse angle. The circumcircle still passes through all three vertices in all cases.
How does the circumradius of a regular hexagon relate to its side length?
For a regular hexagon, the circumradius R equals the side length s exactly: R = s / (2 sin(pi/6)) = s / (2 times 0.5) = s. This is a unique property of the hexagon: it can be divided into six equilateral triangles with the circumcenter as a common vertex. This is why honeycomb cells tile perfectly and why a hex wrench key fits exactly inside a bolt socket of the same size.
What is Euler's inequality for circumradius and inradius?
Euler's inequality states R is greater than or equal to 2r, where R is the circumradius and r is the inradius of any triangle, with equality only for equilateral triangles. Euler's stronger result gives OI squared = R(R minus 2r), where O is the circumcenter and I is the incenter. For a 3-4-5 right triangle: R = 2.5, r = 1, so OI squared = 2.5 times 0.5 = 1.25, meaning OI = 1.118 units.
What is the Law of Sines and how does it relate to circumradius?
The Law of Sines states a / sin A = b / sin B = c / sin C = 2R, where R is the circumradius. This means every ratio of a side to the sine of its opposite angle equals the circumdiameter. The relation provides the fastest way to compute R when you know any side and its opposite angle: R = a / (2 sin A). It is also used in navigation, astronomy, and surveying to solve triangles with angle-side-angle or angle-angle-side data.
Can the circumscribed circle formula be used for irregular polygons?
No. A unique circumscribed circle exists only if all vertices lie on a common circle (a cyclic polygon). All triangles are cyclic. All regular polygons are cyclic. But most irregular quadrilaterals and higher polygons are not cyclic (a quadrilateral is cyclic only if its opposite angles sum to 180 degrees). For non-cyclic polygons, no single circle passes through all vertices, so the circumradius formula does not apply.
What is the circumscribed circle of a square?
For a square with side s: R = s times sqrt(2) / 2 = s / sqrt(2). This is half the diagonal (diagonal = s sqrt(2)). For a square with side 6 cm: R = 6 times sqrt(2) / 2 = 4.243 cm. The circumcircle has area = pi times R squared = pi times 18 = 56.55 sq cm. This is also the minimum circular cross-section that fully encloses a square peg of the same dimensions.
Why does the circumradius increase as a regular polygon gets more sides?
For fixed side length s, R = s / (2 sin(pi/n)). As n increases, pi/n decreases, sin(pi/n) decreases, and R increases. A regular triangle (n=3) has R = s / sqrt(3) = 0.577s. A square (n=4) has R = s sqrt(2)/2 = 0.707s. A hexagon (n=6) has R = s. A 12-gon has R = 1.932s. As n approaches infinity, the polygon approaches a circle and R approaches infinity for fixed s (because the polygon needs a larger and larger circle to fit sides of the same fixed length).