Incircles and Circumcircles of Polygons
A circumscribed circle (circumcircle) passes through all vertices of a polygon, and an inscribed circle (incircle) touches all its sides from inside; every triangle and every regular polygon has both, but other polygons may have only one or neither. In brief: a quadrilateral has a circumcircle if and only if its opposite angles add up to 180° (cyclic quadrilateral), and it has an incircle if and only if the sums of opposite sides are equal, a + c = b + d (tangential quadrilateral).
These circles connect polygons with circles and give formulas for the area and for lengths. A rectangle has a circumcircle but no incircle (unless it is a square), and a rhombus has an incircle but no circumcircle (unless it is a square). This guide explains the conditions, the formulas for triangles, quadrilaterals and regular polygons, and gives worked examples and a table of radii.
What are the circles of a triangle?
Every triangle has a unique circumcircle with center O, the intersection of the perpendicular bisectors, and a unique incircle with center I, the intersection of the angle bisectors.
- Circumradius: R = abc/(4A), where A is the area. In a right triangle, R is half the hypotenuse.
- Inradius: r = A/s, where s is the semi-perimeter. In a right triangle with legs a, b and hypotenuse c: r = (a + b − c)/2.
- Euler’s relation: the distance d between O and I satisfies d2 = R(R − 2r), so R ≥ 2r, with equality for equilateral triangles.
Example 1: the 13-14-15 triangle
The area is 84 and s = 21. Then R = (13 · 14 · 15)/(4 · 84) = 2730/336 = 8.125 and r = 84/21 = 4. Check Euler: R ≥ 2r, and 8.125 ≥ 8 ✓. The distance OI = √(8.125 · (8.125 − 8)) = √1.0156 ≈ 1.008.
Example 2: a right triangle 5-12-13
R = 13/2 = 6.5 and r = (5 + 12 − 13)/2 = 2. Check: A = 30, s = 15, r = 30/15 = 2 ✓.
When does a quadrilateral have a circumcircle?
A convex quadrilateral can be inscribed in a circle (cyclic quadrilateral) if and only if its opposite angles are supplementary: ∠A + ∠C = 180° and ∠B + ∠D = 180°. The reason: the inscribed angle theorem shows that opposite angles subtend arcs adding to 360°. Another test: an exterior angle equals the opposite interior angle.
- Rectangles, squares and isosceles trapezoids are cyclic.
- A general parallelogram or a rhombus is not cyclic, since opposite angles are equal and would have to be 90°.
- Ptolemy’s theorem: in a cyclic quadrilateral, the product of the diagonals equals the sum of the products of opposite sides, ef = ac + bd.
- Brahmagupta’s formula for the area: A = √((s − a)(s − b)(s − c)(s − d)), with s the semi-perimeter.
Example 3
A cyclic quadrilateral has sides 3, 4, 5, 6. Then s = 9 and A = √(6 · 5 · 4 · 3) = √360 ≈ 18.97. For a rectangle 3 by 4: Ptolemy gives d2 = 3 · 3 + 4 · 4 = 25, so d = 5 ✓.
Example 4: angles
A cyclic quadrilateral has angles A = 75° and B = 100°. Then C = 105° and D = 80°. Both pairs add up to 180°.
When does a quadrilateral have an incircle?
A convex quadrilateral has an inscribed circle (tangential quadrilateral) if and only if the sums of the opposite sides are equal: a + c = b + d (Pitot’s theorem and its converse). The tangent segments from a vertex to the circle are equal, which gives the condition.
- Squares, rhombuses and kites are tangential.
- A rectangle that is not a square is not tangential: for sides l and w, l + l ≠ w + w.
- The area is A = r · s, where r is the inradius and s is the semi-perimeter.
- A quadrilateral that is both cyclic and tangential is called bicentric. A square is bicentric, and so is a right kite.
Example 5
A quadrilateral with sides 7, 4, 6, 3 in this order: 7 + 6 = 13 and 4 + 3 = 7, so there is no incircle. With sides 5, 3, 4, 6: 5 + 4 = 9 and 3 + 6 = 9, so there is an incircle.
Example 6: a rhombus
A rhombus with diagonals 6 and 8 has side 5 and area 24. The perimeter is 20 and s = 10, so r = A/s = 2.4. The incircle has the intersection of the diagonals as center.
What are the radii of regular polygons?
A regular n-gon with side a has circumradius R = a/(2 sin(180°/n)) and inradius (apothem) r = a/(2 tan(180°/n)). They are related by r = R cos(180°/n). For n = 6, r/R = cos 30° = √3/2. The area is A = (n/2)R2 sin(360°/n) = (1/2) · P · r. The ratio of the circle areas is the same for every regular polygon of given n: R2/r2 = 1/cos2(180°/n).
Example 7: hexagon with side 10
R = 10 and r = 5√3 ≈ 8.66. The area is (1/2) · 60 · 8.66 = 259.8.
Example 8: a square
With side s, R = s√2/2 ≈ 0.707s and r = s/2. For s = 2, R = √2 ≈ 1.414 and r = 1. The square has area 4, the incircle has area π ≈ 3.14 and the circumcircle has area 2π ≈ 6.28.
Radii of regular polygons with side 1
| Sides n | Polygon | Circumradius R | Inradius r |
|---|---|---|---|
| 3 | triangle | 0.5774 | 0.2887 |
| 4 | square | 0.7071 | 0.5000 |
| 5 | pentagon | 0.8507 | 0.6882 |
| 6 | hexagon | 1.0000 | 0.8660 |
| 8 | octagon | 1.3066 | 1.2071 |
| 10 | decagon | 1.6180 | 1.5388 |
| 12 | dodecagon | 1.9319 | 1.8660 |
As n grows, R and r both approach each other, and the polygon approaches a circle. For side a, multiply the values by a. The radii for a hexagon with side 10 are therefore 10 and 8.66.
Where do these results come from?
Euclid’s Elements (Book IV) shows how to inscribe and circumscribe circles about triangles and to inscribe regular polygons in circles. Ptolemy’s theorem, from the second century, was used for computing chord tables in his Almagest. Brahmagupta’s formula for cyclic quadrilaterals is from the seventh century, and the relation R ≥ 2r for triangles was found by Euler in the eighteenth century.
Where are they used today?
The circumradius determines the smallest circular hole through which a polygonal plate fits, and the inradius gives the largest circle that fits inside a polygon, as with the largest pipe that fits in a triangular duct. Navigation by three beacons uses the circumcircle idea, and tangential polygons appear in the design of gear and bearing layouts.
Which mistakes are common?
Assuming every polygon has both circles
Triangles and regular polygons do. A general quadrilateral needs special conditions.
Mixing up the tests
The circumcircle test uses angles (opposite angles sum to 180°). The incircle test uses sides (opposite side sums are equal).
Confusing R and r
R is for the circle through the vertices (the larger one) and r is for the circle touching the sides.
Using R = a/2 for any right triangle
For a right triangle R is half of the hypotenuse, not half of an arbitrary side.