Constructions (Circles)
345. The Hexagon's Inner Circle · finding the perfect inscribed circle
IP is perpendicular to AB when P is the foot of the perpendicular from I.
The inscribed circle of a regular hexagon touches the middle of each side. Its radius — the apothem — is the perpendicular distance from the centre to a side, so IP ⊥ AB at the touch point.
What this lesson covers
Try to break it
Drag P along side AB. ∠IPB shifts as P moves. Only at the midpoint of AB — the foot of the perpendicular from the incentre I — does ∠IPB read exactly 90°. That foot is where the inscribed circle touches AB, and the segment IP at that point is the inradius.
How you build it
Lock a regular hexagon by drawing just its first side, drop the apothem from the centre I to a side, then draw the incircle through that foot.
- Draw side AB; the regular hexagon ABCDEF and its centre I complete automatically.
- Drop a perpendicular from centre I to side AB. The foot P is where the incircle touches AB, and IP is the apothem.
- With centre I, draw the incircle through P — its radius is the apothem IP.
The proof, step by step
Prove that the constructed regular hexagon has all six sides touching the circle.
- In a regular hexagon, all interior angles are 120° and all sides are equal in length.
- The bisectors of any two adjacent interior angles intersect at a point I that is equidistant from all sides.
- The perpendicular distance from I to any side is the inradius. A circle with this radius centered at I touches all sides exactly once.
Worked example
A regular hexagon has side 4.6 cm. What is the radius of its inscribed circle? (Use √3 ≈ 1.73)
Radius = side × cos(30°) = 4.6 × (√3/2) ≈ 4.6 × 0.866 ≈ 3.99 cm.
- 2.30 cm
- 3.99 cm — correct
- 4.60 cm
- 5.20 cm