A regular hexagon is inscribed in a circle of radius r. Find the length of the sides of the hexagon and the distance of each side from the centre of the circle.
Step-by-step solution
To find: Side of the hexagon and the distance of each side from the centre
Idea: The six equal sides are equal chords, so they subtend equal angles at the centre, 60° each. A triangle with two sides r and a 60° angle between them is equilateral.
- The 6 sides are equal chords, so they subtend equal angles at the centre O: 360° ÷ 6 = 60° each.½ mark
- Take one side AB. OA = OB = r, so ∠OAB = ∠OBA = (180° − 60°) ÷ 2 = 60°. ΔOAB is equilateral, so AB = r.1 mark
- The perpendicular OM from O to AB bisects it: AM = r/2. In right ΔOMA: OM2 = r2 − r2/4 = 3r2/4.1 mark
- Distance OM = (√3/2)r. Every side is the same distance from the centre (equal chords are equidistant).½ mark
Check: For r = 2: side 2, distance √3 ≈ 1.73; check 12 + (√3)2 = 4 = 22 ✓.
Answer to write in the exam
Each side subtends 360° ÷ 6 = 60° at the centre O (equal chords subtend equal angles).
OA = OB = r ⇒ ∠OAB = ∠OBA = 60° ⇒ ΔOAB is equilateral
∴ Side AB = r
OM ⟂ AB ⇒ AM = r/2 (perpendicular from the centre bisects the chord)
OM2 = r2 − (r/2)2 = 3r2/4
∴ Distance of each side from the centre = (√3/2)r
Common mistakes that cost marks
- Dividing 360° by 6 and then halving again: each side subtends 60°, not 30°, at the centre.
- Giving the distance as r/2 (that is half the side, not the distance).
- Writing √3r/2 as √(3r)/2; the root applies only to 3.
How this can come in the exam
A regular hexagon is inscribed in a circle of radius 8 cm. Its perimeter is
- 24 cm
- 48 cm
- 16π cm
- 96 cm
Show answer
(B) 48 cm
Each side equals the radius, 8 cm, so the perimeter is 6 × 8 = 48 cm.
Try one yourself
A regular hexagon is inscribed in a circle of radius 10 cm. Find its side and the distance of each side from the centre.
Show answer
Side = 10 cm; distance = (√3/2) × 10 = 5√3 ≈ 8.66 cm.
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