A regular hexagon is inscribed in a circle of radius r. Find the length of the sides of the hexagon Class 9

A regular hexagon is inscribed in a circle of radius r. Find the length of the sides of the hexagon Class 9

Question 1.

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. Class 9

Solution:

Let the circle have centre O and radius r. In a regular hexagon inscribed in a circle, all central angles are equal

∴ ∠AOB = 60°

Consider ∆AOB:

OA = OB = r ...[Radii of same circle]

∠AOB = 60°

So, ∆AOB is equilateral.

∴ AB = OA = r

∴ Length of each side of hexagon r.

∴ AM = $\frac{\mathrm{AB}}{2}=\frac{r}{2}$

... [perpendicular drawn from centre bisect the chord]

In ∆OMA:

OA² = OM² + AM² ... [Baudhayana-Pythagoras theorem]

∴ r² = OM² + $\left(\frac{r}{2}\right)^2$

∴ r² = OM² + $\frac{r^2}{4}$

∴ OM² = r² - $\frac{r^2}{4}=\frac{3 r^2}{4}$

∴ OM = $\frac{\sqrt{3}}{2} r$

∴ Distance of each side from centre is $\frac{\sqrt{3}}{2} r$ r