A regular hexagon is inscribed in a circle of radius . Find the length of the sides of the hexagon and the distance of each side from the centre of the circle.
(Distance of each side from the centre) r
(Length of the sides) r
Explanation
A regular hexagon inscribed in a circle divides the circle into 6 equal arcs, making the central angle 60 degrees. The triangle formed by two radii and a side is an equilateral triangle, so the side length equals the radius. The distance from the centre to a side (which acts as a chord) is calculated using the perpendicular bisector property and the Pythagorean theorem, similar to the chord-distance problems provided in the context.
Solution Steps
Step 1: A regular hexagon has 6 sides, so the central angle subtended by each side is 360° / 6 = 60°.
Step 2: The triangle formed by the centre and the endpoints of a side has two sides equal to the radius r and an included angle of 60°, making it an equilateral triangle. Thus, the side length is r.
Step 3: To find the distance from the centre to a side (a chord), draw a perpendicular from the centre to the side. This perpendicular bisects the side, so half the chord length is r/2.
Step 4: Apply the Pythagorean theorem: r² = d² + (r/2)². Solving this gives d² = r² - r²/4 = 3r²/4, so the distance d = (/2)r.