40
How the side of a regular polygon relates to the radii of its circumscribed and inscribed circles
Textbook: pp. 112–113
GoalKnow the formulas relating the side a of a regular n-gon with the radii R and r (R = a/(2sin(180°/n)), r = a/(2tg(180°/n))) and apply them to the triangle, square and hexagon.
New words
circumradius R · tashqi aylana radiusi Rinradius r · ichki aylana radiusi rside of the n-gon · n burchakning tomoni180°/n · 180°/n
Explanation
Let O be the centre of a regular n-gon and C the midpoint of side AB. In the right triangle ACO, ∠AOC = 180°/n and AC = a/2, so R = OA = a/(2·sin(180°/n)) and r = OC = a/(2·tg(180°/n)) = R·cos(180°/n). Conversely a = 2R·sin(180°/n) = 2r·tg(180°/n). Triangle (n = 3): R = a/√3, r = a/(2√3), R = 2r. Square (n = 4): R = a/√2, r = a/2, R = r√2. Hexagon (n = 6): R = a, r = a√3/2. With these formulas the side of a regular polygon inscribed in a circle is found from R: the inscribed square has side R√2, the triangle R√3, the hexagon R.
Worked examples
For a = 10: the square has R = 10/√2 = 5√2 and r = 5; the hexagon has R = 10 and r = 5√3.
Inscribed in a circle with R = 8: the triangle has side 8√3, the square 8√2, the hexagon 8.
Class activity
“Formula trio”: complete a table of a, R, r relations for the triangle, square and hexagon and compare the ratios R : r (2, √2, 2/√3).
Practice
1
What is the side of a regular hexagon inscribed in a circle of radius 6?
6
2
What is the side of the square inscribed in a circle of radius 6?
6√2
3
What is the radius of the incircle of a regular hexagon with side 6?
3√3
4
Why is R = a in a regular hexagon?
The central angle is 60°, so the centre and two adjacent vertices form an equilateral triangle.