Polygons Problems

Regular polygons problems with detailed solutions.

Problem 1: A 6 sided regular polygon (hexagon) is inscribed in a circle of radius 10 cm, find the length of one side of the hexagon.

6 sided polygon (hexagon) problem 1

Solution to Problem 1:

  • Angle AOB is given by

    angle (AOB) = 360o / 6 = 60o

  • Since OA = OB = 10 cm, triangle OAB is isosceles which gives

    angle (OAB) = angle (OBA)

  • So all three angles of the triangle are equal and therefore it is an equilateral triangle. Hence

    AB = OA = OB = 10 cm.

Problem 2: A circle of radius 6 cm is inscribed in a 5 sided regular polygon (pentagon), find the length of one side of the pentagon.(approximate your answer to two decimal places).

6 sided polygon (hexagon) problem 1

Solution to Problem 2:

  • Let t be the size of angle AOB, hence

    t = 360o / 5 = 72o

  • The polygon is regular and OA = OB. Let M be the midpoint of AB so that OM is perpendicular to AB. OM is the radius of the inscribed circle and is equal to 6 cm. Right angle trigonometry gives

    tan(t / 2) = MB / OM

  • The side of the pentagon is twice MB, hence

    side of pentagon = 2 OM tan(t / 2) = 8.7 cm (answer rounded to two decimal places)

Problem 3: Find the area of a dodecagon of side 6 mm. (approximate your answer to one decimal place).

Solution to Problem 3:

  • A dodecagon is a regular polygon with 12 sides and the central angle t opposite one side of the polygon is given by.

    t = 360o / 12 = 30o

  • We now use the formula for the area when the side of the regular polygon is known

    Area = (1 / 4) n x2 cot (180o / n)

  • Set n = 12 and x = 6 mm

    area = (1 / 4) (12) (6 mm)2 cot (180o / 12)

    = 403.1 mm2 (approximated to 1 decimal place).

Problem 4:Show that if the number of sides n of a polygon inscribed inside a circle of radius R, is very large then the area of the polygon may be approximated by the area of the circumscribed circle with radius R. (HINT: If angle x is very small and is in radians, then sin x may be approximated by x).

Solution to Problem 4:

  • The area of a regular polygon with n sides may be given in terms of R by

    area = (1/2) n R 2 sin (2 pi / n)

  • If n is large, then 2 Pi /n is very small and sin (2 pi/n) may be approximated by 2 pi / n so that the area may be approximated by

    area = (1/2) n R 2 (2 pi / n)

    = pi R 2

  • which is the area of the circle.

    For more on the above question, see the interactive tutorial in regular polygons.

More references to triangles and geometry.

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Updated: February 2015

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