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Problem of the Week
Problem C and Solution
One Dot at a Time


Priya is drawing a polygon on a piece of wood. First she hammers a nail into the piece of wood, calling this point \(O\). Then she attaches one end of a piece of string to the nail, and the other end to a pencil. She pulls the string tight and makes a dot on the wood, calling this point \(A\). Keeping the string tight, she rotates it \(20^\circ\) clockwise and makes another dot, calling this point \(B\). She then connects points \(A\) and \(B\) with a straight line.

Triangle AOB with dashed lines for sides OA and OB and a solid line for side AB. Angles OAB measures 20 degrees.

She repeats this process, rotating the string \(20^\circ\) clockwise, making a dot, and connecting this point to the previous point with a straight line each time, until she has gone all the way around the circle and completed the polygon.

Five triangles with the same size and shape as AOB share vertex O and form a figure that looks similar to five side by side triangular slices of a pie with centre O. The outer edge of the pie is made up of straight line segments, each the same length as AB. For each triangular slice, the angle at centre O measures 20 degrees. An arrow indicates that more triangles can be added to the figure.

  1. How many sides does Priya’s completed polygon have?

  2. What is the sum of all the interior angles in the polygon?


  1. Each time the process is repeated, another congruent triangle is created. Each of these triangles has a \(20^\circ\) angle at \(O\), the centre of the circle. Since a complete rotation at the centre is \(360^\circ\), that means there are \(360\div 20=18\) triangles formed. Since each triangle has one edge on the side of the polygon, it follows that the polygon has \(18\) sides. An \(18\)-sided polygon is called an octadecagon, from octa meaning \(8\) and deca meaning \(10\).

  2. Since the distance between each dot and point \(O\) (the nail) is always the same, it follows that the two sides of each congruent triangle that connect to point \(O\) are equal. Therefore, the congruent triangles are all isosceles, and the angles that are not at point \(O\) are all equal. The angles in a triangle sum to \(180^\circ\), so after the \(20^\circ\) angle is removed, there is \(160^\circ\) remaining for the other two angles. It follows that each of the other two angles in each triangle measures \(160^\circ \div 2=80^\circ\). The following diagram illustrates this information for the two adjacent triangles \(AOB\) and \(BOC\).

    Triangle AOB and triangle BOC share side OB. Sides OA, OB, and OC are marked as being equal in length. Angles AOB and BOC both measure 20 degrees. Remaining angles OAB, OBA, OBC, and OCB all measure 80 degrees.

    Each interior angle in the polygon is formed by an \(80^\circ\) angle from one triangle and the adjacent \(80^\circ\) angle from the next triangle. It follows that each interior angle measures \(80^\circ+80^\circ=160^\circ\). Thus, there are \(18\) interior angles in the octadecagon and each angle measures \(160^\circ\).

    A regular polygon with 18 vertices labelled with letters A through S. The angle at each vertex measures 160 degrees.

    Therefore, the sum of all the interior angles in the octadecagon is \(18\times 160^\circ=2880^\circ\).