Inscribed in a Circle

An equilateral triangle and regular hexagon are inscribed in the same circle.   Find the ratio of the area of the triangle to that of the hexagon.
Source: 12/7/2009


First, rotate the equilateral triangle around center C of the circle so that vertex A coincides with a vertex of the regular hexagon. Draw segment \overline{CD}.

Show that triangle ACB is congruent to triangle ADB
Triangle ACB is congruent to triangle ADB by ASA (angle side angle):

\angle 1=\angle 2=30^{o}    (see why below)
AB=AB    reflexive property
\angle 3=\angle 4=30^{o}    (see why below)

Since point C is equidistant from the endpoints A \textup{ and } B of segment \overline{AB}, C is on the the perpendicular bisector of \overline{AB}. Furthermore, we can prove that \overline{CM} is also the angle bisector of \angle C by proving that right triangle AMC is congruent to right triangle BMC.
Since \angle C=120^{o} (center angle intercepting the same arc as the inscribed angle of an equilateral triangle), half of it is equal to 60^{o}. Thus, \angle 1=30^{o} and \angle 3=30^{o}.
Similarly, we can prove that \angle 2=30^{o} and \angle 4=30^{o}.

Unit area
Let the area of triangle ACB be the unit area. Then, we can visually determine that the equilateral triangle is made up of 3 units and the regular hexagon is made up of 6 units.

Thus, the ratio of the area of the equilateral triangle to the area of the regular hexagon is

3:6 \textup{ or } 1:2.

Answer: 1:2.

About mvtrinh

Retired high school math teacher.
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One Response to Inscribed in a Circle

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