Question

A circle is inscribed in a regular octagon. The same circle circumscribes a regular hexagon. Find the ratio of the areas of the circle, the hexagon and the octagon.

• A. \(2\pi : 3\sqrt{3} : 16(\sqrt{2}-1)\)
• B. \(\pi : 3\sqrt{3} : 4(\sqrt{2}-1)\)
• C. \(\dfrac{2\pi}{3} : 2\sqrt{3} : 4(\sqrt{2}-1)\)
• D. None of these

Hint:

Write each polygon's area using the circle's radius, as the octagon's apothem and the hexagon's circumradius, then simplify tan(22.5 degrees) to root 2 minus 1.

Answer 1

The Correct Option is A

Step 1: Fix a convenient radius.
Instead of carrying the letter $r$ through the whole problem, set the common circle's radius to $r = 1$ and work out each area as a plain number, then read off the ratio at the end.

Step 2: Find the octagon's area.
The circle of radius 1 is inscribed in the octagon, so the octagon's apothem is 1. Using $\text{Area} = n a^2\tan(\pi/n)$ with $n=8$, $a=1$:
\[ \text{Area}_{\text{octagon}} = 8\tan(22.5^{\circ}) = 8(0.4142) \approx 3.3137 \]

Step 3: Find the hexagon's area.
The same circle of radius 1 circumscribes the hexagon, so the hexagon's circumradius is 1. Using $\text{Area} = \frac{n}{2}R^2\sin(2\pi/n)$ with $n=6$, $R=1$:
\[ \text{Area}_{\text{hexagon}} = 3\sin(60^{\circ}) = 3\left(\frac{\sqrt{3}}{2}\right) \approx 2.598 \]

Step 4: Find the circle's area and compare.
\[ \text{Area}_{\text{circle}} = \pi(1)^2 \approx 3.1416 \]
So the three areas, in order circle, hexagon, octagon, are about $3.1416 : 2.598 : 3.3137$. Doubling each to clear the decimals into neat surds gives about $6.283 : 5.196 : 6.627$.

Step 5: Match against the answer choices.
Now evaluate option (1), $2\pi : 3\sqrt{3} : 16(\sqrt{2}-1)$, as numbers: $2\pi \approx 6.283$, $3\sqrt{3} \approx 5.196$, and $16(\sqrt{2}-1) \approx 6.627$. These match the doubled values from Step 4 exactly.

Final Answer:
The ratio of circle, hexagon and octagon areas is $2\pi : 3\sqrt{3} : 16(\sqrt{2}-1)$. \[ \boxed{2\pi : 3\sqrt{3} : 16(\sqrt{2}-1)} \]