Question 10

Let's find the area of the hexagon and the circle. Then we will calculate their ratio.

Step 1 — Area of the hexagon

A regular hexagon has six equal sides. When inscribed in a circle, its side length is equal to the circle's radius $r$. We can divide the hexagon into six equilateral triangles. Each triangle has a side length of $r$. The area of one equilateral triangle is given by the formula $\frac{\sqrt{3}}{4} \times (\text{side})^2$.

$\text{Area of one triangle} = \frac{\sqrt{3}}{4} r^2$

The total area of the hexagon is six times this.

$\text{Area of hexagon} = 6 \times \frac{\sqrt{3}}{4} r^2$

$= \frac{6\sqrt{3}}{4} r^2$

$= \frac{3\sqrt{3}}{2} r^2$

$\boxed{\text{Area of hexagon} = \frac{3\sqrt{3}}{2} r^2}$

Step 2 — Area of the circle and the ratio

The area of a circle with radius $r$ is a standard formula.

$\text{Area of circle} = \pi r^2$

Now, let's find the ratio of the area of the hexagon to the area of the circle.

$\text{Ratio} = \frac{\text{Area of hexagon}}{\text{Area of circle}}$

$= \frac{\frac{3\sqrt{3}}{2} r^2}{\pi r^2}$

We can cancel out $r^2$ from the numerator and denominator.

$= \frac{3\sqrt{3}}{2\pi}$

Let's calculate the approximate value. We know that $\sqrt{3} \approx 1.732$ and $\pi \approx 3.14159$.

$\text{Ratio} \approx \frac{3 \times 1.732}{2 \times 3.14159}$

$\approx \frac{5.196}{6.28318}$

$\approx 0.827$

$\boxed{\text{Ratio} = \frac{3\sqrt{3}}{2\pi} \approx 0.827}$

Step 3 — Comparing with Question 8

The question asks why this answer is twice the answer to Question 8. If our answer is $\frac{3\sqrt{3}}{2\pi}$, then half of this value would be $\frac{1}{2} \times \frac{3\sqrt{3}}{2\pi} = \frac{3\sqrt{3}}{4\pi}$. Let's think about what shape might give this ratio. Consider an equilateral triangle inscribed in a circle of radius $r$. The side length $s$ of such a triangle is $r\sqrt{3}$. The area of this equilateral triangle is $\frac{\sqrt{3}}{4}s^2$.

$\text{Area of inscribed triangle} = \frac{\sqrt{3}}{4} (r\sqrt{3})^2$

$= \frac{\sqrt{3}}{4} (3r^2)$

$= \frac{3\sqrt{3}}{4} r^2$

The ratio of the area of this inscribed equilateral triangle to the area of the circle is:

$\text{Ratio (triangle to circle)} = \frac{\frac{3\sqrt{3}}{4} r^2}{\pi r^2}$

$= \frac{3\sqrt{3}}{4\pi}$

This value is exactly half of the ratio we found for the hexagon. So, Question 8 was likely asking for the ratio of the area of an equilateral triangle inscribed in a circle to the area of the circle.

Answer

(i) The ratio of the area of the hexagon to the area of the circle is $\frac{3\sqrt{3}}{2\pi} \approx 0.827$. (ii) The answer is exactly twice the answer to Question 8 because Question 8 likely asked for the ratio of the area of an equilateral triangle inscribed in a circle to the area of the circle, which is $\frac{3\sqrt{3}}{4\pi}$.