Question 8

We will find the area of the circle and the area of the inscribed equilateral triangle. Then we will calculate their ratio.

Step 1 — Area of the circle

Let's find the area of the circle first.

The radius of the circle is given as $r$.

The formula for the area of a circle is $\pi$ times radius squared.

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

$\boxed{\pi r^2}$

Step 2 — Side length of the triangle

Now, let's find the side length of the equilateral triangle.

The triangle is inscribed in the circle.

The center of the circle is also the centroid of the triangle.

The radius $r$ connects the center to a vertex.

Let $s$ be the side length of the equilateral triangle.

Let $h$ be the height of the triangle.

The centroid divides the median (height) in a 2:1 ratio.

So, the radius $r$ is two-thirds of the height $h$.

$r = \frac{2}{3}h$

We can find the height $h$.

$h = \frac{3}{2}r$

For an equilateral triangle, the height $h$ is also related to its side $s$.

$h = \frac{\sqrt{3}}{2}s$

Now we can find the side length $s$.

Let's set the two expressions for $h$ equal.

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

Multiply both sides by 2.

$\sqrt{3}s = 3r$

Divide both sides by $\sqrt{3}$.

$s = \frac{3}{\sqrt{3}}r$

We can simplify this expression.

$s = \sqrt{3}r$

$\boxed{\sqrt{3}r}$

Step 3 — Area of the triangle

Next, let's find the area of the equilateral triangle.

The formula for the area of an equilateral triangle is $\frac{\sqrt{3}}{4}$ times side squared.

We found the side length $s$ in the previous step.

The side length $s$ is $\sqrt{3}r$.

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

Substitute the value of $s$.

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

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

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

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

Step 4 — Ratio of areas

Finally, let's find the ratio of the areas.

We need the ratio of the triangle's area to the circle's area.

Area of triangle is $\frac{3\sqrt{3}}{4}r^2$.

Area of circle is $\pi r^2$.

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

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

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

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

Now, let's calculate the approximate value.

We know $\sqrt{3} \approx 1.732$.

We know $\pi \approx 3.14159$.

$= \frac{3 \times 1.732}{4 \times 3.14159}$

$= \frac{5.196}{12.56636}$

$\approx 0.41347$

Rounding to three decimal places, we get 0.413.

$\boxed{\frac{3\sqrt{3}}{4\pi} \approx 0.413}$

Answer

The ratio of the area of the triangle to the area of the circle is $\frac{3\sqrt{3}}{4\pi}$. This ratio is approximately 0.413.