Measuring Space: Perimeter and Area | Exercise 6.3

Question 10

A hexagon is inscribed in a circle of radius $r$ . Show that the ratio of the area of the hexagon to the area of the circle is equal to $\dfrac{3\sqrt{3}}{2\pi} \approx 0.827$ . Can you see why the answer is exactly twice the answer to Question 8?

Solution

Understand the Question

A regular hexagon inscribed in a circle of radius $r$ is made up of $6$ identical equilateral triangles, each with side length equal to the radius $r$ .
The area of an equilateral triangle with side length $r$ is $\dfrac{\sqrt{3}}{4}r^2$ , so the area of the hexagon is $6 \times \dfrac{\sqrt{3}}{4}r^2 = \dfrac{3\sqrt{3}}{2}r^2$ .
The area of the circle is $\pi r^2$ .
Dividing the area of the hexagon by the area of the circle gives the required ratio $\dfrac{3\sqrt{3}}{2\pi} \approx 0.827$ .
An inscribed equilateral triangle has an area of $\dfrac{3\sqrt{3}}{4}r^2$ , which is exactly half of the inscribed hexagon's area.

Step 1 · Find the Area of the Hexagon

A regular hexagon inscribed in a circle of radius $r$ consists of $6$ congruent equilateral triangles, each of side length $r$ . Diagram 1

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

\begin{aligned} \text{Area of hexagon} &= 6 \times \dfrac{\sqrt{3}}{4} r^2 \\[0.6em] &= \dfrac{6\sqrt{3}}{4} r^2 \\[0.6em] &= \dfrac{3\sqrt{3}}{2} r^2 \end{aligned}

Step 2 · Calculate the Ratio of Hexagon Area to Circle Area

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

Ratio of the area of the hexagon to the circle:

\begin{aligned} \text{Ratio} &= \dfrac{\text{Area of hexagon}}{\text{Area of circle}} \\[0.8em] &= \dfrac{\frac{3\sqrt{3}}{2} r^2}{\pi r^2} \\[0.8em] &= \dfrac{3\sqrt{3}}{2\pi} \end{aligned}

Approximating with $\sqrt{3} \approx 1.732$ and $\pi \approx 3.14159$ :

\begin{aligned} \text{Ratio} &\approx \dfrac{3 \times 1.732}{2 \times 3.14159} \\[0.8em] &\approx \dfrac{5.196}{6.28318} \\[0.6em] &\approx 0.827 \end{aligned}

Step 3 · Compare with the Inscribed Equilateral Triangle

For an equilateral triangle inscribed in the same circle of radius $r$ , the side length is $s = r\sqrt{3}$ :

\begin{aligned} \text{Area of inscribed triangle} &= \dfrac{\sqrt{3}}{4} (r\sqrt{3})^2 \\[0.6em] &= \dfrac{\sqrt{3}}{4} (3r^2) \\[0.6em] &= \dfrac{3\sqrt{3}}{4} r^2 \end{aligned}

Ratio for the inscribed triangle:

\begin{aligned} \text{Ratio (triangle to circle)} &= \dfrac{\frac{3\sqrt{3}}{4} r^2}{\pi r^2} \\[0.8em] &= \dfrac{3\sqrt{3}}{4\pi} \end{aligned}

Since $\dfrac{3\sqrt{3}}{2\pi} = 2 \times \dfrac{3\sqrt{3}}{4\pi}$ , the ratio for the regular hexagon is exactly twice that of the inscribed equilateral triangle.

Answer

The ratio is $\dfrac{3\sqrt{3}}{2\pi} \approx 0.827$ , which is exactly twice the answer to Question 8 because the area of an inscribed regular hexagon is twice the area of an inscribed equilateral triangle in the same circle.

Common Mistakes

Side Length Assumption: Forgetting that for an inscribed regular hexagon, each side length equals the radius $r$ of the circle.
Inverting the Ratio: Calculating $\dfrac{\text{Area of circle}}{\text{Area of hexagon}}$ instead of $\dfrac{\text{Area of hexagon}}{\text{Area of circle}}$ .
Premature Rounding: Rounding $\sqrt{3}$ or $\pi$ too early in intermediate calculations, which can lead to an inaccurate decimal result.

Question 10

More questions in Exercise 6.3