A regular hexagon is inscribed in a circle of radius . Find the length of the sides of the hexagon and the distance of each side from the centre of the circle.

Question

Homi · Accepted Answer

Let's define the dimensions of a small rectangle.

Step 1 — Define dimensions

Let the length of each small rectangle be $l$ . Let the width of each small rectangle be $w$ . From the figure, the top row has 4 rectangles. These rectangles are oriented with their length horizontally. So, the total width of the top row is $4l$ . The height of the top row is $w$ . The bottom row has 5 rectangles. These rectangles are oriented with their width horizontally. So, the total width of the bottom row is $5w$ . The height of the bottom row is $l$ .

Diagram 1

Step 2 — Formulate equations

The total width of the large rectangle must be the same for both rows. So, we set the widths equal. $4l = 5w$ The total height of the large rectangle is the sum of the heights of the two rows. Total height $= w + l$ . The area of the large rectangle is given as $72 \text{ cm}^2$ . We know that Area = Total width $\times$ Total height. $72 = (4l)(w+l)$

Step 3 — Solve for dimensions

From the first equation, we can express $w$ in terms of $l$ . $w = \frac{4}{5}l$ Now, we substitute this into the area equation. $72 = 4l\left(\frac{4}{5}l + l\right)$ Let's combine the terms inside the parenthesis. $72 = 4l\left(\frac{4l+5l}{5}\right)$ $72 = 4l\left(\frac{9l}{5}\right)$ Multiply the terms on the right side. $72 = \frac{36l^2}{5}$ To solve for $l^2$ , we multiply both sides by 5 and divide by 36. $36l^2 = 72 \times 5$ $36l^2 = 360$ $l^2 = \frac{360}{36}$ $l^2 = 10$ We take the square root of both sides to find $l$ . $l = \sqrt{10}$ Now, we find the value of $w$ using $w = \frac{4}{5}l$ . $w = \frac{4}{5}\sqrt{10}$ So, the dimensions of each small rectangle are $\sqrt{10}$ cm and $\frac{4}{5}\sqrt{10}$ cm.

Step 4 — Calculate perimeter

The perimeter of each small rectangle is $2(l+w)$ . $P = 2\left(\sqrt{10} + \frac{4}{5}\sqrt{10}\right)$ Let's find a common denominator for the terms inside the parenthesis. $P = 2\left(\frac{5\sqrt{10}}{5} + \frac{4\sqrt{10}}{5}\right)$ Now, we add the terms. $P = 2\left(\frac{9\sqrt{10}}{5}\right)$ Finally, we multiply by 2.

$\boxed{\frac{18\sqrt{10}}{5} \text{ cm}}$

Answer

The perimeter of each small rectangle is $\frac{18\sqrt{10}}{5}$ cm.

Question 19

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