### Video Transcript

A flower bed is designed as a regular hexagon with an area of 54 times the square root of three meters squared. Find the side length of the hexagon, giving the answer to the nearest meter.

Let’s think about what we know about the area of a regular 𝑛-sided polygon. It’s equal to 𝑛 times 𝑥 squared over four times the cot of 180 over 𝑛, where 𝑥 is the value of the side length. And this formula is written such that we’re operating with cotangent in degrees. If the flower bed that we’re dealing with is a regular hexagon, it’s a regular six-sided polygon. This means we know the value for 𝑛. We’ve also been given the area of this polygon. This means we have enough information to solve for the missing side length 𝑥.

We’ll plug in 54 times the square root of three for the area and six for 𝑛. We can reduce six over four to three-halves. And then 180 divided by six is equal to 30. The tan of 30 degrees is an angle we usually memorize. We know that that is equal to one over the square root of three. And since the cotangent is the inverse of tangent, we can say that the cot of 30 degrees is therefore equal to the square root of three. From there, we can divide both sides of this equation by the square root of three so that we have 54 is equal to three-halves times 𝑥 squared.

We had a term of the square root of three on either side of this equation, and they both canceled out. To get 𝑥 by itself, we’ll multiply both sides of the equation by two-thirds. 54 multiplied by two-thirds is 36. And on the right side of the equation, we’re left with 𝑥 squared. To solve for 𝑥, we’ll then need to take the square root of both sides of the equation. The square root of 𝑥 squared is 𝑥. And for the square root of 36, we’ll only be interested in the positive square root since we’re dealing with distance. And that means 𝑥 must be equal to six.

Since our area was given in meters squared, the side length needs to be given in meters. Since our regular hexagon has an area of 54 times the square root of three meters squared, its side length must be equal to six meters.