## Real World Applications – Algebra I

### Topic

The United States’ Pentagon – How long is each wall?

### Student Exploration

The United States’ Pentagon is the world’s largest office building. It houses the country’s governmental offices and military forces are based. The building itself is a regular pentagon, meaning that all five sides are of equal length. If each outer wall of the Pentagon is long, we can apply our knowledge of the addition of radicals to find the total perimeter of the building.

Since we know that every wall is the same length and that there are five walls total, we add five times.

Since all of the numbers under the radical sign are the same, we treat it the same as adding like terms. We add the “32”, and keep the radicals the same.

Since we’re adding the same side to itself five times, we can also multiply one wall by 5 to get the same answer, instead of adding it to itself five times. We would have .

We also know that by now, we should be able to simplify radicals. can be simplified. 828’s greatest common factor that’s a perfect square is 36. Using a factor tree or any other method, we have

Let’s say that for the sake of this exercise we wanted the height of the Pentagon to match the horizontal length of each wall. So, each face of the Pentagon building would be a square. We could calculate this by multiplying to itself, or . To find the area of each face of the Pentagon, we would multiply everything together. Multiply , and multiply the numbers in the radical signs. Since we’re multiplying radicals, we can either write or . Either way, when you multiply a radical by itself, the result is the inside of the root sign. In this case, . So, the face of the Pentagon wall would be .