### Video Transcript

Given that the area of each square
on the chessboard is 81 square centimeters, find the diagonal length of the
chessboard.

This chessboard is composed of 64
congruent squares arranged in eight rows of eight. The diagonal length of the
chessboard, which we will denote by capital ๐ท, is therefore equal to eight times
the diagonal length of each individual square, which weโll denote by lowercase
๐. Weโre given that the area of each
square on the chessboard is 81 square centimeters. And we can also recall that the
area of a square is equal to half the square of the length of its diagonal, ๐
squared over two.

So, combining these two pieces of
information, we can form an equation. Lowercase ๐ squared over two is
equal to 81. Multiplying both sides of this
equation by two, we find that ๐ squared is equal to 162. And then square rooting, we find
that ๐ is equal to the square root of 162, which in simplified form is equal to
nine root two. So we found the length of the
diagonal of each of the smaller squares from which this chessboard is composed.

To find the diagonal length of the
chessboard itself, we need to multiply this value by eight. ๐ท is equal to eight multiplied by
nine root two, which in exact form is 72 root two. And the units for this are
centimeters. So, by recalling that the area of a
square is equal to half the square of the length of its diagonal, we found that the
diagonal length of the chessboard in exact form is 72 root two centimeters.