Question Video: Finding the Diagonal Length of a Square Using Its Area | Nagwa Question Video: Finding the Diagonal Length of a Square Using Its Area | Nagwa

Question Video: Finding the Diagonal Length of a Square Using Its Area Mathematics • Second Year of Preparatory School

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Given that the area of each square on the chessboard is 81 cm², find the diagonal length of the chessboard.

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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 𝐷, is therefore equal to eight times the diagonal length of each individual square, which we’ll denote by lowercase 𝑑. We’re 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, 𝑑 squared over two.

So, combining these two pieces of information, we can form an equation. Lowercase 𝑑 squared over two is equal to 81. Multiplying both sides of this equation by two, we find that 𝑑 squared is equal to 162. And then square rooting, we find that 𝑑 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. 𝐷 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.

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