Find the area of a square, to the nearest hundredth, if half the length of its diagonal is 3.62 centimeters.
If we have a square and we want to know its area, then we take the length of one of the sides 𝑠. And we square that value. We’re interested in the area of a square. If this is the diagonal of our square, we’ve been given the value of half of the diagonal, which is 3.62 centimeters.
In order to solve the problem, we need to remember something about the diagonal of a square. The length of the diagonal of a square is equal to its side length multiplied by the square root of two. Since we know half of the diagonal in our square is 3.62 centimeters, we can multiply that value by two to find the length of that diagonal. The diagonal is then 7.24 centimeters.
Since we know that the diagonal is equal to the side length times the square root of two, we can use the length of the diagonal to find the length of the side. The length of the side is the length of the diagonal divided by the square root of two. And so the side length is 7.24 over the square root of two centimeters.
Now, we could go ahead and calculate at this step. But we’re not really trying to find the side. We’re trying to find the area. And the area equals the side squared. And so we can plug in 7.24 over the square root of two in for 𝑠. And then we have 7.24 over the square root of two squared, which will be equal to 7.24 squared over the square root of two squared. We know that the square root of two squared equals two. So we can solve for 7.24 squared divided by two.
When we do that, we get 26.2088. And this is a measure of centimeters squared. If we want to round to the nearest hundredth, that’s the second decimal place. And so we look to the right to the third decimal place where there is an eight. And so the zero in the hundredths place becomes a one. And everything to the left of the hundredths place stays the same. The area of this square is then 26.21 centimeters squared.
At this point, it’s worth noting that there’s another way to solve this problem. We started this problem with our most common formula for finding the area of a square. But we can actually have this formula in terms of the diagonal. The area of a square is also equal to one-half times the diagonal squared. If you recognize this formula, then you could take half the diagonal times two to get the total diagonal 7.24. And you could plug it in for the length of the diagonal so that you have area equals one-half times 7.24 squared, which will again give us 26.2088. And we know that that rounds to 26.21 centimeters squared.
Both methods give us the area of this square with half a diagonal of 3.62 centimeters is equal to 26.21 centimeters squared.