Question Video: Using Square Roots to Find Side Lengths and Areas of Squares Mathematics

A square has a side length of ๐‘™ cm and an area of 63 cmยฒ. Find the area of a square whose side length is 6๐‘™ cm.

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Video Transcript

A square has a side length of ๐‘™ centimeters and an area of 63 square centimeters. Find the area of a square whose side length is six ๐‘™ centimeters.

Remember, the area of a rectangle is calculated by multiplying its width by its height. A square is simply a rectangle whose sides are the same length. And so, we can say that the area of a square is its side length multiplied by itself or its side length squared. Now, weโ€™re told that our square has an area of 63 square centimeters. Weโ€™re also told that its side length is ๐‘™. So, replacing area with 63 and side length with ๐‘™, we form an equation. We get 63 is equal to ๐‘™ squared.

Now, the question wants us to find the area of a square whose side length is six ๐‘™ centimeters. So, what weโ€™re going to do is begin by calculating the value of ๐‘™. In other words, weโ€™re going to solve this equation for ๐‘™. To do so, we perform an inverse operation. Currently, ๐‘™ is being squared. The opposite of squaring is square rooting a number. This essentially undoes the previous operation.

And so, if we square both sides of our equation, we get simply ๐‘™ on the right-hand side. Then, the left-hand side is equal to the square root of 63. Now, itโ€™s worth recalling that when we find the square root in an equation, we look to find the positive and negative square roots of the number. But this is a side length, so we canโ€™t have a negative value. And ๐‘™ is equal to the square root of 63. We might be interested in simplifying this radical or surd. But in fact, weโ€™re not quite done with it. So, weโ€™ll leave it as it is for now.

Our new square has a side length of six ๐‘™ centimeters. We calculated ๐‘™ to be equal to the square root of 63. So, six ๐‘™ must be six times this. Itโ€™s six root 63. Since this is the new side length of our square, the area of this square is this value squared. Itโ€™s six root 63 times six root 63. We distribute the two over both parts of this value. So, we get six squared times the square root of 63 squared. Six squared is 36.

And of course, squaring and finding the square root are inverse operations of one another. They undo the other operation. So, the square root of 63 squared is just 63. This means the area of our square is 36 times 63, which is 2,268. And since weโ€™re working in centimeters, the units here are square centimeters.

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