Video: Deciding Proportionality of the Side Length of a Square to Its Area

Video Transcript

Is the length of one side of the given figure proportional to its area?

So, let’s have a look at the shape in this question. We can see that there are four right angles and two sides labeled the same length. So, we must have a square. We’re asked if the length of one side is proportional to the area. So, let’s recall how to find the area of a square. And that is that the area of a square is equal to the length times the length, or the length squared. So, the area of our square is 𝑆 squared.

Let’s recall proportion. If we have two quantities 𝐴 and 𝐵 which are proportional, then that means from one situation to another, both quantities have been multiplied by the same number. We know that in one situation, the area of our square is equal to 𝑆 squared. Let’s imagine another situation where we double the length of our sides. In this case, the area of our second square, or square two, would be equal to two 𝑆 times two 𝑆, which is four 𝑆 squared. We can note that the area of our first square, which we could call square one, was equal to 𝑆 squared. So, the area of square two is equal to four times the area of square one.

Now, let’s imagine another situation where we multiply the length of our square by three. So, in this case, the area of square three would be equal to three 𝑆 times three 𝑆, which is nine 𝑆 squared. And given that our first square was equal to 𝑆 squared, then this means that the area of square three is equal to nine times the area of square one. So, if we consider these values as fractions of the length over the area, in the first situation we have the length 𝑆 over 𝑆 squared. We then double the lengths, so the fraction would be two 𝑆 over the area of four 𝑆 squared. And in our final situation, we had three 𝑆 as the length over the area of nine 𝑆 squared.

These two quantities would be proportional if we can say that they are multiplied by the same number. However, going from the first fraction to the second fraction would mean the numerator was multiplied by two and the denominator was multiplied by four. We can also see that from the first fraction to the third fraction, we multiplied the numerator by three and the denominator by nine, which means that these have not been multiplied by the same number. So, the answer to the question, is the length of one side of this figure proportional to its area, is no.