Given a rhombus whose diagonals’ lengths are 5.3 centimeters and 3.8 centimeters and a square whose diagonal length is 5.3 centimeters, which one is smaller in area?
So at first, we might think, okay, we need to know how to find the area of a rhombus. That area is 𝑝 times 𝑞 divided by two, where 𝑝 and 𝑞 are the diagonals. And when we find the area of a square, we usually say the area equals 𝑠 squared, one side length squared. Before we move on, I want us to think about the relationship between a square and a rhombus. But let’s start first with a parallelogram. One type of parallelogram is a rectangle. Another type of parallelogram is a rhombus. And a square is both a rectangle and a rhombus. And while all of this is really interesting information, for this problem, we’re only interested in this fact that a square is a rhombus. And that means that any rules that apply to a rhombus will also apply to a square, even the formula for finding the area.
We know that a square has two diagonals. If we label one of them 𝑝 and one of them 𝑞, we can say that 𝑝 equals 𝑞. The diagonals are equal to one another. And then we can say that the area of a square can also be found by squaring the length of its diagonal and dividing by two. And this is true because a square is a rhombus. Now, we are ready to actually consider the problem in hand. We have a rhombus and a square. The rhombus has a diagonal of 3.8 centimeters and a diagonal of 5.3 centimeters. And so the area of this rhombus will be equal to 3.8 times 5.3 divided by two.
Our square has a diagonal of 5.3 centimeters. And we know this means both diagonals will be equal to 5.3 centimeters. And so we can say that the area of the square will be equal to 5.3 squared divided by two. 3.8 times 5.3 divided by two equals 10.07. Since this is area we’re dealing with centimeters squared, 5.3 squared divided by two equals 14.045 which again will be centimeters squared. Our question wants to know which shape is smaller in area. As 10.07 is smaller than 14.045, the rhombus is smaller in area than the square.