Lesson Video: Area of a Rhombus Mathematics

Video Transcript

In this video, we will learn how to find the area of a rhombus using the length of its diagonals. We’ll recall first that a rhombus is any quadrilateral in which all four sides are of equal length. So, for example, a square is an example of a rhombus, but not all rhombuses or rhombi are squares. The interior angles aren’t always right angles. So a rhombus can also look like this. As long as all four sides are the same length, it’s a rhombus.

As a rhombus is in fact a special case of a parallelogram, its area can be calculated from the length of its base and its perpendicular height using the formula area equals base times height, or 𝑏ℎ. What we’re going to introduce and practice here though is another method for finding the area of a rhombus, using instead the lengths of its diagonals. So here we have a rhombus in which we’ve labeled the vertices 𝐴, 𝐵, 𝐶, and 𝐷. We can add in the two diagonals of the rhombus, which are the two line segments that connect opposite corners together. So these are the line segments 𝐴𝐶 and 𝐵𝐷.

A key property of all parallelograms — so, in particular, it’s true for rhombuses — is that their diagonals bisect one another. So, in our diagram, 𝐴𝐸 is the same length as 𝐸𝐶 and 𝐷𝐸 is the same length as 𝐸𝐵. But another key property of rhombuses, which isn’t true in general for other parallelograms, is that the diagonals also intersect at right angles. Each of the diagonals divides the rhombus into two congruent triangles.

Let’s focus on the diagonal 𝐵𝐷, which divides the rhombus into triangles 𝐴𝐵𝐷 and 𝐶𝐵𝐷. These two triangles are congruent because they each have two sides, which are the same length as the sides of the original rhombus. And then they have a shared side 𝐵𝐷. So, by the side-side-side congruency condition, they’re identical to one another. The area of the rhombus is therefore twice the area of each individual triangle. Focusing just on the upper triangle then, we can say that the area of the rhombus 𝐴𝐵𝐶𝐷 is twice the area of triangle 𝐴𝐵𝐷.

Now, we can find the area of a triangle using the formula base multiplied by perpendicular height over two. The base is 𝐵𝐷, and the perpendicular height is 𝐴𝐸. Remember, we said that the diagonals of a rhombus are perpendicular to one another. So 𝐴𝐸 is the perpendicular height of this triangle. But remember, we also said that the diagonals of a rhombus bisect one another. So the length of 𝐴𝐸 is half of the length of 𝐴𝐶. The area of triangle 𝐴𝐵𝐷 then is therefore one-quarter of 𝐵𝐷 multiplied by 𝐴𝐶. And so the area of the rhombus 𝐴𝐵𝐶𝐷, which is twice the area of triangle 𝐴𝐵𝐷, is two times a quarter times 𝐵𝐷 times 𝐴𝐶, which is a half of 𝐵𝐷 multiplied by 𝐴𝐶.

Now, remember that 𝐵𝐷 and 𝐴𝐶 are the diagonals of this rhombus. So we found a formula for calculating its area using the lengths of its diagonals. It’s one-half of their product. We can therefore generalize this result using 𝑑 one and 𝑑 two to represent the lengths of the two diagonals. The area of a rhombus with diagonals of length 𝑑 one and 𝑑 two units is equal to 𝑑 one 𝑑 two over two, which in words we can think of as the area of a rhombus is equal to half the product of the lengths of its diagonals. We’ll now consider some examples in which we apply this formula.

The figure shows a rhombus within a rectangle. Find the area of the rhombus to two decimal places.

Looking at the diagram, we notice that the vertices of the rhombus are each at the midpoint of one of the rectangle sides. We know this because these line markers indicate that, for example, line segment 𝐴𝑋 is the same length as line segment 𝑋𝐷. From this, we can deduce that the diagonals of the rhombus, that’s 𝑋𝑍 and 𝑌𝑇, are each parallel to one side of the rectangle. And so it follows that they’re also the same length as the rectangle sides. So 𝑋𝑍 is 15.8 centimeters and 𝑌𝑇 is 30.3 centimeters.

We can then recall that the area of a rhombus is equal to half the product of the lengths of its diagonals. If the lengths of the diagonals are 𝑑 one and 𝑑 two, then the area of a rhombus is 𝑑 one multiplied by 𝑑 two over two. So the area of the rhombus 𝑋𝑇𝑍𝑌 is equal to the length of 𝑌𝑇 multiplied by the length of 𝑋𝑍 over two. That’s 30.3 multiplied by 15.8 over two, which is 293.37 square centimeters.

Now, we can observe that the area of the rectangle 𝐴𝐵𝐶𝐷 is equal to its length multiplied by its width. That’s 30.3 multiplied by 15.8, and this is double the area of the rhombus. This illustrates a general result, which is that the area of a rhombus drawn inside a rectangle such that each vertex of the rhombus is at the midpoint of one of the rectangle sides is half the area of the rectangle enclosing it.

Let’s now consider another example in which we’re given the area of a rhombus and the relationship between its diagonals and asked to calculate their length.

One diagonal of a rhombus is twice the length of the other diagonal. If the area of the rhombus is 81 square millimeters, what are the lengths of the diagonals?

In this question, we’re required to relate the area of a rhombus to the lengths of its diagonals. So we recall that the area of a rhombus is equal to half the product of the lengths of its diagonals. It’s equal to 𝑑 one 𝑑 two over two, where 𝑑 one and 𝑑 two are the lengths of the diagonals. We’ve been given that the area of this rhombus is 81 square millimeters, but not the lengths of either diagonal. However, we do know that one diagonal is twice the length of the other. So we could let diagonal length 𝑑 one be equal to twice the length of diagonal 𝑑 two.

We can then form an equation. Replacing 𝑑 one with its expression in terms of 𝑑 two and the area of the rhombus with 81, we have the equation two 𝑑 two multiplied by 𝑑 two over two is equal to 81. The twos in the numerator and denominator of this fraction will cancel each other out. And we’re left with 𝑑 two multiplied by 𝑑 two, or 𝑑 two squared, is equal to 81. To find the value of 𝑑 two, we take the square root of both sides of this equation, taking only the positive value as 𝑑 two represents a length. So 𝑑 two is equal to the positive square root of 81, which is nine.

So we found the length of one diagonal. And to find the length of the other, we need to double this value. 𝑑 one is two times nine, which is 18. So, by recalling that the area of a rhombus is half the product of the lengths of its diagonals, which we used to set up and then solve an equation, we found that the lengths of the diagonals of this rhombus are nine millimeters and 18 millimeters.

In our next example, we’ll solve a problem involving a rhombus and a square of equal area.

A rhombus and a square have the same area. If the square’s perimeter is 44 and one of the diagonals of the rhombus is 10, how long is the other diagonal to two decimal places?

So we have a rhombus and a square which have the same area. And we’re given some other information about each shape. First, the perimeter of the square is 44 units. And secondly, one of the diagonals of the rhombus is of length 10 units. As we’re told that the areas of these two shapes are the same, this must be key information. So let’s begin by calculating the area of a square. We know that the area of a square is its side length squared. As we know that the perimeter of this square is 44 units, we know that four times the side length is equal to 44. And then dividing both sides of this equation by four, we find that the side length of the square is 11 units. So its area is 11 squared, which is 121 square units.

We now know that the area of both the square and the rhombus is 121 square units. And we want to use this information in conjunction with the fact that one diagonal of the rhombus is of length 10 units to calculate the length of the other diagonal. We should recall that the area of a rhombus is half the product of the lengths of its diagonals, 𝑑 one 𝑑 two over two. So, as we already know the length of one diagonal is 10 units and the area is 121 square units, we have that 10 multiplied by the length of the second diagonal over two is equal to 121. Simplifying, we find that five multiplied by the length of the second diagonal is 121. And then dividing both sides of this equation by five, we find that the length of the second diagonal is 121 over five or 24.2 units. We were asked to give the answer to two decimal places though, so that’s 24.20 units.

The example we just considered involved calculating the area of a square, which we did by squaring its side length, a formula we’ve been familiar with for many years. But as a square is simply a special type of rhombus in which all the interior angles are right angles, we can also calculate the area of a square using the formula we’ve introduced in this video. That would be 𝑑 one 𝑑 two over two, where 𝑑 one and 𝑑 two are the lengths of the square’s diagonals. However, there’s something special about squares, which is that the diagonals are of equal length. So, instead of using the letters 𝑑 one and 𝑑 two, we can simply use the letter 𝑑 to represent both. And the formula then becomes that the area of a square is equal to 𝑑 squared over two. We can state this as a general result. The area of a square is equal to half the square of the length of its diagonal. Let’s now consider an example in which we apply this formula.

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.

Let’s now consider one final example in which we’ll apply the formulae we’ve introduced to compare the areas of a square and a rhombus.

Determine the difference in area between a square having a diagonal of 10 centimeters and a rhombus having diagonals of two centimeters and 12 centimeters.

Although it’s not essential to answering this problem, we’ll begin by sketching the two shapes. We have a square with a diagonal of length 10 centimeters and a rhombus with diagonals of lengths two centimeters and 12 centimeters. We then need to recall how to find the area of each of these shapes from the length of their diagonal. The area of a square with a diagonal of length 𝑑 units is 𝑑 squared over two. The area is half the square of the length of the diagonal. So the area of this square, which has a diagonal of 10 centimeters, is 10 squared over two. That’s 100 over two, which is 50 square centimeters.

The area of a rhombus, on the other hand, with diagonals of lengths 𝑑 one and 𝑑 two is 𝑑 one multiplied by 𝑑 two over two. It’s half the product of the lengths of its diagonals. So the area of this rhombus, which has diagonals of lengths 12 centimeters and two centimeters, is 12 multiplied by two over two, which simplifies to 12 square centimeters. To find the difference in area, we subtract the area of the smaller shape, that’s the rhombus, from the area of the larger shape, that’s the square, 50 minus 12, which gives a difference in area of 38 square centimeters.

Let’s now summarize the key points from this video. The area of a rhombus is half the product of the lengths of its diagonals. If these diagonals have lengths 𝑑 one and 𝑑 two units, then we can express this as the area of a rhombus is equal to 𝑑 one multiplied by 𝑑 two over two. A square is a special type of rhombus in which all the interior angles are right angles and the two diagonals are of equal length. As a result, the area of a square is half the square of the length of its diagonal. So the area of a square with diagonals of length 𝑑 units is equal to 𝑑 squared over two. We can use these formulae to solve a variety of problems concerning the areas of rhombuses and the areas of squares.