Lesson Video: Finding the Area of a Rhombus Using Diagonals Mathematics • 6th Grade

In this video, we will learn how to find the area of a rhombus in terms of its diagonal lengths as half the product of these lengths.

17:09

Video Transcript

In this video, we’ll look at how we find the area of a rhombus using the diagonals. We’ll start by looking at the properties of a rhombus. We’ll recap how we find the area using the base and height, and then we’ll prove how we can find the area using these diagonals. Let’s begin with a rhombus.

The mathematical definition of a rhombus is that it’s a quadrilateral with all four sides equal in length. So, we could draw a few rhombuses like this so long as in each rhombus, we know that all the four sides will be equal in length. You may already know that there’s one way we can find the area of a rhombus using the base and height. That’s by saying that it’s equal to the base multiplied by the height. We can think of this visually if we imagine a rectangle drawn with the same base and height. If we imagine this exterior section of triangle being cut off and placed in this empty section in our rectangle, then we can see how the area of this rhombus with a base 𝑏 and height ℎ would be the same as the area of a rectangle with the length and width of the base and height.

So, let’s say that we’re given a rhombus 𝐴𝐵𝐶𝐷. And instead of being given the lengths of the base and height, we’re given the length of the diagonals instead. How could we find the area using these lengths? Sometimes, it’s helpful to rotate this rhombus so we can get a better visual idea. We know that the four sides will all be the same length, and we’re going to consider this rhombus in terms of two triangles. We have triangle 𝐴𝐵𝐶 at the top and triangle 𝐴𝐷𝐶 underneath. We can say that side 𝐴𝐵 is equal in length to side 𝐴𝐷 as they’re two lengths of the rhombus. In the same way, the side 𝐵𝐶 is equal to the side 𝐶𝐷. And as 𝐴𝐶 is a common side, then the length of 𝐴𝐶 is equal in both triangles.

What we’ve shown here is that there are three pairs of corresponding sides congruent. Don’t worry if you haven’t done too much on the congruency of triangles. But what we’ve shown here is that triangle 𝐴𝐵𝐶 is congruent with triangle 𝐴𝐷𝐶. This means that these two triangles are exactly the same shape and size.

Let’s return to the main purpose of this. We really want to find the area of this rhombus 𝐴𝐵𝐶𝐷. We’ve split it into two triangles, so we could say that the area of the rhombus is equal to the area of triangle 𝐴𝐵𝐶 plus the area of triangle 𝐴𝐷𝐶. We’ve just shown that 𝐴𝐷𝐶 and 𝐴𝐵𝐶 are equal. So, this is really like finding the area of 𝐴𝐵𝐶 plus the area of 𝐴𝐵𝐶, or alternatively it’s two multiplied by the area of triangle 𝐴𝐵𝐶. And how do we work out the area of the triangle 𝐴𝐵𝐶? We’d need to recall that the area of a triangle is equal to a half times the base times the height. In triangle 𝐴𝐵𝐶, the base will be the length of 𝐴𝐶 and the height will be half of 𝐵𝐷. We can simplify what’s inside the parentheses to give us two times a quarter times 𝐴𝐶 times 𝐵𝐷. This will then simplify to give us a half times 𝐴𝐶 times 𝐵𝐷. And what exactly are 𝐴𝐶 and 𝐵𝐷? Well, they’re the lengths of the diagonals.

This means we’ve proven that there’s another formula for the area of a rhombus using the diagonals. When 𝑑 sub one and 𝑑 sub two are the lengths of the diagonals, to find the area, we work out 𝑑 sub one times 𝑑 sub two over two. We could, of course, also give this formula as a half times 𝑑 sub one times 𝑑 sub two. As we go through this video, we’ll be looking at questions and applying the formula for the area using the diagonals. But of course, it’s always worth remembering that the other formula using the base and height also exists. So, let’s look at some questions.

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

We can recall that a rhombus has four sides of equal length. We notice that the length and width of this rectangle will also correspond with the lengths of the diagonals in the rhombus. So, we’ll have a diagonal of 30.3 centimeters and a diagonal of 15.8 centimeters. In order to find the area of this rhombus, we remember the formula that the area of a rhombus is equal to 𝑑 sub one times 𝑑 sub two over two, where 𝑑 sub one and 𝑑 sub two are the lengths of the diagonals.

Plugging in the values of 30.3 and 15.8 for the lengths of our diagonals into the formula gives us 30.3 times 15.8 over two. Notice that because we’re multiplying, it doesn’t matter which order we write the diagonals in. As we’re asked for an answer to two decimal places, we could go ahead and plug this into our calculator. But of course, it’s always nice to simplify our calculation where we can. Here, we could take out the common factor of two from 15.8 and two. This will give us an answer of 239.37, and the units here will be the squared units of squared centimeters. As our answer already has just two decimal places, then we won’t need to do any rounding. So, the answer is 239.37 square centimeters.

Let’s have a look at another question. This time, we’re not given a diagram.

A diagonal of a rhombus has length 2.1, while the other one is four times as long. What is its area?

We start by remembering that a rhombus is a quadrilateral with all four sides equal in length. So, when we model our rhombus, we’ll need to have four equal sides. Instead of being given any information about the length of the sides of this rhombus, we’re given information about the diagonals. We’re given that one of these diagonals has a length of 2.1 units, and the other one is four times as long. Four multiplied by 2.1 will give us 8.4. Looking at our diagram, we can see that there’s a shorter diagonal and a longer diagonal. And therefore, the shorter one will be 2.1 units, and the longer one will be 8.4 units.

There are two formulas that we can use to find the area of a rhombus. One involves the base and the perpendicular height, and the other one involves the diagonals. As we’re only given the length of the diagonals here, it would be sensible to use that formula. So, we remember that the area of a rhombus is equal to 𝑑 sub one multiplied by 𝑑 sub two over two, where 𝑑 sub one and 𝑑 sub two are the lengths of the two diagonals. We can plug in the values of our two diagonals to give us 2.1 multiplied by 8.4 over two.

Simplifying our calculation, we’ll need to work out 2.1 multiplied by 4.2, which we can do without a calculator. We work out 21 multiplied by 42 using whatever multiplication method we choose. And then, as our two values had a total of two decimal places, then so will our answer. We weren’t given any length units in the question, but as we’ve worked out an area, we would be using square units. And so, our answer is that the area of this rhombus is 8.82 square units.

In our next question, we’ll find the area of a rhombus given on a coordinate grid.

Determine the area of the rhombus 𝐴𝐵𝐶𝐷. Unit length equals one centimeter.

On the coordinate grid, we have this rhombus 𝐴𝐵𝐶𝐷. As it’s a rhombus, we know that the four sides will all be of the same length. We’re asked to find the area of this rhombus, which is the amount of space within the shape. When we’re finding the area of a rhombus, we have a choice of two different formulas. The first formula for the area of a rhombus tells us that we multiply the two diagonals and divide by two. With the second formula, we would have the base multiplied by the perpendicular height.

In order to establish which area formula we should use, we’ll need to look and see which lengths we’re given. In order to use the second formula, the base would be the length of one of the sides, and we need to find the perpendicular height. As we’re not going to be physically measuring these lengths with a ruler, we’d need to use something like the Pythagorean theorem to find these lengths.

Let’s see if it would be easier to use our first formula. Could we find the length of the diagonals? Well, yes, we can, using the grid. Our horizontal line 𝐴𝐶 goes from negative eight to two on the 𝑥-axis, which means that it will be 10 units long. In fact, we’re told that each unit is one centimeter. So, 𝐴𝐶 will be 10 centimeters. The diagonal 𝐷𝐵 goes from negative five to negative nine on the 𝑦-axis, so it will be four centimeters long. Using the formula that involves the diagonals, we plug in our two values, which gives us 10 multiplied by four over two. We can simplify this first or work out 10 times four is 40 divided by two, which gives us 20. And the units here will be the square units of squared centimeters. We can give our answer that the rhombus 𝐴𝐵𝐶𝐷 has an area of 20 square centimeters.

Let’s have a look at another question.

In the rhombus 𝐴𝐵𝐶𝐷, the side length is 8.5 centimeters, and the diagonal lengths are 13 centimeters and 11 centimeters. Find the length of line segment 𝐷𝐹. Round your answer to the nearest 10th.

We can begin this question by recognizing that a rhombus is a quadrilateral that has all four sides of the same length. We’re told that this side length is 8.5 centimeters, so we can label this on the diagram. We can also label the two diagonals. One of them is 13 centimeters, and one is 11 centimeters. It’s always nice to see if we can get these in the correct positions. And as the diagonal 𝐴𝐶 looks longer than the length of 𝐵𝐷, then it will be 13 centimeters. We’re asked to find the length of this line segment, 𝐷𝐹. If we look at the diagram, we should notice that this length of 𝐷𝐹 is in fact the perpendicular height of the rhombus. So, how could we link the diagonals of the rhombus with the perpendicular height? Well, we can in fact do this using the formulas for the area of a rhombus.

The first formula, we should remember, is that the area of a rhombus is calculated by multiplying the two diagonals 𝑑 sub one and 𝑑 sub two and then halving it. The second formula tells us that the area of a rhombus is equal to the base multiplied by the perpendicular height. As we’re given the lengths of the diagonals in this question, let’s fill these values in to our first formula. We therefore calculate 11 multiplied by 13 divided by two. As we’re asked for our answers to the nearest 10th, we can assume that we’re allowed to use a calculator. So, we can give our answer as 71.5 square centimeters.

Now, we found the area of a rhombus, we can plug our value in to the second formula. On the left-hand side, we’ll have the area as 71.5. The base will be the length of the rhombus, which is 8.5 centimeters. And we’re trying to work out the unknown perpendicular height, which we can leave as ℎ. In order to find the value of ℎ, we would divide both sides of our equation by 8.5, which gives us 8.41176 and so on is equal to ℎ. As we need to round our answer to the nearest 10th, we would check our second decimal digit to see if it’s five or more. And as it isn’t, then our value of ℎ would round to 8.4 centimeters. We know that the length of line segment 𝐷𝐹 is the same as the perpendicular height of the rhombus. So, our answer is that the length of line segment 𝐷𝐹 is 8.4 centimeters.

Let’s look at one final question involving a square and a rhombus.

Two plots of land have the same area. One is a square, and the other is a rhombus with diagonals of lengths 48 meters and 35 meters. What is the perimeter of the square plot? Give your answer to two decimal places.

In this question, it might be helpful to draw some diagrams to visualize the problem. We’re told that there’s two plots of land, and one is a square, and the other is a rhombus. Let’s draw the square. We know that this will be a quadrilateral with four sides of the same length and all the interior angles will be 90 degrees. When it comes to the rhombus, we know that this will be a quadrilateral with all four sides the same length. On this diagram, we can put the double markation on the lines so that we don’t get confused into thinking the lengths of this rhombus will be the same as the lengths of the square.

The other information that we’re given is the lengths of the diagonals on the rhombus. The longer one is 48 meters, and the shorter one is 35 meters. We’re also told that these two plots of land, the square and the rhombus, have the same area. And we’re asked to find the perimeter of the square plot. We can remember that the perimeter of a shape is the distance around the outside. We could do this if we had the length of the side of the square, but we don’t; we’ll need to calculate it. Let’s see if we can find the area of the rhombus. To do this, we’ll need to recall a certain formula.

We can find the area of a rhombus using the two diagonals 𝑑 sub one and 𝑑 sub two by multiplying 𝑑 sub one and 𝑑 sub two and then dividing by two. We can simply plug in our two diagonals of 35 and 48 to work out 35 times 48 over two. It’s always good to simplify a calculation when we can. We’re asked to give our answer to two decimal places here, so we can assume that a calculator would be allowed.

Either by using a calculator or by a written method, we’ll get our answer of 840. And as it’s an area, our units will be square meters. We were told that the areas were the same, which means that the square will also have an area of 840 square meters. We’ll need to remember that the area of a square is equal to the length squared. And this time, we’re plugging in the fact that the area is 840. So, we have 840 is equal to 𝐿 squared. In order to find the value of 𝐿, we would take the square root of both sides of our equation. So, 𝐿 is equal to the square root of 840. As we’re dealing with a length, the units will be in meters.

It’s always tempting to pick up our calculator and find the decimal value for this. But as we still need to find the perimeter, we’ll keep our answer in the square root form. We remember that the perimeter of the square will be the distance around the outside edge, which means that we’ll be working out four multiplied by the square root of 840. Using our calculator, we get the decimal value of 115.93101 and so on meters. Rounding to two decimal places means we check our third decimal digit to see if it’s five or more. And as it isn’t, then our answer rounds down to give us the perimeter of the square is 115.93 meters.

We can now summarize what we’ve learned in this video. We began this video by recalling that a rhombus is a quadrilateral with all four sides equal in length. We recalled the first formula that the area of a rhombus is equal to the base multiplied by the perpendicular height. We then proved that the area of a rhombus is also equal to half of the product of the diagonals. We can write this formula as 𝑑 sub one multiplied by 𝑑 sub two over two or 𝑑 sub one and 𝑑 sub two are the length of the diagonals.

Finally, we can use either of these formulas to find the area of a rhombus, depending on the information that we’re given about the lengths. As we saw in one of the questions, sometimes we’ll need to use both to find some missing length information. We’ll need to remember both of these formulas, but to be careful, because it’s very easy to get confused and work out the base times height and incorrectly divide by two. But we only divide by two when we’re using the formula with the diagonals.

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