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.
