Video Transcript
So In this video, we’re looking at
Heron’s formula. And what we’re gonna do is learn
how to use Heron’s formula to find the area of a triangle.
And what Heron’s formula is, is a
way to find the area of a triangle when the length of all three sides are known. And as well as trying to find the
area of a triangle using Heron’s formula, we’re also gonna find the area of
composite shapes using Heron’s formula. Now, you might think it sounds a
bit funny, “Heron’s formula.” It isn’t, in fact, made up by a
heron. Heron’s formula, which is also
sometimes known as Hero’s formula, is named after the hero of Alexandria who
actually devised this formula. Okay, before we actually get
started and ask some questions to show us how to use it, what we need to do is first
of all see what it means.
So, what Heron’s formula is, is 𝐴,
the area, is equal to the square root of 𝑠 multiplied by 𝑠 minus 𝑎 multiplied by
𝑠 minus 𝑏 multiplied by 𝑠 minus 𝑐, where 𝑎, 𝑏, and 𝑐 are the sides of our
triangle. Well, that might be all well and
good. But what’s 𝑠? This isn’t something we’ve seen
before. Well, 𝑠 is, in fact, the
semiperimeter. So, what this means is half of the
perimeter of our triangle, and we have a formula for this as well. And that formula is that 𝑠, the
semiperimeter, is equal to 𝑎 plus 𝑏 plus 𝑐 divided by two. Well, this makes sense because 𝑎
plus 𝑏 plus 𝑐 will give us our perimeter of our triangle. And if we divide it by two, that
would be half of the perimeter or the semiperimeter.
But why would we use Heron’s
formula? Well, Heron’s formula is really
useful because it means we can find the area of a triangle when all we know are all
three side lengths. So, we don’t have to calculate any
other sides or we’d have to find a perpendicular height or we have to find another
side using Pythagorean theorem or an angle, nothing like that. We can just have the length of all
three sides and use it to find the area of our triangle.
Okay, great. So now, we know what Heron’s
formula is. And we know how to use it. Let’s get on and have a look at
some examples. In our first example, we’re just
gonna look at an example where we have to just plug three values into Heron’s
formula.
The area of the triangle whose side lengths are three centimeters, six centimeters, and seven centimeters equals
something cm squared. So, in this question, what we’re
asked to do is find the area of the triangle. And what we’re given are the three
side lengths. And what we know is that if we have
the three side lengths and we want to find the area, what we can use is Heron’s
formula. And what Heron’s formula tells us
is that if we have triangle 𝑎𝑏𝑐, then the area is equal to the square root of 𝑠
multiplied by 𝑠 minus 𝑎 multiplied by 𝑠 minus 𝑏 multiplied by 𝑠 minus 𝑐, where
𝑠 represents the semiperimeter, which is half the perimeter of our triangle. And we’ve got a formula for that;
𝑠 is equal to 𝑎 plus 𝑏 plus 𝑐 over two.
Okay, great. So now we’ve remembered what
Heron’s formula is and what the formula for the semiperimeter is. We can use this to find the area of
our triangle. So first of all, we can find our
𝑠, our semiperimeter, cause this is equal to three plus six plus seven over two,
which is gonna be equal to 16 over two, which is eight. So, we know that the semiperimeter
would be eight centimeters. So now we’ve got the
semiperimeter. What we can do is plug this into
the Heron’s formula for the area. So, what we get is the area is
equal to the square root of eight multiplied by eight minus three multiplied by
eight minus six multiplied by eight minus seven. And because of the way that
formula’s set up, it doesn’t matter which one of our sides are 𝑎, 𝑏, or 𝑐, which
is gonna be equal to the square root of eight multiplied by five multiplied by two
multiplied by one, which is gonna give us root 80.
And then, what we’re going to do is
simplify our root 80. And what we’re going to get is root
16 multiplied by root five. And that’s because we apply one of
our radical or surd rules, which is that root 𝑎 multiplied by root 𝑏 is equal to
root 𝑎𝑏. And therefore, what we can say that
the area of the triangle whose side lengths are three centimeters, six centimeters,
and seven centimeters is four root five centimeters squared.
So that was our first example. We’re now gonna take a look at the
next example which is gonna be similar, just using some slightly different
notation.
𝐴𝐵𝐶 is a triangle, where 𝐵𝐶
equals 28 centimeters, 𝐴𝐶 equals 20 centimeters, and 𝐴𝐵 equals 24
centimeters. Find the area of 𝐴𝐵𝐶 giving the
answer to the nearest square centimeter.
So, the first thing we’ve done is
just drawn a quick sketch to visualize what’s happening. So, what we have is a triangle. And in that triangle, we have three
sides, and we know each of their lengths. So, therefore, if we know the
lengths of each of the sides of our triangle and we want to find the area, then what
we’re going to do is use Heron’s formula. Well, if we quickly remind
ourselves of Heron’s formula, if we have a triangle lengths 𝑎, 𝑏, and 𝑐, then the
area of this triangle is equal to the square root of 𝑠 multiplied by 𝑠 minus 𝑎
multiplied by 𝑠 minus 𝑏 multiplied by 𝑠 minus 𝑐, where 𝑠 is the semiperimeter,
so half of the perimeter of our triangle. And we could find that by adding
together each of the sides, so 𝑎 plus 𝑏 plus 𝑐, and then dividing it by two.
So, the first thing we’re gonna do
is find out 𝑠. And to find this, what we’re gonna
do is add together the three side lengths, so 28 plus 20 plus 24, and then divide
this by two. And this is gonna give us a
semiperimeter of 36 centimeters. So, therefore, the area is gonna be
equal to the square root of 36 multiplied by 36 minus 28 multiplied by 36 minus 20
multiplied by 36 minus 24, which is gonna be equal to the square root of 55296. Well, this is equal to
235.1510153. However, is this the final
answer? Well, no because if we look back at
the question, we can see that we want the answer to the nearest square
centimeter. So, therefore, we can say that to
the nearest square centimeter, the area of the triangle is 235 centimeters
squared.
So far, what we’ve seen is a couple
of examples of how to use Heron’s formula to find the area of a triangle. Now, what we’re gonna take a look
at is example where we’re gonna use Heron’s formula to find the area of a
rhombus.
The perimeter of the given rhombus
is 292 centimeters and the length of 𝐴𝐶 is 116 centimeters. Use Heron’s formula to calculate
the area of the rhombus, giving the answer to three decimal places.
Well, the first thing we need to do
is remind ourselves of Heron’s formula. So, what we’ve got is that if we
have a triangle 𝐴𝐵𝐶 and the area is equal to the square root of 𝑠 multiplied by
𝑠 minus 𝑎 multiplied by 𝑠 minus 𝑏 multiplied by 𝑠 minus 𝑐, where 𝑠 is the
semiperimeter which we can find by adding together each of the sides of the
triangle, so 𝑎 plus 𝑏 plus 𝑐, and then dividing it by two. So, this is useful. Well, we take a look at our diagram
and what we’ve got is a rhombus. Well, what we can do is if we take
a look at the diagonal across our rhombus is see that our rhombus can be divided
into two identical triangles. However, we still only have one
side length of our triangle. But what we can do is we can use
the perimeter of the rhombus to help us work out what the other side lengths
are. And that’s because in a rhombus
each of the sides are the same length.
So therefore, if we call each of
the side lengths of our rhombus 𝑥, we can say that 𝑥 is gonna be equal to 292
divided by four. Well, this is equal to 73. So therefore, we could say that
each of the side lengths of our rhombus is 73 centimeters. And more importantly, if we look at
the orange triangle, we’ve now got a triangle where we know all three sides, 116,
73, and 73. Well, then, the first thing we want
to do is work out what the semiperimeter is. And we do that by adding together
our side lengths, so 116 plus 73 plus 73, and then dividing it by two. And this gives us a semiperimeter
of 131 centimeters.
Okay, great. So now what we can do is substitute
this and our side lengths into our formula, so our Heron’s formula. And when we do that, we get that
the area is equal to the square root of 131 multiplied by 131 minus 116 multiplied
by 131 minus 73 multiplied by 131 minus 73, which is equal to 2571.0425 et
cetera. We might think, “Wow, great! We found the area. Can we finish at this point?” But no. And why is that? Well, if we take a look back at our
rhombus, we can see that, in fact, it’s two identical triangles. So, therefore, we need to multiply
this by two. And when we do that, we get
5142.08518 et cetera. And then if we look back at the
question for the degree of accuracy we want the answer left in, we can see we want
it to three decimal places. So, therefore, after rounding, we
can say that the area of the rhombus is 5142.085 centimeters squared, and we set
that to three decimal places.
So great, we’ve looked at a
selection of questions so far. We’ve got two more examples
now. The first one is gonna have a look
at a composite figure, so more complex composite figure. And we’ll be using Heron’s formula
to find the area of it, and that hits one of the objectives of the lesson. And then the final problem is a
problem-solving question, where we look to find the radius of a circle.
Find the area of the figure below
using Heron’s formula, giving the answer to three decimal places.
So, the first thing we do is we
remind ourselves about Heron’s formula. And Heron’s formula tells us what
the area is of a triangle if we know all three side lengths. So, for instance, if we’ve got the
triangle 𝑎𝑏𝑐, then the area is equal to the square root of 𝑠 multiplied by 𝑠
minus 𝑎 multiplied by 𝑠 minus 𝑏 multiplied by 𝑠 minus 𝑐, where 𝑠 is a
semiperimeter which we can find by adding together the all three side lengths, 𝑎
plus 𝑏 plus 𝑐, and then dividing it by two.
Well, if we take a look at our
shape, it’s a composite made up of two triangles, triangle 𝐴 and triangle 𝐵. So, in fact, what we’re gonna do is
we’re gonna start with triangle 𝐵. And that’s because for triangle 𝐵,
we know all three sides. So, the first thing we can do is
find the semiperimeter of triangle 𝐵. And that’s gonna be equal to all
three side lengths added together divided by two, so 20 plus 23 plus 16 over two,
which is gonna give us a semiperimeter of 29.5 centimeters.
Okay, great. So, now, what we can do is
substitute this into Heron’s formula to find the area of triangle 𝐵. So, therefore, the area of the
triangle is gonna be equal to the square root of 29.5 multiplied by 29.5 minus 20
multiplied by 29.5 minus 23 multiplied by 29.5 minus 16, which is gonna be equal to
156.818166 et cetera centimeters square. Okay, great. We don’t round at this point
because we don’t want any rounding errors before we get to the final answer. So, that’s the area of triangle
𝐵.
Now, let’s move on to triangle
𝐴. Well, for triangle 𝐴, we only have
two side lengths, and what we need to do is find the other side length and that’s so
that we can find the area of the triangle. I’m gonna call the other side
length 𝑥. And because we’ve got a right
triangle, what we can do is use the Pythagorean theorem. And what the Pythagorean theorem
tells us is that if we have a right triangle, then we have 𝑎 squared plus 𝑏
squared equals 𝑐 squared, where 𝑐 is the hypotenuse or longest side. But what we’re gonna do is
rearrange this because what we’re trying to find is the shorter side. So, what we’re gonna get is that 𝑥
squared is equal to 20 squared minus 16 squared. So, 𝑥 is gonna be equal to root
144. So, therefore, the length of 𝑥 is
gonna be 12 centimeters.
So, what we now know is all three
lengths of our triangle. But what we could do is use Heron’s
formula. However, because of the type of
triangle we have, what we’re gonna instead use is the area of a triangle is equal to
a half the base times the height cause we know the base and we know the
perpendicular height. So, therefore, the area of triangle
𝐴 is gonna be equal to a half multiplied by 12 multiplied by 16, which is equal to
96 centimeters squared. So, now, the final stage to find
the total area is gonna be to add this to the area of triangle 𝐵. Well, when we do that, we’re gonna
get 252.8181 et cetera. However, the question wants the
answer to three decimal places, which is gonna give a final answer of 252.818
centimeters squared to three decimal places.
Okay, great. Now, we’re gonna move on to our
final example.
The lengths of a triangle are 12
centimeters, five centimeters, and 11 centimeters. Find the radius of the interior
circle touching the sides using the formula 𝑟 equals the area of the triangle
𝐴𝐵𝐶 over 𝑝, where 𝑝 is half of the triangle’s perimeter.
Well, to be able for us to use the
formula for the radius of the circle, what we need to do is find the area of the
triangle. And what we’re told are three side
lengths of our triangle: 12, five, and 11. Well, if we have three side lengths
of a triangle, then what we can use is Heron’s formula to find the area. And Heron’s formula tells us that
the area is equal to the square root of 𝑠 multiplied by 𝑠 minus 𝑎 multiplied by
𝑠 minus 𝑏 multiplied by 𝑠 minus 𝑐, where 𝑎, 𝑏, and 𝑐 are the side lengths of
our triangle. And 𝑠 is equal to 𝑎 plus 𝑏 plus
𝑐 over two because it’s the semiperimeter of our triangle. Well, actually, this ties in with
the formula we have for the radius because we’re told that the formula for the
radius is equal to the area of the triangle divided by 𝑝, where 𝑝 is half of the
triangle’s perimeter. Well, in fact, this is the same as
𝑠 because they both mean the semiperimeter or half of the triangle’s perimeter.
So, the first thing we want to do
is find out the area of the triangle. And to do that, firstly, we need to
find the semiperimeter or 𝑝, the half-perimeter of the triangle. Well, this is gonna be equal to 12
plus five plus 11 over two. So, this is gonna be equal to 14
centimeters. So great, what we can do now is
substitute this into Heron’s formula. And when we do that, we’re gonna
get the area is equal to the square root of 14 multiplied by 14 minus 12 multiplied
by 14 minus five multiplied by 14 minus 11, which is equal to root 756. And if we simplify this, we get six
root 21. And we’re gonna keep it in surd
form to keep accuracy, and the units for this will be centimeters squared because
it’s our area.
Okay, great. So, now, we have everything we need
to substitute into the formula to find the radius. What we’ve done is that we sketched
what we’re trying to find because what we’re trying to find is the radius of the
interior circle which touches the sides of our triangle. So, we can say that the radius is
equal to six root 21 over 14. And we know that because 14 was our
𝑠, our semiperimeter. And we’d already said that this is
the same as 𝑝. So, therefore, this is gonna give
us our final answer, which is the radius of the interior circle is three over seven
multiplied by root 21 centimeters.
Okay, great, we looked at a range
of different examples covering all the different objectives of the lesson. But what we’re gonna do now is just
have a look at the key points once again. Well, the first and main key point
is what Heron’s formula is. Heron’s formula allows us to find
the area of a triangle when we know all three side lengths. And we get that with the formula 𝐴
is equal to the square root of 𝑠 multiplied by 𝑠 minus 𝑎 multiplied by 𝑠 minus
𝑏 multiplied by 𝑠 minus 𝑐, where 𝑠 is the semiperimeter. So that is half of the perimeter of
the triangle. And we get that by adding the sides
together, so 𝑎 plus 𝑏 plus 𝑐, and then dividing it by two.
And what we also found in the
lesson was an interesting adaptation of Heron’s formula, and that was to find the
radius of an interior circle which touches the sides. And that formula was that 𝑟 is
equal to the area of the triangle, which we could find using Heron’s formula,
divided by 𝑝, or in fact 𝑠 because it’s the semiperimeter or half-perimeter of the
triangle.
