Video Transcript
In this video, we are going to see
how we can calculate the area of a triangle by using its base and height. We’ll see that there are in fact
two equivalent ways in which we can write the formula to find the area of a
triangle. But in order to see how we
calculate the area of a triangle, let’s start by reviewing how we find the area of
another geometric shape, a rectangle.
Let’s take this rectangle, which is
drawn on squared paper. By counting the squares, we could
see that the length of the rectangle is seven units and the width of the rectangle
is four units. That means there are 28 squares in
the rectangle. We would say that the area of the
rectangle is 28 square units. And in fact, we can recall that we
can calculate the area of any rectangle if we know its length and width, even if
it’s not drawn on squared paper.
To find the area of a rectangle, we
multiply the length by the width. So the area of this rectangle with
its length of six centimeters and width of five centimeters is found by calculating
five times six. That’s 30. And the area units in this case
would be square centimeters. So now, let’s think about how we
might relate rectangles to triangles.
Let’s draw a right triangle that
has a base of five units and a height of four units. Now, to find the area of this
triangle, we could try counting squares. But there are lots of parts of
squares along the diagonal that become tricky to add up accurately. So, instead, let’s draw a rectangle
around the triangle that has the same length and height. That means that it will have a
length of five units and a height of four units. The area of the triangle will be
part of the area of the rectangle.
We can work out that the area of
the rectangle is five times four, which is 20 square units. And can we see what portion or
fraction of this rectangle that the triangle is? It’s half of it. The area of this triangle is half
of 20 square units, so that’s 10 square units.
Of course, we don’t always have
triangles drawn for us on squared paper. However, we can still apply the
same method to find the area of a triangle. For example, this right triangle
has a base of nine centimeters and a height of six centimeters. Two copies of the triangle combine
to make a rectangle whose area is double the area of the original triangle.
The area of the rectangle around it
is calculated by multiplying the length by the height. So that’s nine times six, which
gives 54 square centimeters. Therefore, the area of the triangle
is half of this. Half of 54 is 27, so the area is 27
square centimeters. So, now that we have seen how to
find the areas of right triangles, what about the areas of triangles which don’t
have right angles?
Here is a triangle that has a base
of 10 centimeters and a perpendicular height of eight centimeters. In the same way as before, we can
complete a rectangle around it, which has a length of 10 centimeters and a height of
eight centimeters. But is the area of the triangle
still going to be half the area of the rectangle?
Let’s shade in the right-hand part
of the triangle. Notice how we’ve got a small
triangle and a small rectangle. And the area of this small triangle
is half the area of the small rectangle. And we can do the same for the
remaining part of the triangle on the left. Its area is equal to half of this
other rectangle. So the area of the triangle is half
of the rectangle that surrounds it.
Let’s now put together a formula
that will help us find the area of a triangle without having to draw in rectangles
every time. The area of a triangle is equal to
the base multiplied by the height divided by two. Remember that the base times the
height is equivalent to finding the length times the width to find the area of the
rectangle.
Notice that the height must always
be the perpendicular height. This won’t necessarily be the
length of a side in the triangle, especially if it isn’t a right triangle. We have to draw a line through the
tip of the triangle which is 90 degrees to the base and measure the height using
that line.
Now, we can also write this formula
in a different way, and it means the same thing. The area of a triangle is equal to
one-half times the base times the perpendicular height. So, for our example, we could write
that the area of the triangle is one-half times 10 times eight. And we can work that out in a few
different ways. We could multiply the 10 and eight
first, which is 80, and then halve it to give 40 square centimeters. Or we could halve the 10 first,
which is five, and then multiply the five by eight. That’s also 40 square
centimeters.
If one of the base or perpendicular
height is an even number, then we might find it easier to halve it first before
multiplying by the other dimension. So let’s now see how we can put
these formulas into practice in the following example.
𝐴𝐵𝐶𝐷 is a rectangle. Find the area of triangle
𝐴𝐵𝐹.
In the diagram, we can see that we
have a rectangle with vertices 𝐴, 𝐵, 𝐶, and 𝐷. However, we are interested in this
triangle 𝐴𝐵𝐹. That’s the triangle that has the
vertices 𝐴, 𝐵, and 𝐹.
Let’s recall the formula to find
the area of a triangle. The area of a triangle is equal to
one-half times the base times the height. And that height must be the
perpendicular height. But in this triangle, we don’t know
the perpendicular height, or do we?
We know that the rectangle has a
height of five centimeters. And since this is a rectangle, then
the distance from the base to the top will always be five centimeters. So the line segment 𝐴𝐵 is five
centimeters below 𝐹, which is the top of the triangle. We can show this on our diagram
with a line segment drawn from 𝐹 perpendicular to 𝐴𝐵.
So now we know that the triangle
has a base of eight centimeters and a perpendicular height of five centimeters. This means that the area of
triangle 𝐴𝐵𝐹 is one-half times eight times five. One-half of eight is four, and four
times five is 20. Not forgetting the units for the
area, we can give the answer that the area of triangle 𝐴𝐵𝐹 is 20 square
centimeters, since both of our lengths were given in centimeters.
So far, we’ve seen how to find the
area of a right triangle and triangles which are not right triangles. But sometimes we have triangles
where the top of the triangle is not above the base, a triangle that might look
something like this. The important thing to remember is
that our formula to find the area of a triangle is always the same. And the height must always be the
perpendicular height. The base will be the same as we saw
before. It is one of the sides of the
triangle. But what about the perpendicular
height?
Well, we can’t use this side length
in orange because this isn’t perpendicular to the base. And we can’t draw a perpendicular
line up from the edge of the base like this because this isn’t the full height of
the triangle. Instead, to find the perpendicular
height, we need to consider the perpendicular distance from the top of the triangle
to a line which is an extension of the base. We can then use the formula with
the base and perpendicular height.
Let’s see how this works for a
triangle with some given measurements. Here, we have a triangle drawn with
one side of length four centimeters and another side of length 13 centimeters. It has a perpendicular height of
five centimeters from the four-centimeter base. Remember that if we have one of the
side lengths of a triangle and its perpendicular height from that side, we can work
out the area. So we can take the base of this
triangle as four centimeters and the perpendicular height from this side as five
centimeters.
Therefore, the area of the triangle
can be calculated as one-half times four times five. Four multiplied by five is 20, and
then we need to halve this. This gives us an area of 10 square
centimeters.
Notice how we also had this length
of 13 centimeters. Sometimes, we do have extra
information on a triangle diagram. Or we can work out the length of
some of the other sides of the triangle. But that doesn’t mean that we have
to use this extra information. Remember that to find the area of a
triangle, we just need to use the base length and the perpendicular height from that
side.
Another important thing to note
with triangles shaped like this is that this extra length that we extend from the
base to the perpendicular height should not be included as part of the base
length. For example, if we knew that this
length was eight centimeters, it would not make the base a total of 12
centimeters. The base length of the triangle
would still be four centimeters.
Now, before we finish with this
video, there is an important point to really highlight about the base and the
perpendicular height of a triangle. We are used to seeing triangles
drawn like this, with one of the sides placed horizontally. And so if we want to work out the
area, we’d multiply the value of the base by that of the perpendicular height to the
base and halve it.
But we can pick any side to be the
base of a triangle. So we could take this side to be
the base, in which case we must find the perpendicular height from that base, even
if we have to extend the base to find the perpendicular height. We can call any of the three sides
of a triangle the base. So, when we are talking about the
perpendicular height of a triangle, we really mean the perpendicular height from
that chosen base. So, no matter which orientation a
triangle is drawn in or which side we call the base, we can still calculate the area
of a triangle by using one of the formulas we have seen.
Let’s now summarize the key points
of this video. We saw that we can write a formula
to calculate the area of a triangle in two different but equivalent ways, either as
the area of a triangle equals the base times the height divided by two or the area
of a triangle equals one-half times the base times the height. The important thing to remember is
that we always use the perpendicular height from the side chosen to be the base to
calculate the area. Whether the triangles have a right
angle, don’t have a right angle, or even if we need to find a perpendicular height
using an extended line from the base, we can still apply one of the formulas above
to find the area of a triangle.
