# Lesson Video: Area of a Triangle Mathematics • 6th Grade

In this video, we will learn how to calculate the area of a triangle given its base length and height.

12:53

### 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.