Lesson video

Hi, I'm Mr. Bond.

And in this lesson, we're going to learn how to find the area of a triangle using the formula a half ab sin C.

Let's start by recapping a method that we already know to find the area of a triangle.

We know that the area of a triangle is equal to a half multiplied by the base multiplied by the height.

So for this triangle, the base is six centimetres, and the height is four centimetres.

The height needs to be the perpendicular height.

And we know that four centimetres is the perpendicular height because it's a right-angle triangle.

So we need to perform this calculation.

A half multiplied by six is equal to three, and three multiplied by four is equal to 12.

So the area of this triangle is 12 centimetres squared.

Now we're going to find the area of this triangle.

Look at both triangles and think, what's the same, and what's different? Hopefully, you've noticed that two of the lengths are the same, four centimetres and six centimetres, but the angle is different.

The triangle on the left is a right-angle triangle.

And the triangle on the right is not.

The angle that was 90 degrees is now 70 degrees.

Think about how this will affect the height of the triangle.

It's going to be slightly shorter.

So what will we expect will happen to the area? It's going to be slightly less.

Okay.

So in order for us to find the area of this triangle, we need to know its height, marked on here with a dotted line.

That dotted line shows the perpendicular height.

So we've effectively split that triangle up into two right-angle triangles.

Let's show that triangle on the left separately, because this will now help us work out the height.

Using trigonometry, we can say that sin 70 is equal to the height divided by four.

And this comes from the fact that sin theatre is equal to opposite over hypotenuse.

So to solve the h, we can multiply both sides of the equation by four to give four sin 70 is equal to h.

Now we know that the area of a triangle is equal to one half multiplied by the base multiplied by the perpendicular height.

So now for this triangle, we know that that's going to be a half multiplied by six, which is the length of the base, multiplied by four sin 70, which we've just worked out is the height.

Performing this calculation on our calculators gives that the area is equal to 11.

3 centimetres squared to three significant figures.

So we've just done a specific example of finding the area of a triangle.

Now we're going to generalise by using variables for the lengths, a, b, and c.

And also a variable for the angle C.

Let's think about how we worked out the height last time.

We said that sin C is equal to the height divided by b.

So to solve for the height, we need to rearrange the formula, and multiply both sides of the equation by b.

So the height is equal to b sin C.

We know that the area for any triangle is given by a half multiplied by the base multiplied by the perpendicular height.

For our triangle, the base is equal to a, and we've just worked out that the perpendicular height is equal to b sin C.

Substituting those into our formula gives this.

The area is equal to a half, multiplied by a, multiplied by b sin C.

When we write a formula, we don't normally use the multiplication sign.

So let's rewrite this.

The area of a triangle is equal to a half ab sin C.

We've just discovered another formula for finding the area of a triangle.

Let's use the new formula we've just learned about to find the area of this triangle.

We know that the area is equal to a half ab sin C.

which means a half multiplied by a multiplied by b multiplied by sin C.

So for this triangle, our value for a is 12.

Our value for b is 16.

And our value for the angle C, which is the included angle between a and b, is 28 degrees.

Substituting those into the formula, gives this.

And when we use our calculator to perform this calculation, we get 45.

1 metres squared to three significant figures.

Here's a question for you to try.

Pause the video to complete your task and resume the video when you've finished.

Here's the solution.

So this was just a case of doing exactly what we did in the previous example.

Substituting a is equal to either four or nine, b is equal to the other one, and c is equal to 86 degrees.

Here's another question for you to try.

Again, pause the video to complete your task, and resume the video when you're finished.

Here's the answer.

You'll see that I've given two different options for the answer.

This is because the units given per each length were mixed.

So we had 1.

3 metres and 90 centimetres.

So you could have converted metres to centimetres or centimetres to metres in order to find the final answer.

Here's something that I'd like you to have a think about.

Would the formula is still work, does the formula, the area is equal to a half ab sin C still work for right angle triangles? Pause the video to have a think, and resume the video when you're finished.

What did you think? You could have tried using it for some right angle triangles.

Yes, it works for all triangles.

What about when the value of C is equal to 90 degrees? Well, when C is equal to 90 degrees, sin C is equal to one.

So the formula becomes area equals half a b, where a and b are the base and perpendicular height.

Here's an example where I'd like you to see if you can spot the mistake.

Take a moment to have a read through the working and see if you can spot the mistake.

Did you manage to find it? The mistake is here.

They've substituted the value for the angle C as 48 degrees.

But this isn't the angle C, it's in fact, the angle b.

When looking for our angle C, it needs to be the angle that's included between a and b.

Or, we can think about it as the angle that's opposite sin C.

So, to go through this again, the formula's correct.

The area is equal to a half multiply by a multiplied by b multiplied by sin C.

Our values for a and b are 8.

9 and 8.

1.

But our value for the angle C is 77 degrees.

So this is the correct substitution.

And when we perform this calculation on our calculator, we'll get that the area is equal to 35.

1 centimetres squared to three significant figures.

Here's today's final question for you to try.

Pause the video to complete the task and resume the video when you've finished.

Here's the answer.

In this question for part a, it was a little bit of revision of the sin and co-sin rules.

So you could have used either the sin or the co-sign rule to find the length AC.

And then in part b, we needed to find the area of triangle ABC.

Now, the easiest way to do this is just using the information that was given, and using a and b is 7.

5 metres and nine metres, and our value for the angle C, 35 degrees.

That's all for this lesson.

Thanks for watching.