Everyone, you may remember that to calculate the area of any triangle, you'll always need the height. You may recognize the one-half base times height formula. It works even for non-right triangles. As long as you're given the height and base, you can find the area. But some questions will give you angles and sides, and you won't actually be given the height. But don't worry because in these types of problems, I'm going to show you how you can find it by using the sine of a known angle. So, what I'm going to show you in this video is that we're going to work out the areas of these two triangles, and they're exactly the same. We're just going to use different information to get there, and then I'll show you three new equations that you'll need to solve for the area. Let's get started here.

In this first example, we will have one-half if we have the base and the height of this triangle, then we could just use the familiar one-half base times height formula to find the area. So, really, what this becomes here is one-half. The base of this triangle is 8, and the height of this triangle is just equal to 3. When you work this out, this ends up being 12. Pretty straightforward. That's the area of this triangle. By the way, it doesn't necessarily have to be b as the base. You could have a or c. All these things could be potentially different letters. But the area of this is 12.

What about this triangle over here? The key difference is that we actually don't have that height that's already drawn for us, but the principle is still the same. We're still just going to use one-half of the base times height. What's the base of this triangle? Well, actually, it's the same thing as the triangle on the left. The base really is just b. So I'm just going to write that over here. What about the height of this triangle? Well, here's where it's a little bit different because whenever you want to draw the height of a triangle, you're just going to take the base that you've drawn, and you're basically going to draw a perpendicular line up to one of the corners. And this effectively sort of turns this into a right triangle. So, what is the value of this height over here, which I'll write as h? Well, the idea here is that if you have the sides, then basically what you can do is turn this into a right triangle, and we have an angle and the hypotenuse of this triangle so we can use sohcahtoa. So, the sine of this over here, we're going to use the sine of angle a which is equal to h over the hypotenuse, which is c. If I rearrange for this, what I can find is that h=c⋅sin(a). So, basically, what happens here is that if I have the hypotenuse and angle, I can figure out the height, and this effectively becomes the height of my triangle.

So, really what happens is that the base becomes the letter b, and the height of this triangle actually becomes c⋅sin(a). So b is the base and then c⋅sin(a) is the height. Alright. So let's see actually what this works out to be. This is going to be area equals one-half, and I'm done my base is still b exactly like it was on the left. And then now what happens is instead of 3, I'm just going to plug in c⋅sin(a), which is 6 times the sine of 30 degrees. When you work this out, what you're actually going to get is that this actually equals 12 exactly like what it did on the left side. So the areas of these two triangles are exactly the same. This actually makes total sense because if you solve for the height of this triangle, this is really just 6 times the sine of 30, which is actually just equal to 3. So using different information, we found the height of this triangle is still 3, and, therefore, the areas are going to be exactly the same. That's really all there is to it. Alright. Now, remember that the positions of these variables will be in different places throughout your triangles.

So there are sort of three different variations of this formula, kind of like how there was for the law of cosines. I'm just going to go ahead and show you those. Sometimes you'll have BC sin A. Another variation that you might see is AC sin of big B. And the last one you'll see here is AB sin C. So what you'll notice here is that the angle is always going to be the last the third letter that is not in these other two over here. And it's always two lowercase letters and the sign of an angle over here. Alright. Now, remember that all of these things will be multiplied, so you can shuffle these letters around, but this is the idea behind finding the area of a non-right triangle. Thanks for watching, and let's get some practice.