Question Video: Deriving the Formula for Calculating the Area of a Triangle Using Sine Mathematics • First Year of Secondary School

Video Transcript

In the given figure, find an expression for the height ℎ in terms of 𝑎, 𝑏, and 𝑐. Find an expression for the area of the triangle in terms of 𝑎, 𝑏, and 𝑐.

Let’s begin by finding an expression for the height ℎ in our triangle. If we look carefully, we can partition this triangle somewhat. And in doing so, we create a smaller triangle with an angle of 𝑐 degrees and a length of 𝑏.

Now in fact, this length 𝑏 represents the length of the hypotenuse of this right-angle triangle. And we know that it’s a right-angle triangle because angles on a straight line sum to 180 degrees. And since we have an angle of 90 degrees, we can find the angle on the other side of the dotted line by subtracting 90 from 180. That gives us 90 degrees. And therefore, we have a right-angle triangle. And since 𝑏 is the longest side in the triangle, it’s the side opposite the right angle we know it’s a hypotenuse.

We can also say that the side in this triangle labelled as lowercase ℎ is the opposite side of the triangle. It’s the side opposite the included angle 𝑐 degrees. And when we notice that we know the length of the opposite and the hypotenuse, we can see that we need to use the sine ratio, where sin of 𝜃 is equal to opposite over hypotenuse.

Substituting what we know about our right-angle triangle into this formula, we see that sin 𝑐 is equal to ℎ, which is the opposite, over the hypotenuse, which is 𝑏. So sin 𝑐 is equal to ℎ over 𝑏. And since the aim of this question is to find an expression for the height ℎ, we need to make ℎ the subject. And we do so by multiplying both sides of our equation by 𝑏. ℎ over 𝑏 multiplied by 𝑏 is simply ℎ. And sin 𝑐 multiplied by 𝑏 is written 𝑏 sin 𝑐.

And we see that in fact this question was a little bit misleading. An expression for the height ℎ in terms of 𝑎, 𝑏, and 𝑐 doesn’t actually include 𝑎. It’s simply 𝑏 sin 𝑐.

Next, we’re going to find an expression for the area of the triangle in terms of 𝑎, 𝑏, and 𝑐. This time, we recall the formula for area of a triangle. It’s a half multiplied by its base multiplied by its perpendicular height. The base of our triangle is 𝑎 units. And the perpendicular height is ℎ. So we could say that the area is a half 𝑎ℎ.

But of course, we’re looking to find an expression for the area of the triangle in terms of 𝑎, 𝑏, and 𝑐. We just saw that ℎ is equal to 𝑏 sin 𝑐. So we can see that the area is a half 𝑎 multiplied by 𝑏 sin 𝑐. And if we fully simplify this expression, the area of our triangle is a half 𝑎𝑏 sin 𝑐.