Video: Finding the Area of a Triangle Using Trigonometry

Finding the Area of a Triangle Using Trigonometry

14:12

Video Transcript

In this video, we’ll see how we can find the area of a triangle using trigonometry. To do this, we’ll need to know the lengths of two sides and their included angle. But before we do that, let’s consider a rectangle. We know that a rectangle has four 90-degree angles. It has a length and a width. And the area is equal to the length times the width. If we draw a diagonal across this rectangle, we’ve divided it in half. And each of these triangles have an area of one-half the length times the width.

Let’s look more closely at the right triangle. When we’re dealing with the triangle, that width becomes a height. And instead of calling it the length, we call it the base. And so, we say that the area of a triangle is one-half the height times the base. The key to using this formula is that the height is perpendicular to the base. If we’re not given a perpendicular height, we can’t use this formula.

Let’s say we’re dealing with a triangle that looks like this. We could call this side the base. But the height of this triangle is the perpendicular distance from the base to the vertex opposite that base. This distance is the height. It’s still true that the area of this triangle is one-half times the height times the base. That formula is true because every triangle is half of a rectangle.

If we double this triangle and move the pieces around, we have a rectangle. This proves that the formula area equals one-half times height times base will find the area of any triangle as long as you have the perpendicular height. But this video is going to help us learn how to solve the area of triangles when we aren’t given the height. In these cases, we need trigonometry to come to the rescue.

Okay, let’s go back. Area equals one-half the height times the base. And we know that the height is a perpendicular intersection of the base. Now, in this triangle, our base is 𝑎. So, let’s just leave that there. We want to use some trigonometry to help us find the height. The height of this triangle actually makes the larger triangle divided into two smaller triangles. And the two smaller triangles that make up the larger triangle are both right-angled triangles.

If we call the point where the height intersects the base point 𝑃, we could say triangle 𝐴𝑃𝐶 is a right triangle. Let’s just move that information up a bit. We can also say that the line segment 𝐴𝑃 is the height of the larger triangle 𝐴𝐵𝐶. We can say that side 𝑏 is the hypotenuse of the smaller triangle, 𝐴𝑃𝐶. And now, let’s consider angle 𝐶. Opposite angle 𝐶 is the line segment 𝐴𝑃 or what we’ve labelled as ℎ.

And we’ve already said that side 𝑏 is the hypotenuse. If we know an opposite and a hypotenuse in a right triangle, what trig value should we be thinking of? We can choose from the sine, cosine, or tangent relationships. But the sine relationship is the one that is the opposite over the hypotenuse. This means we can say that sin of angle 𝜃 equals ℎ over 𝑏. This means that the height over the side length 𝑏 will be equal to the sin of the angle 𝜃.

We’re trying to solve for the height. If we multiply both sides of this equation by the side length 𝑏, then we can say that the height is equal to the sin of 𝜃 times 𝑏. And in case of this triangle, we know that 𝜃 is the sin of angle 𝐶. We can now substitute in for the height, sin of 𝐶 times 𝑏. Another way to find the area of a triangle is one-half times sin 𝐶 times 𝑏 times 𝑎. We can rearrange it to its more common form, the area of a triangle is found by one-half times 𝑎 times 𝑏 times sin of 𝐶.

Before we move on, let’s just walk through this one more time. For our triangle, 𝐴𝐵𝐶, we can find its area if we know the lengths of two sides and the value of its included angle. If we knew side length 𝑎 and 𝑏 and the value of angle 𝐴, we could not use this rule. Nor could we use this rule if we knew the value of side length 𝑎 and side length 𝑏 and the angle 𝐵. We must know the included angle, the angle between the two side lengths we know. To use this formula, we must have two side lengths and an included angle. It does not matter which two sides, as long as the angle we’re given is included between those two sides. Now, let’s move on to some examples.

Which of the following is a formula that can be used to find the area of a triangle? A) one-half 𝑎𝑏 cos 𝐶, B) one-half 𝑎𝑏 sin 𝐶, C) one-third 𝑎𝑏 sin 𝐶, D) one-fourth 𝑎𝑏 cos 𝐶, or E) one-fourth 𝑎𝑏 sin 𝐶.

If we sketch a triangle and label it 𝐴, 𝐵, and 𝐶, the side length opposite vertex 𝐴 is usually labelled with a lower case 𝑎. The side length opposite vertex 𝐵 is labelled with a lower case 𝑏. And we label lower case 𝑐 the side length opposite vertex 𝐶. We have to remember that a triangle is half of a rectangle. And so, it’s unlikely that options C through E would be the answer.

We noticed that options A and B are dealing with the angle at vertex 𝐶, that’s this angle, and the lengths 𝑎 and 𝑏. At this point, we recognize that we have two sides and an included angle. And we know that the height of this triangle will be equal to 𝑏 times the sin of 𝐶. To use trigonometry to solve for the area of a triangle, we take one-half times 𝑎 times 𝑏 times sin of 𝐶, which is option B here.

For this example, we need to apply the formula we’ve been talking about.

In the given figure, work out the area of the triangle to two decimal places.

In this figure, we were given two side lengths and an included angle. Since we have this information, we can use the formula 𝐴 equals one-half times 𝑎𝑏 times sin of 𝐶 to find the area, where 𝑎 and 𝑏 are side lengths and 𝐶 is an included angle. The area of this triangle is found by one-half times 10 times seven times sin of 136 degrees. If you plug this into your calculator, you get 24.3130429 continuing. If your calculator did not give you this answer, you should check and make sure that your calculator is set to degree mode and not to radians.

To get our final answer, though, we want it rounded to two decimal places. There’s a one in the second decimal place. To the right of that, in the third decimal place, the thousandths place, is a three. This means we should round down to 24.31. We weren’t given any units, so we can just say that the area of this triangle is 24.31 units squared.

This example might look just as simple as the last one on the surface, but it’s going to take a few more steps to find the area.

The figure shows a triangular field with sides 670 meters, 510 meters, and 330 meters. Find the area of the field giving the answer to the nearest square meter.

At this point, we’re not given the angle measure of any of the angles inside this triangle. That means we can’t check for a perpendicular distance that could be the height. Before we do anything here, we’re going to have to find at least one of the angles. Because we know all three sides, there is one rule we can use here. We can use an application of the cosine rule, which tells us 𝑐 squared is equal to 𝑎 squared plus 𝑏 squared minus two 𝑎𝑏 times cos of 𝐶.

Now, we actually want to find the measure of angle 𝐶. We don’t know what that is. But we can rearrange this rule so that the cos of 𝐶 is the term that we’re looking for. We can subtract 𝑎 squared and 𝑏 squared from both sides of the equation. Then, we have the statement 𝑐 squared minus 𝑎 squared minus 𝑏 squared equals negative two 𝑎𝑏 cos of 𝐶. And if we divide both sides by negative two 𝑎𝑏, we can say that cos of 𝐶 is equal to 𝑐 squared minus 𝑎 squared minus 𝑏 squared over negative two 𝑎𝑏.

Our goal is to find 𝐶. And the 𝑏 side is equal to 510 meters. And the 𝑎 side is equal to 670 meters. And our third side is 𝑐. 𝑐 squared is 330 squared minus 670 squared minus 510 squared over negative two times 670 times 510. All of this will equal the cos of angle 𝐶. If you enter all of that into your calculator, you will get 0.878109453 continuing is equal to the cos of angle 𝐶.

We need to be careful here. Angle 𝐶 is not equal to 0.878109453 continuing degrees. The cos of angle 𝐶 is equal to this decimal value. To find angle 𝐶, we need to take the inverse cos of 0.878109453 continuing. On most calculators, you can hit cos inverse of the answer, of the previous answer. Angle 𝐶 is equal to 28.58485793 continuing degrees. If your calculator did not give you this answer, you should check and make sure that you’re calculating in degrees and not in radians.

Now that we have a value for one of the angles, we have the value of two side lengths, and an included angle. Which means we can use the area is equal to one-half times 𝑎 times 𝑏 times sin of 𝐶, where 𝑎 and 𝑏 are side lengths and 𝐶 is an included angle. To find the area of this triangle, we want to say the area is equal to one-half times 670 times 510 times the sin of the answer already in your calculator. Why would we do this? This gives us the most accurate answer before we round.

It’s calculating the sin of 28.58485793 continuing. When we do that, we get 81744.85833 continuing. And this is the point where we want to round to the nearest square meter. We wanna round to the nearest whole number. We have a four in the ones place, and the digit to the right of that is an eight, telling us we need to round up. That four rounds up to a five, and everything to the left of the five stays the same. So, we have 81745 square meters. The area of this playing field is 81745 meters squared when we round to the nearest square meter.

Let’s do a quick recap of the key points of finding the area of a triangle using trigonometry. The area of a triangle can be found by multiplying one-half times side length 𝑎 times side length 𝑏 times the sin of the included angle 𝐶. So, we say that this formula requires us to have two sides and an included angle.

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