In this explainer, we will learn how to find the area of a triangle using the lengths of two sides and the sine of the included angle.

We have known from early on in our mathematics journey how to find the area of a triangle using its base and perpendicular height. However, we are somewhat limited by this method as these two measurements may not always be given. In this explainer, we will extend our knowledge by introducing the trigonometric formula for the area of triangles, which we now derive.

Consider triangle in which we know the lengths of two sides and and the measure of the angle between them, angle . We refer to this as the included or enclosed angle. The known information is represented in bold on the figure below.

In order to apply the usual formula for the area of a triangle, we need to know the lengths of its base and perpendicular height. We draw in a line from vertex that is perpendicular to the base , which we will label as .

The area of triangle , using the usual formula, is . We need to consider how to express the perpendicular height in terms of the known side lengths and the known angle. Consider triangle in the figure above. As this is a right triangle, we can apply the sine ratio to express in terms of and . Recalling that the sine ratio is the length of the opposite side divided by the length of the hypotenuse, we have

Rearranging by multiplying by gives

We have now expressed the perpendicular height of the triangle in terms of side and angle , both of which we assumed to be known. We can now substitute this expression for into the usual formula for the area of a triangle to give the trigonometric formula

### Definition: The Trigonometric Formula for the Area of Triangles

The trigonometric formula for the area of triangles is where and are the lengths of two sides and is the measure of the included angle.

This formula is valid in both degrees and radians and can be applied to any triangle. It would be possible to work through the process of explicitly calculating the perpendicular height using trigonometry for each triangle and then applying the more basic formula, but the trigonometric formula combines these steps for us and is therefore more efficient.

It is important to note that this formula can be applied whenever we know any two sides of a triangle and their included angle. For the triangle above, we could equivalently express the formula as or as

However, it is better not to be overly concerned about the exact letters used and instead to understand what they represent in terms of the relative positioning of the sides and angle.

Let us now demonstrate how to apply this formula to calculate the area of a triangle given the lengths of two sides and the measure of the included angle.

### Example 1: Using the Trigonometric Formula for the Area of Triangles

is a triangle, where , , and . Find the area of , giving your answer to three decimal places.

### Answer

It is helpful to produce a sketch of triangle as shown below (not to scale).

From our sketch, it is clear that the information we have been given consists of the lengths of two of the triangleβs sides and the measure of their included angle. We recall the trigonometric formula for the area of a triangle:

Substituting the side lengths of 15 cm and 25 cm and the included angle of and evaluating gives

The area of triangle , to three decimal places, is
123.011 cm^{2}.

In the previous example, we were explicitly given two side lengths and their included angle. In other problems, we may be given a slightly different set of information. It may then be necessary to calculate the required lengths and angles using other geometric properties, such as the angle sum in a triangle. We will now consider an example of this.

### Example 2: Using the Trigonometric Formula for Areas of Triangles to Find the Area of an Isosceles Triangle

An isosceles triangle has two sides of length 48 cm and base angles of . Find the area of the triangle, giving the answer to three decimal places if necessary.

### Answer

We begin by sketching the triangle (not to scale). We recall that the base angles of an isosceles triangle are the angles formed by each of the equal sides with the third side.

We now recall the trigonometric formula for the area of a triangle:

We know the lengths of two of the triangleβs sides and we can calculate the measure of their included angle, angle on our diagram, using the sum of angles in a triangle. By subtracting the measures of the other two angles from , we obtain

Substituting the two side lengths of 48 cm and the included angle of into the trigonometric formula for the area of a triangle and evaluating gives

As this is an integer value, there is no need to round our answer to three decimal places.

The area of the triangle is
576 cm^{2}.

In the previous problem, the included angle was one of the special angles for which the values of the three trigonometric ratios can be expressed exactly in terms of quotients and radicals. The use of such angles enables us to answer problems like these when we do not have access to a calculator.

We now summarize the key steps to follow when applying the trigonometric formula for the area of triangles.

### How To: Calculating the Area of Triangles Using the Trigonometric Formula

- Identify a pair of side lengths and the included angle.
- It may be necessary to calculate any of these values using other information given in the question, such as using the sum of angles in a triangle or angles on a straight line.
- Substitute the values into the formula where and represent the lengths of the sides and represents the included angle.

We can also work backward when given the area of a triangle, one side length, and the measure of one angle to determine the length of the second side which encloses the angle. This will require us to form and solve an equation, as we will demonstrate in our next example.

### Example 3: Finding the Length of a Side of a Triangle given Its Area, the Length of a Side, and the Measure of an Angle

is a triangle where
,
,
and the area of the triangle is
cm^{2}.
Find the length of giving the
answer to two decimal places.

### Answer

We begin by sketching triangle using the information given in the question.

Next, we recall the trigonometric formula for the area of a triangle:

We recall that and represent the lengths of any two sides and represents the included angle, so for our triangle we can express the area using sides and and the included angle of as

By substituting for the area and 18 for , we can form an equation with only one unknown:

We recall that and solve our equation for by first canceling a factor of from each side and then isolating :

The length of to two decimal places is 16.44 cm.

The problems we have seen so far have each been related to a single triangle. It is also possible to apply the trigonometric formula to calculate the areas of compound shapes involving triangles. We may need to use other skills, such as right triangle trigonometry, to calculate the missing lengths we need, as we will see in our next example.

### Example 4: Finding the Area of a Compound Shape Using the Trigonometric Formula for the Area of Triangles

Find the area of the figure below giving the answer to three decimal places.

### Answer

The compound shape in the figure consists of two triangles, triangle and triangle . Let us consider triangle first. This is an equilateral triangle with a side length of 34 m and hence each of the interior angles are . We recall the trigonometric formula for the area of a triangle: where and represent the lengths of two sides and represents the included angle. In triangle , every side length is 34 m and every angle is , so substituting these values into the formula gives

Recalling that , we have

Next, we consider triangle , which is a right triangle. We are given the measure of one other angle and we can deduce that the length of its hypotenuse, , is 34 m. In order to apply the trigonometric formula for the area of a triangle, we first need to calculate the length of the second side that encloses angle , side .

In relation to angle , side is the adjacent. Applying right triangle trigonometry, we have

Rearranging gives

We are now able to apply the trigonometric formula for the area of triangles using sides and and the included angle :

The total area of the compound shape is the sum of the areas of the two triangles:

The area of the figure, to three decimal places, is
750.844 m^{2}.

In the previous example, we applied the trigonometric formula to calculate the area of a right triangle using the lengths of two of its sides and their included angle, which in this case was not the right angle. It is interesting to note what happens if we apply the trigonometric formula using the right angle and the two sides that enclose it. These two sides are the base and perpendicular height of the triangle, as shown in the figure below.

Applying the trigonometric formula for the area of a triangle, we obtain

Recalling that , this simplifies to which is consistent with the usual formula for the area of a triangle using its base and perpendicular height. Thus, we have shown that if applied to a right triangle in this way, the trigonometric formula reduces to the area formula we are already familiar with.

We have seen how we can apply the trigonometric formula for the area of triangles to compound shapes, but it can also be applied to calculate the areas of certain other geometric shapes, such as parallelograms. If such shapes can be divided into triangles, then, provided we are given the necessary set of information, we can use this formula to find their area as the sum of the areas of the triangles they contain. Let us now consider an example in which we apply this formula to calculate the area of a parallelogram.

### Example 5: Finding the Area of a Parallelogram Using the Trigonometric Formula for the Area of Triangles

is a parallelogram, where , , and . Find the area of , giving the answer to the nearest square centimetre.

### Answer

We begin by sketching the parallelogram, as shown below.

Usually when calculating the area of a parallelogram, we apply the formula

However, we are not given the perpendicular height of this parallelogram. Instead, we recognize that as is a parallelogram, each of its diagonals divide it up into two congruent triangles. Let us add the diagonal to our sketch.

As triangles and are congruent, they have the same area. The area of the parallelogram can therefore be calculated as twice the area of triangle , in which we know the lengths of two sides and the measure of their included angle. We can therefore apply the trigonometric formula for the area of a triangle:

The area of the parallelogram is twice this:

Simplifying and evaluating gives

The area of , to the nearest
square centimetre, is
397 cm^{2}.

Let us finish by recapping some key points from this explainer.

### Key Points

- The area of any triangle can be calculated using the lengths of two of its sides and the sine of their included angle.
- The trigonometric formula for the area of triangles is where and are the lengths of two sides and is the measure of the included angle.
- When given the area of a triangle and two pieces of information from the side lengths and and the angle , the trigonometric formula can be used to find the missing side or angle measure.
- The trigonometric formula can also be used to calculate the areas of other geometric shapes or compound shapes which can be divided into triangles.