Question Video: Using the Trigonometric Formula for the Area of a Triangle to Find the Area of a Triangle Mathematics

A small triangular flower bed has measurements as shown in the diagram. Calculate its area in square feet. Give your answer to two decimal places.

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Video Transcript

A small triangular flower bed has measurements as shown in the diagram. Calculate its area in square feet. Give your answer to two decimal places.

In this problem, we are asked to find the area of a triangle. It isn’t a right triangle, and we haven’t been given its base length and perpendicular height. So, we’ll recall instead the trigonometric formula for the area of a triangle. In any triangle 𝐴𝐡𝐢, where the lowercase letters π‘Ž, 𝑏, and 𝑐 represent its three side lengths and the uppercase letters 𝐴, 𝐡, and 𝐢 represent the opposite angles, then the area of such a triangle can be found using the formula a half π‘Žπ‘ sin 𝑐.

Notice that the information we need in order to apply this formula is two side lengths and then the measure of their included angle, which is sometimes referred to as their enclosed angle. It’s the angle between the two side lengths we know that we also need to know. In this problem, we are given the length of two sides; they are 4.5 and four feet, respectively. But we don’t know the measure of the angle between them. We are given the measures of the other two angles, though, and so we can work out the measure of the final angle in the triangle, using the fact that the angles in any triangle sum to 180 degrees. Subtracting the measures of the other two angles from 180 degrees then, we find that the measure of the third angle is 106 degrees.

We now have all the information we need in order to apply the trigonometric formula for the area of a triangle. We may find it helpful, though, to label the triangle using the letters π‘Ž, 𝑏, and 𝑐, but this isn’t essential. Substituting four and 4.5 for the two known side lengths and 106 degrees for the measure of their included angle, we have that the area of this triangle is equal to a half multiplied by four multiplied by 4.5 multiplied by sin of 106 degrees. Simplifying, we have nine sin of 106 degrees. And we can then evaluate this on a calculator, ensuring that it is in degree mode. That gives 8.651 continuing. The question specifies that we should give our answer to two decimal places. So, as the figure in the third decimal place is a one, we round down to 8.65.

By applying the trigonometric formula for the area of a triangle then, we found that the area of this triangular flower bed to two decimal places is 8.65 square feet.

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