Typically to find the area of a triangle, the measurements of its height and base are used. On the other hand, if the lengths of two sides and the measure of their included angle are known, what should be the process to find the area of the triangle?In the lengths of and are and inches, respectively. The measure of varies from to
Izabella wants to finish her math homework before she goes out to buy some fabric for her table. She is given an equilateral triangle whose side length is unit.
Izabella's homework problem is to find the area of this triangle. Help her find the answer.
Since is an equilateral triangle, the measure of each of its interior angles is Consequently, measures
In the previous example, it was shown that trigonometric ratios can be used to find the area of a triangle given the lengths of two sides and the measure of their included acute angle. What happens if the included angle is obtuse?In the lengths of and are and inches, respectively. The measure of varies from to
Izabella finished her homework and is ready to go. On her way out, she notices an empty triangular region in the garden.
Her father wants to fill this region with topsoil. To figure out how much topsoil he needs, he asks Izabella to find the area of the region. Help them find the area. If it is necessary, round the answer to the nearest square foot.
Begin by drawing the external altitude of the side.
There is another way of finding the area of a triangle that can be derived from the previous examples.
Because is a right triangle, the height of the triangle can be related to the sine of using the sine ratio. Next, substitute the expression found for into the general formula for the area of a triangle.
The first formula was obtained. To obtain the second formula, notice that is also a right triangle. Therefore, the sine ratio can be applied again, this time to connect and By substituting this expression into the general formula for the area of a triangle, the second formula can be obtained.
To deduce the third formula, the altitude from or should be drawn. In this case, the altitude from will be arbitrarily drawn and labeled with a length of Because is obtuse, the altitude will lie outside the triangle.
In this case, the length of the base is and the height is Since is a right triangle, the sine ratio can be used to connect and Since and form a linear pair, they are supplementary angles. Recall that the sine of an angle is equal to the sine of its supplementary angle. With this information, and using the Substitution Property of Equality, a formula connecting and can be written. Multiplying both sides of the last equation by it is obtained that Finally, substitute this expression for into the formula for the area of
While sitting on the bus on the way to the fabric store, Izabella began daydreaming about a formula for the area of a triangle involving the sine ratio. She dreams about applying this formula to find the area of the Bermuda Triangle which she remembers from the movie Gulliver's Travels.
To find how much fabric Izabella needs, the surface area of the table must be found. Therefore, begin by finding the included angle.