Download We are all familiar with the formula for the area of a triangle, , where

Name: _______________________________
Common Core Geometry – Honors
Date: ________________
Area of a Triangle Using Trig
AIM: How do we use trigonometry to determine the area of a triangle?
Do Now: A man is walking his dog on level ground in a straight line with the dog's
favorite tree. The angle of elevation from the man's present position to the top of a
nearby telephone pole is 30º. The angle of elevation from the tree to the top of the
telephone pole is 45º. If the telephone pole is 40 feet tall, how far is the man with the
dog from the tree? Express answer to the nearest tenth of a foot.
Homework: Worksheet
We are all familiar with the formula for the area
of a triangle,
,
where b stands for the base and h stands for the
height drawn to that base.
(the lettering used is of no importance)
In the triangle at the right, the area could be expressed as:
Now, let's be a bit more creative and look at the diagram again. By using the right
triangle on the left side of the diagram, and our knowledge of trigonometry, we can
state that:
This tells us that the height, h, can be expressed as bsinC.
If we substitute this new expression for the
height, we can write the triangle area
formula as:
(where a and b are adjacent sides and C is the included
angle)
Example:
Given the triangle below, find its area. Express the area rounded to three decimal
places.
Given the parallelogram shown at the right,
find its EXACT area.
If we are looking for an EXACT answer, we
do NOT want to round our value for sin
60º. We need to remember that the sin 60º
(from our 30º- 60º- 90º reference triangle)
is
.
Now, the diagonal of a parallelogram divides the
parallelogram into two congruent triangles. So
the total area of the parallelogram will be
double the area of one of the triangles formed by
a diagonal.
square units.
We discovered, due to the doubling, that the area of a parallelogram is really just
Parallelogram
(where a and b are adjacent sides and C is the included angle)
Class Work:
1. In  ABC, AB = 10, AC = 8, and m<A = 45º. Find the area of  ABC, to the
nearest tenth of a square unit.
2. In an isosceles , the two equal sides each measure 24 meters, and they include
an angle of 30º. Find the area of the isosceles triangle, to the nearest sq. meter.
3. In ABC, AB = 12 meters and AC = 20 meters. If the area of the triangle is 77
sq. meters, find the measure of <A, to the nearest degree.
4. In a rhombus, each side is 15, and one angle is 130º. Find the area of the
rhombus, to the nearest square unit.
5. A farmer has a triangular field where two sides measure 450 yards and 320
yards. The angle between these two sides measures 80º. The farmer wishes to
use an insecticide that costs $4.50 per 100 sq. yards or any part of 100 yds. What
will it cost to use this insecticide on this field?
6. A triangle has two sides of 30 meters and 26 meters, and the angle between them
is an obtuse angle. If the area of the triangle is 300 sq. meters, find the measure
of the obtuse angle (to the nearest degree.)