5.4: Derivation of the Triangle Area Formula
While in the lunch room with your friends one day, you're discussing different ways you can use the things you've learned in math class. You tell your friends that you've been learning a lot about triangles, such as how to find their area. One of your friends looks down at your plate and starts to smile.
"Alright," he says. "If you're so good at things involving triangles, I dare you to find something simple. Tell me the area of your slice of pizza." He points down at the pizza on your plate.
The pizza is shaped like a triangle. But unfortunately its not a right triangle. The outer edge is 5 inches long, and the long sides are 7 inches long. The angle between the edge and the long side of the slice is
Read through this Concept, and at its end, you'll be able to answer your friend's challenge.
Watch This
James Sousa Example: Determine the Area of a Triangle Using the Sine Function
Guidance
We can use the area formula from Geometry,
In
We can use a similar method to derive all three forms of the area formula, regardless of the angle:
The formula
Example A
In
Solution: Using our new formula,
Example B
The Pyramid Hotel recently installed a triangular pool. One side of the pool is 24 feet, another side is 26 feet, and the angle in between the two sides is
Solution: In order to find the cost of the cover, we first need to know the area of the cover. Once we know how many square feet the cover is, we can calculate the cost. In the illustration above, you can see that we know two of the sides and the included angle. This means we can use the formula
The cost of the cover will be
Example C
In
Solution: Using our new formula,
Vocabulary
Oblique Triangle: An oblique triangle is a triangle that does not have
SAS Triangle: An SAS triangle is a triangle where two sides and the angle in between them are known quantities.
Guided Practice
1. A farmer needs to replant a triangular section of crops that died unexpectedly. One side of the triangle measures 186 yards, another measures 205 yards, and the angle formed by these two sides is
What is the area of the section of crops that needs to be replanted?
2. The farmer goes out a few days later to discover that more crops have died. The side that used to measure 205 yards now measures 288 yards. How much has the area that needs to be replanted increased by?
3. Find the perimeter of the quadrilateral at the left If the area of
Solutions:
1. Use
2.
3. You need to use the
Second, you need to find sides
The perimeter of the quadrilateral is 50.55.
Concept Problem Solution
where in this case, one of the sides is equal to 5, the other is equal to 7, and the angle is
Practice
Find the area of each triangle.

△ABC if a=13, b=15, andm∠C=71∘ . 
△ABC if b=8, c=4, andm∠A=67∘ . 
△ABC if b=34, c=29, andm∠A=138∘ . 
△ABC if a=3, b=7, and \begin{align*}m\angle C=80^\circ\end{align*}.  \begin{align*}\triangle ABC\end{align*} if a=4.8, c=3.7, and \begin{align*}m\angle B=43^\circ\end{align*}.
 \begin{align*}\triangle ABC\end{align*} if a=12, b=5, and \begin{align*}m\angle C=20^\circ\end{align*}.
 \begin{align*}\triangle ABC\end{align*} if a=3, b=10, and \begin{align*}m\angle C=50^\circ\end{align*}.
 \begin{align*}\triangle ABC\end{align*} if a=5, b=9, and \begin{align*}m\angle C=14^\circ\end{align*}.
 \begin{align*}\triangle ABC\end{align*} if a=5, b=7, and c=11.
 \begin{align*}\triangle ABC\end{align*} if a=7, b=8, and c=9.
 \begin{align*}\triangle ABC\end{align*} if a=12, b=14, and c=4.
 A farmer measures the three sides of a triangular field and gets 114, 165, and 257 feet. What is the measure of the largest angle of the triangle?
 Using the information from the previous problem, what is the area of the field?
Another field is a quadrilateral where three sides measure 30, 50, and 60 yards, and two angles measure \begin{align*}130^\circ\end{align*} and \begin{align*}140^\circ\end{align*}, as shown below.
 Find the area of the quadrilateral. Hint: divide the quadrilateral into two triangles and find the area of each.
 Find the length of the fourth side.
 Find the measures of the other two angles.
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Here you'll learn how to derive and apply a formula for the area of a triangle that involves the sine function.