Finding the area of a trianglezor!!!

Section 6-3: Area of a Triangle

This section is pretty simple. It pretty much centers around two equations for the area of a triangle:

- Area = ½ (b)(c)sinA
- Hero’s Formula: Area = sqrt(s(s-a)(s-b)(s-c)), where s is the semiperimeter, ½ (a+b+c)

Area = ½ (b)(c)sinA

Proving this formula is a combination of the area formula ½ (b)(h) and the definition of sine.

Since the height (h) is not given as information, we must find a new way to calculate the height. We are, however, given the angle A.

c x SinA = h/c x c
(c)sinA = h

By substituting the new h formula into the old area formula, we get:

Area = ½ (b)(c)sinA

Here’s an example problem:
Find the area of this triangle.

Because we are not given an angle of the triangle, we cannot plug in the information straight into the triangle area formula. First, we must find an angle using the law of cosines. For the sake of this example, we will use angle A.

15^2 = 17^2 + 18^2 – 2(17)(18)cosA

We are solving for cosA, so the formula boils down to:

cosA = (17^2 + 18^2 – 15^2) / ((2)(17)(18))
cosA = .63398…
cos-1A = 50.655°

Now, we can substitute the given information into the area formula:

Area = ½ (17)(18)sin(50.655°)
Area = 118.32 units2

Hero’s Formula

Now some crazy guy named Hero decided to make our lives easier if we were to find ourselves in the situation described above, that is, if we were given only the sides of the triangle and no angle. Instead of going all the way through the law of cosines to figure out the angle, we can use this formula:

Area = sqrt(s(s-a)(s-b)(s-c)), where s is the semiperimeter, ½ (a+b+c)

Using the same triangle in the sample problem above, the formula would work out to be:

Area = sqrt(25(25-15)(25-17)(25-18))
Area = sqrt(25(10)(8)(7))
Area = 118.32 units2

Voila! The same answer that we got above.

So… that’s about it folks! Here is a link that has more details on finding the area of a triangle. (O, and I’m sorry about the sqrt() signs, mathtype isn’t working on my computer)

Now for my personalization. I thought long and hard about this one, but I decided nothing would make me happier than to prove that girls are evil. So here’s something that proves that girls are evil.

Girls = Time x Money
Time is money, so Time = Money
Girls = Money x Money, or Money2
Money is the root of all evil, so Money = sqrt(evil)
Girls = (sqrt(evil))2
Girls = evil

REMINDER!!!! KAORI – YOU ARE NEXT!!!! AREN’T YOU HAPPY THAT I MADE THIS REMINDER PINK JUST FOR YOU!?!?!?!?