# 8.2 Non-right triangles: law of cosines  (Page 4/8)

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## Heron’s formula

Heron’s formula finds the area of oblique triangles in which sides $\text{\hspace{0.17em}}a,b\text{,}$ and $\text{\hspace{0.17em}}c\text{\hspace{0.17em}}$ are known.

$\text{Area}=\sqrt{s\left(s-a\right)\left(s-b\right)\left(s-c\right)}$

where $\text{\hspace{0.17em}}s=\frac{\left(a+b+c\right)}{2}\text{\hspace{0.17em}}$ is one half of the perimeter of the triangle, sometimes called the semi-perimeter.

## Using heron’s formula to find the area of a given triangle

Find the area of the triangle in [link] using Heron’s formula.

First, we calculate $\text{\hspace{0.17em}}s.$

$\begin{array}{l}\begin{array}{l}\\ s=\frac{\left(a+b+c\right)}{2}\end{array}\hfill \\ s=\frac{\left(10+15+7\right)}{2}=16\hfill \end{array}$

Then we apply the formula.

$\begin{array}{l}\begin{array}{l}\\ \text{Area}=\sqrt{s\left(s-a\right)\left(s-b\right)\left(s-c\right)}\end{array}\hfill \\ \text{Area}=\sqrt{16\left(16-10\right)\left(16-15\right)\left(16-7\right)}\hfill \\ \text{Area}\approx 29.4\hfill \end{array}$

The area is approximately 29.4 square units.

Use Heron’s formula to find the area of a triangle with sides of lengths $\text{\hspace{0.17em}}a=29.7\text{\hspace{0.17em}}\text{ft},b=42.3\text{\hspace{0.17em}}\text{ft},\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}c=38.4\text{\hspace{0.17em}}\text{ft}.$

Area = 552 square feet

## Applying heron’s formula to a real-world problem

A Chicago city developer wants to construct a building consisting of artist’s lofts on a triangular lot bordered by Rush Street, Wabash Avenue, and Pearson Street. The frontage along Rush Street is approximately 62.4 meters, along Wabash Avenue it is approximately 43.5 meters, and along Pearson Street it is approximately 34.1 meters. How many square meters are available to the developer? See [link] for a view of the city property.

Find the measurement for $\text{\hspace{0.17em}}s,\text{\hspace{0.17em}}$ which is one-half of the perimeter.

$\begin{array}{l}s=\frac{\left(62.4+43.5+34.1\right)}{2}\hfill \\ s=70\text{\hspace{0.17em}}\text{m}\hfill \end{array}$

Apply Heron’s formula.

$\begin{array}{l}\text{Area}=\sqrt{70\left(70-62.4\right)\left(70-43.5\right)\left(70-34.1\right)}\hfill \\ \text{Area}=\sqrt{506,118.2}\hfill \\ \text{Area}\approx 711.4\hfill \end{array}$

The developer has about 711.4 square meters.

Find the area of a triangle given $\text{\hspace{0.17em}}a=4.38\text{\hspace{0.17em}}\text{ft}\text{\hspace{0.17em}},b=3.79\text{\hspace{0.17em}}\text{ft,}\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}c=5.22\text{\hspace{0.17em}}\text{ft}\text{.}$

about 8.15 square feet

Access these online resources for additional instruction and practice with the Law of Cosines.

## Key equations

 Law of Cosines $\begin{array}{l}{a}^{2}={b}^{2}+{c}^{2}-2bc\mathrm{cos}\text{\hspace{0.17em}}\alpha \hfill \\ {b}^{2}={a}^{2}+{c}^{2}-2ac\mathrm{cos}\text{\hspace{0.17em}}\beta \hfill \\ {c}^{2}={a}^{2}+{b}^{2}-2abcos\text{\hspace{0.17em}}\gamma \hfill \end{array}$ Heron’s formula

## Key concepts

• The Law of Cosines defines the relationship among angle measurements and lengths of sides in oblique triangles.
• The Generalized Pythagorean Theorem is the Law of Cosines for two cases of oblique triangles: SAS and SSS. Dropping an imaginary perpendicular splits the oblique triangle into two right triangles or forms one right triangle, which allows sides to be related and measurements to be calculated. See [link] and [link] .
• The Law of Cosines is useful for many types of applied problems. The first step in solving such problems is generally to draw a sketch of the problem presented. If the information given fits one of the three models (the three equations), then apply the Law of Cosines to find a solution. See [link] and [link] .
• Heron’s formula allows the calculation of area in oblique triangles. All three sides must be known to apply Heron’s formula. See [link] and See [link] .

## Verbal

If you are looking for a missing side of a triangle, what do you need to know when using the Law of Cosines?

two sides and the angle opposite the missing side.

If you are looking for a missing angle of a triangle, what do you need to know when using the Law of Cosines?

Explain what $\text{\hspace{0.17em}}s\text{\hspace{0.17em}}$ represents in Heron’s formula.

$\text{\hspace{0.17em}}s\text{\hspace{0.17em}}$ is the semi-perimeter, which is half the perimeter of the triangle.

#### Questions & Answers

what is f(x)=
I don't understand
Joe
Typically a function 'f' will take 'x' as input, and produce 'y' as output. As 'f(x)=y'. According to Google, "The range of a function is the complete set of all possible resulting values of the dependent variable (y, usually), after we have substituted the domain."
Thomas
Sorry, I don't know where the "Â"s came from. They shouldn't be there. Just ignore them. :-)
Thomas
Darius
Thanks.
Thomas
Â
Thomas
It is the Â that should not be there. It doesn't seem to show if encloses in quotation marks. "Â" or 'Â' ... Â
Thomas
Now it shows, go figure?
Thomas
what is this?
i do not understand anything
unknown
lol...it gets better
Darius
I've been struggling so much through all of this. my final is in four weeks 😭
Tiffany
this book is an excellent resource! have you guys ever looked at the online tutoring? there's one that is called "That Tutor Guy" and he goes over a lot of the concepts
Darius
thank you I have heard of him. I should check him out.
Tiffany
is there any question in particular?
Joe
I have always struggled with math. I get lost really easy, if you have any advice for that, it would help tremendously.
Tiffany
Sure, are you in high school or college?
Darius
Hi, apologies for the delayed response. I'm in college.
Tiffany
how to solve polynomial using a calculator
So a horizontal compression by factor of 1/2 is the same as a horizontal stretch by a factor of 2, right?
The center is at (3,4) a focus is at (3,-1), and the lenght of the major axis is 26
The center is at (3,4) a focus is at (3,-1) and the lenght of the major axis is 26 what will be the answer?
Rima
I done know
Joe
What kind of answer is that😑?
Rima
I had just woken up when i got this message
Joe
Can you please help me. Tomorrow is the deadline of my assignment then I don't know how to solve that
Rima
i have a question.
Abdul
how do you find the real and complex roots of a polynomial?
Abdul
@abdul with delta maybe which is b(square)-4ac=result then the 1st root -b-radical delta over 2a and the 2nd root -b+radical delta over 2a. I am not sure if this was your question but check it up
Nare
This is the actual question: Find all roots(real and complex) of the polynomial f(x)=6x^3 + x^2 - 4x + 1
Abdul
@Nare please let me know if you can solve it.
Abdul
I have a question
juweeriya
hello guys I'm new here? will you happy with me
mustapha
The average annual population increase of a pack of wolves is 25.
how do you find the period of a sine graph
Period =2π if there is a coefficient (b), just divide the coefficient by 2π to get the new period
Am
if not then how would I find it from a graph
Imani
by looking at the graph, find the distance between two consecutive maximum points (the highest points of the wave). so if the top of one wave is at point A (1,2) and the next top of the wave is at point B (6,2), then the period is 5, the difference of the x-coordinates.
Am
you could also do it with two consecutive minimum points or x-intercepts
Am
I will try that thank u
Imani
Case of Equilateral Hyperbola
ok
Zander
ok
Shella
f(x)=4x+2, find f(3)
Benetta
f(3)=4(3)+2 f(3)=14
lamoussa
14
Vedant
pre calc teacher: "Plug in Plug in...smell's good" f(x)=14
Devante
8x=40
Chris
Explain why log a x is not defined for a < 0
the sum of any two linear polynomial is what
divide simplify each answer 3/2÷5/4
divide simplify each answer 25/3÷5/12
Momo
how can are find the domain and range of a relations
the range is twice of the natural number which is the domain
Morolake
A cell phone company offers two plans for minutes. Plan A: $15 per month and$2 for every 300 texts. Plan B: $25 per month and$0.50 for every 100 texts. How many texts would you need to send per month for plan B to save you money?
6000
Robert
more than 6000
Robert
For Plan A to reach $27/month to surpass Plan B's$26.50 monthly payment, you'll need 3,000 texts which will cost an additional \$10.00. So, for the amount of texts you need to send would need to range between 1-100 texts for the 100th increment, times that by 3 for the additional amount of texts...
Gilbert
...for one text payment for 300 for Plan A. So, that means Plan A; in my opinion is for people with text messaging abilities that their fingers burn the monitor for the cell phone. While Plan B would be for loners that doesn't need their fingers to due the talking; but those texts mean more then...
Gilbert
can I see the picture
How would you find if a radical function is one to one?