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.
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.
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.
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.
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] .
Section exercises
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
Is there any normative that regulates the use of silver nanoparticles?
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?