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
can someone help me with some logarithmic and exponential equations.
In this morden time nanotechnology used in many field .
1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc
2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc
3- Atomobile -MEMS, Coating on car etc.
and may other field for details you can check at Google
Azam
anybody can imagine what will be happen after 100 years from now in nano tech world
Prasenjit
after 100 year this will be not nanotechnology maybe this technology name will be change .
maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
Azam
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
Prasenjit
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
Damian
silver nanoparticles could handle the job?
Damian
not now but maybe in future only AgNP maybe any other nanomaterials
Azam
can nanotechnology change the direction of the face of the world
At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.