Area of a triangle: Difference between revisions

From Citizendium
Jump to navigation Jump to search
imported>David E. Volk
(New page: The area of any triangle can be calculated using the following equation: <math> Area = \left(\frac{1}{2}\right)bc (sin A) = \left(\frac{1}{2}\right)ac (sin B) = \left(\frac{1}{2}\right)ab...)
 
imported>James Yolkowski
(mention most common formula)
 
(4 intermediate revisions by 3 users not shown)
Line 1: Line 1:
The area of any triangle can be calculated using the following equation:
{{subpages}}
There are several ways to compute the area of a [[triangle]].  The most common formula is <math>Area = \frac{1}{2}bh</math>, where <math>b</math> is the ''base'', which can be any side, and <math>h</math> is the ''altitude'' of the triangle, which is the perpendicular distance from the base (or the line containing the base for obtuse triangles) to the corner opposite the base.  This formula does require that the altitude be known; other formulae not having this requirement are:


<math> Area = \left(\frac{1}{2}\right)bc (sin A) = \left(\frac{1}{2}\right)ac (sin B) = \left(\frac{1}{2}\right)ab (sin C) </math>
* <math> Area = \left(\frac{1}{2}\right)bc (sin A) = \left(\frac{1}{2}\right)ac (sin B) = \left(\frac{1}{2}\right)ab (sin C) </math>
* <math> Area = \sqrt{ s (s-a) (s-b) (s-c)} </math>
where <math> s =  \frac{1}{2} (a+b+c) </math>is the [[semiperimeter]] of the triangle. This formula is known as the Heron's formula (or Hero's formula), named after the mathematician, [[Heron of Alexandria]].




[[Image:Triangle.jpg|center|frame|Triangle]]
[[Image:Triangle.jpg|center|frame|Triangle]]


== Right triangles ==
== Right triangles ==


For right triangles, the angle C, opposite the hypotenus (c), is 90 degrees.  The sign of 90 degrees is 1, so the equation reduces to:
For [[right triangle|right triangles]], the angle C, opposite the [[hypotenuse]] (c), is 90 degrees.  The sine of 90 degrees is 1, so the equation reduces to:




<math> Area = \left(\frac{1}{2}\right)ab </math>
<math> Area = \left(\frac{1}{2}\right)ab </math>

Latest revision as of 22:18, 6 December 2009

This article is a stub and thus not approved.
Main Article
Discussion
Related Articles  [?]
Bibliography  [?]
External Links  [?]
Citable Version  [?]
 
This editable Main Article is under development and subject to a disclaimer.

There are several ways to compute the area of a triangle. The most common formula is , where is the base, which can be any side, and is the altitude of the triangle, which is the perpendicular distance from the base (or the line containing the base for obtuse triangles) to the corner opposite the base. This formula does require that the altitude be known; other formulae not having this requirement are:

where is the semiperimeter of the triangle. This formula is known as the Heron's formula (or Hero's formula), named after the mathematician, Heron of Alexandria.


Triangle

Right triangles

For right triangles, the angle C, opposite the hypotenuse (c), is 90 degrees. The sine of 90 degrees is 1, so the equation reduces to: