Editing Equilateral triangle (section)

=== Area ===
[[File:Equilateral triangle with height square root of 3.svg|thumb|The right triangle with a [[hypotenuse]] of <math> 1 </math> has a height of <math> \sqrt{3}/2 </math>, the sine of 60°.]]
The area of an equilateral triangle with edge length <math> a </math> is
<math display="block"> T = \frac{\sqrt{3}}{4}a^2. </math>
The formula may be derived from the formula of an isosceles triangle by [[Pythagoras theorem]]: the altitude <math> h </math> of a triangle is [[Isosceles triangle#Height|the square root of the difference of squares of a side and half of a base]].{{sfnp|Harris|Stocker|1998|p=[https://books.google.com/books?id=DnKLkOb_YfIC&pg=PA78 78]}} Since the base and the legs are equal, the height is:{{sfnp|McMullin|Parkinson|1936|p=[https://books.google.com/books?id=6RA8AAAAIAAJ&pg=PA96 96]}}
<math display="block"> h = \sqrt{a^2 - \frac{a^2}{4}} = \frac{\sqrt{3}}{2}a. </math>
In general, the [[Area_of_a_triangle|area of a triangle]] is half the product of its base and height. The formula for the area of an equilateral triangle can be obtained by substituting the altitude formula.{{sfnp|McMullin|Parkinson|1936|p=[https://books.google.com/books?id=6RA8AAAAIAAJ&pg=PA96 96]}} Another way to prove the area of an equilateral triangle is by using the [[trigonometric function]]. The area of a triangle is formulated as the half product of base and height and the sine of an angle. Because all of the angles of an equilateral triangle are 60°, the formula is as desired.{{citation needed|date=September 2024}}

A version of the [[isoperimetric inequality#Isoperimetric inequality for triangles|isoperimetric inequality for triangles]] states that the triangle of greatest [[area]] among all those with a given [[perimeter]] is equilateral. That is, for perimeter <math> p </math> and area <math> T </math>, the equality holds for the equilateral triangle:{{sfnp|Chakerian|1979}}
<math display="block"> p^2 = 12\sqrt{3}T. </math>