Difference between revisions of "Equilateral triangle"
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| + | <asy> | ||
| + | pair A=(0,50),B=(43.301,-25),C=(-43.301,-25); | ||
| + | draw(A--B--C--A); | ||
| + | label("$60^{\circ}$",(-28,-16)); | ||
| + | draw(anglemark(B,C,A,300)); | ||
| + | label("$60^{\circ}$",(28,-16)); | ||
| + | draw(anglemark(A,B,C,300)); | ||
| + | label("$60^{\circ}$",(0,35)); | ||
| + | draw(anglemark(C,A,B,300)); | ||
| + | </asy> | ||
An '''equilateral triangle''' is a [[triangle]] in which all sides have equal length and all angles have equal measure. Since a triangle has a total of <math>180</math> degrees, each angle of an equilateral triangle has <math>60</math> degrees. | An '''equilateral triangle''' is a [[triangle]] in which all sides have equal length and all angles have equal measure. Since a triangle has a total of <math>180</math> degrees, each angle of an equilateral triangle has <math>60</math> degrees. | ||
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The equilateral triangle is the simplest [[regular polygon]]. It is also the only type of [[polygon]] that such that the [[equilateral]] polygon must also be the [[equiangular]] polygon. | The equilateral triangle is the simplest [[regular polygon]]. It is also the only type of [[polygon]] that such that the [[equilateral]] polygon must also be the [[equiangular]] polygon. | ||
| − | One useful way to manipulate an equilateral triangle is to draw an [[altitude]], which would divide it into two 30-60-90 [[right triangle]]s. Using this, the area of an equilateral triangle can be found in terms of a side: <math>\frac{s^2\sqrt{3}}{4}</math>. | + | One useful way to manipulate an equilateral triangle is to draw an [[altitude]], which would divide it into two 30-60-90 [[right triangle]]s. The altitude of the triangle is <math>\frac{s \sqrt{3}}{2}</math>. Using this, the area of an equilateral triangle can be found in terms of a side: <math>\frac{s^2\sqrt{3}}{4}</math>. |
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| + | Notice that all equilateral triangles are similar (since all equilateral triangles have the same angle measurements, so, by AA similarity, all equilateral triangles are similar). | ||
[[Category:Geometry]] | [[Category:Geometry]] | ||
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Latest revision as of 18:57, 15 September 2022
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![[asy] pair A=(0,50),B=(43.301,-25),C=(-43.301,-25); draw(A--B--C--A); label("$60^{\circ}$",(-28,-16)); draw(anglemark(B,C,A,300)); label("$60^{\circ}$",(28,-16)); draw(anglemark(A,B,C,300)); label("$60^{\circ}$",(0,35)); draw(anglemark(C,A,B,300)); [/asy]](http://latex.artofproblemsolving.com/f/5/6/f56da6a9a5f35768610d03edcc49e679a9aaee48.png)
An equilateral triangle is a triangle in which all sides have equal length and all angles have equal measure. Since a triangle has a total of 
degrees, each angle of an equilateral triangle has 
degrees.
The equilateral triangle is the simplest regular polygon. It is also the only type of polygon that such that the equilateral polygon must also be the equiangular polygon.
One useful way to manipulate an equilateral triangle is to draw an altitude, which would divide it into two 30-60-90 right triangles. The altitude of the triangle is 
. Using this, the area of an equilateral triangle can be found in terms of a side: 
.
Notice that all equilateral triangles are similar (since all equilateral triangles have the same angle measurements, so, by AA similarity, all equilateral triangles are similar).
