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# How can we find the area of equilateral traingle

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VP
Joined: 17 Feb 2010
Posts: 1492
How can we find the area of equilateral traingle [#permalink]

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04 Sep 2010, 17:36
How can we find the area of equilateral triangle if we know just the height of equilateral triangle (say 3/2)?
VP
Joined: 05 Mar 2008
Posts: 1469
Re: How can we find the area of equilateral traingle [#permalink]

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04 Sep 2010, 17:52
seekmba wrote:
How can we find the area of equilateral triangle if we know just the height of equilateral triangle (say 3/2)?

if the height is 3/2 then we can divide the triangle in two 30-60-90 triangles and use the ratio of 1 - sqrt 3 - 2
Math Expert
Joined: 02 Sep 2009
Posts: 39626
Re: How can we find the area of equilateral traingle [#permalink]

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04 Sep 2010, 18:10
seekmba wrote:
How can we find the area of equilateral triangle if we know just the height of equilateral triangle (say 3/2)?

Let say the side of an equilateral triangle is $$b$$.
Attachment:

image017.gif [ 1.54 KiB | Viewed 892 times ]
Now, the height of equilateral triangle is also the median, thus it divides the base into half. Half of the base, $$\frac{b}{2}$$ and height, $$h$$, would be the legs of right triangle and the hypotenuse would be the side of this triangle so $$(\frac{b}{2})^2+h^2=b^2$$ --> $$b^2=h^2*\frac{4}{3}$$ --> $$b=h\frac{2}{\sqrt{3}}$$ --> $$area=\frac{1}{2}*height* base=\frac{1}{2}*h*h\frac{2}{\sqrt{3}}=\frac{h^2}{\sqrt{3}}$$

Or: as the hight divides the equilateral triangle in two 30-60-90 triangles then the sides of these triangles will be in the ratio $$1:\sqrt{3}:2$$, so if height (the side opposite 60 degrees) is $$h$$ then the side of the equilateral triangle (the side opposite 90 degrees) will be $$h*\frac{2}{\sqrt{3}}$$ and the $$area=\frac{1}{2}*h*h\frac{2}{\sqrt{3}}=\frac{h^2}{\sqrt{3}}$$.

For more on triangles check Triangles chapter of Math Book.

Hope it helps.
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Re: How can we find the area of equilateral traingle   [#permalink] 04 Sep 2010, 18:10
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