Is the height of an equilateral triangle the same as the base

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I know that formula for the height of an equilateral triangle is height = (1/2) * √3 * base, so the height and base can never both be integers. But how could I find values that are as close as possible to integers? I happen to know that a base of 15 results in a height of 12.9903811, which is really close to 13. Is a better base that results in an even closer height?

asked Jun 15, 2019 at 13:51

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Use the continued fraction expansion for $\frac {\sqrt{3}}2$. The first $10$ convergents are $$\left \{ 0, 1, \frac 6 {7}, \frac {13}{ 15},\frac {84}{97}, \frac {181}{ 209}, \frac {1170}{1351}, \frac {2521}{ 2911}, \frac {16296}{ 18817}, \frac {35113}{ 40545}\right \} $$

Note that your $13,15$ pair occurs early on.

Using the final one, we note that $$\frac {35113}{ 40545}=0.866025403872240\dots$$ while $$\frac {\sqrt 3}2=0.86602540378443864676\dots$$ and $$40545\times \frac {\sqrt 3}2=35112.99999644\dots$$

answered Jun 15, 2019 at 13:56

lulululu

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Yes, you can get arbitrarily close. A good way is to look at the convergents of the continued fraction for $\frac {\sqrt 3}2$ In this case it is $[0;1,\overline{6,2}]$ or $$\frac 1{1+\frac 1{6+\frac 1{2+\frac 1{6+\frac 1{2+\ldots}}}}}$$ $\frac {13}{15}$ comes from taking just the first $6$ and first $2$, so $$\frac 1{1+\frac 1{6+\frac 12}}=\frac 1{1+\frac 2{13}}=\frac{13}{15}$$ They are particularly close if you stop before a large term, so you could next look at $$\frac 1{2+\frac 1{6+\frac 12}}=\frac 1{2+\frac 2{13}}=\frac{13}{28}\\ \frac 1{1+\frac 1{6+\frac {13}{28}}}=\frac 1{1+\frac {28}{181}}=\frac {181}{209}\\ \left(\frac{181}{209}\right)^2\approx 0.7500057$$

answered Jun 15, 2019 at 14:10

Ross MillikanRoss Millikan

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The height of the equilateral triangle is #h=(asqrt(3))/2#

Explanation:

If you draw a height in an equilateral triangle you can see that the triangle is divided in 2 right angled triangles in which: sides #a# are hypothenuses, height is one cathetus (common for both triangles), the other is equal to #a/2#, so if we use the Pythagorean theorem we get:

#(a/2)^2+h^2=a^2#

#h^2=a^2-(a/2)^2#

#h^2=a^2-a^2/4#

#h^2=3/4a^2#

#h=(asqrt(3))/2#

#h=5sqrt3" m"#
#A=25sqrt3" m"^2#

We can see that if we split an equilateral triangle in half, we are left with two congruent equilateral triangles. Thus, one of the legs of the triangle is #1/2s#, and the hypotenuse is #s#. We can use the Pythagorean Theorem or the properties of #30˚-60˚-90˚# triangles to determine that the height of the triangle is #sqrt3/2s#.

If we want to determine the area of the entire triangle, we know that #A=1/2bh#. We also know that the base is #s# and the height is #sqrt3/2s#, so we can plug those in to the area equation to see the following for an equilateral triangle:

#A=1/2bh=>1/2(s)(sqrt3/2s)=(s^2sqrt3)/4#

Since #s=10#, #A=(10^2sqrt3)/4=(100sqrt3)/4=25sqrt3" m"^2#

Also, since the height is #sqrt3/2s#, we can say the height is #5sqrt3" m"#.

The formula for a regular triangle area is equal to the squared side times the square root of 3 divided by 4:

area = (a² × √3)/ 4

and the equation for the height of an equilateral triangle looks as follows:

h = a × √3 / 2, where a is a side of the triangle.

But do you know where the formulas come from? You can find them in at least two ways: deriving from the Pythagorean theorem (discussed in our Pythagorean theorem calculator) or using trigonometry.

1. Using Pythagorean theorem

• The basic formula for triangle area is side a (base) times the height h, divided by 2:

  area = (a × h) / 2

• Height of the equilateral triangle is derived by splitting the equilateral triangle into two right triangles. See our right triangle calculator to learn more about right triangles.

One leg of that right triangle is equal to height, another leg is half of the side, and the hypotenuse is the equilateral triangle side.

`(a/2)² + h² = a²` 

After simple transformations, we get a formula for the height of the equilateral triangle:

h = a × √3 / 2

• Substituting h into the first area formula, we obtain the equation for the equilateral triangle area:

  area = a² × √3 / 4

2. Using trigonometry

• Let's start with the trigonometric triangle area formula:

  area = (1/2) × a × b × sin(γ), where γ is the angle between the sides.

• We remember that all sides and all angles are equal in the equilateral triangle, so the formula simplifies to:

  area = 0.5 × a × a × sin(60°)

• What is more, we know that the sine of 60° is √3/2, so the formula for equilateral triangle area is:

  area = (1/2) × a² × (√3 / 2) = a² × √3 / 4

  The height of the equilateral comes from the sine definition:

  h / a = sin(60°) so h = a × sin(60°) = a × √3 / 2

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