given an equilateral triangle construct a square?

Question

given an equilateral triangle construct a square haveing equal area - how do i go about doing this?

Answer 1

Let the side of the equilateral triangle = 1. Then what we are looking for is the geometric mean of half of its base times its altitude, or √((1/2)((1/2)√3)). To find this length through construction:

1) First mark the midpoint of the base of the triangle
2) Using a compass, mark a distance on the base from the midpoint equal to the altitude of the triangle
3) Find mark the midpoint of the combined lengths of half of the base and the length equal to the altitude
4) Draw a semicircle using this midpoint as the center and half of the combined lengths as the radius
5) Draw a line perpendicular and through the original midpoint of the triangle, so that it intersects the semicircle
6) The distance from the original midpoint to the intersection is the length of the side of the desired square
7) Complete the construction and form the square, using this distance which is the geometric mean discussed above

Answer 2

So, what I assume you mean is that you have an equilateral triangle with sides lengths of "s" and you want a square with sides "x" with equal area. So, we'll take the equations for the area and set them equal:

Area of equilateral triangle: A = (s² √3) / 4
Area of square: A = x²

x² = (s² √3) / 4
==> square root both sides
x = s*(√(√3)) / 2
==> simplify double radicals as exponent of 1/4
x = ½ s (3^(1/4)) ... ANSWER
==> or approximate the constant
x = (0.658037006)s

So, the side length "x" of the square will relate to the side length "s" of the triangle by the above factor.

Answer 3

Area of equilateral triangle = (s^2)(sqrt3)(1/4)

Area of square = s^2

Please don't be confused with the s for the square and the triangle because their values are different. I'm just using them the way most teachers would.

So to clear things up, I'll assign s1 as the side for the triangle and s2 fo the square.

s1 = side for triangle

s2 = side for square

(s1)^2 * (sqrt3)(1/4)

(s2)^2

Since you want their area to be equal; set them equal.

(s1)^2 * (sqrt3)(1/4) = (s2)^2

Multiply by 4 on both sides.

(s1)^2 * (sqrt3) = 4(s2)^2

Divide sqrt3

(s1)^2 = (4/sqrt3)(s2)^2

Square root both sides.

s1 = 1/4th rt3(2)(s2)

Clearly, you can assign any value for s1. Just make sure s2 would match.

If you wanted to go from s2 to s1:

(s2)^2 = (s1)^2 (sqrt3) (1/4)

s2 = (s1)(4th rt 3)(1/2)