# The perimeter of the square is 12.8 m. find the perimeter of a triangle with the same area,

The perimeter of the square is 12.8 m. find the perimeter of a triangle with the same area, one of which is 6.4 m long. which figure has a large perimeter.

1. Find the side of the square:

4d = P1 = 12.8;
d = 12.8: 4 = 3.2.
2. Square area:

S = a ^ 2 = 3.2 ^ 2 = 10.24.

3. If a triangle is rectangular with legs a = 6.4 and b, then its area is:

1/2 * ab = S, hence:
b = 2S / a = 2 * 10.24 / 6.4 = 3.2;
c = √ (a ^ 2 + b ^ 2) = √ (6.4 ^ 2 + 3.2 ^ 2) = 3.2√5.
P2 = a + b + c = 6.4 + 3.2 + 3.2√5 = 3.2 (√5 + 3) ≈ 16.8.
4. Compare the perimeter of a square with the perimeter of an equilateral triangle with the same area:

S = h ^ 2 * √3 / 4, where h is the side of the triangle;
S = d ^ 2;
h ^ 2 * √3 / 4 = d ^ 2;
h ^ 2 = 4d ^ 2 / √3;
h = 2d / 3 ^ (1/4);
Pkv. = 4d;
Ptr. = 3h = 6d / 3 ^ (1/4)
Psq. / Ptr. = 4d / 3h = 4d: (6d / 3 ^ (1/4)) = 4d * 3 ^ (1/4)) / 6d = 2 * 3 ^ (1/4) / 3 ≈ 0.88 <1;
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