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An equilateral triangle whose area and perimeter are equal has the area of exactly the square root of 432. Let's say we have an equilateral triangle with a
by cbr 10y ago
An equilateral triangle whose area and perimeter are
equal has the area of exactly the square root of 432.
Let's say we have an equilateral triangle with a perimeter of 3x units. Then each side is x units, and the area is sqrt(3) * x^2 / 4 square units. So we set:
3x = sqrt(3) * x^2 / 4
3 = sqrt(3) * x / 4
3*4/sqrt(3) = x
12/sqrt(3) = x
Then the area is:
sqrt(3) * (12/sqrt(3))^2 / 4
sqrt(3) * 144 / 12
sqrt(3) * 12
sqrt(3) * sqrt(144)
sqrt(432)
Fun!
- castratikron 10y agoI saw that in the article, too, but it didn't make sense to me. One has units of area and the other has units of length. How can they possibly be equal? That's like saying a 700 foot wall is the same size as a 700 square foot apartment.
- cbr 10y agoThat's what made it jump out to me too! But the key thing is that one number is in linear units and the other is in square units. So for any linear unit you choose (mm, in, km, furlongs, ...) an equilateral triangle with perimeter sqrt(432) units will have area sqrt(432) square units.
- nitrogen 10y agoIt is the intersection between a line and a parabola.