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#### 单位正方形内接等边三角形的面积是多少？

Equilateral Triangle In A Square

《谨慎决定》
I’m Presh talwalkar.

An equilateral triangle is inscribed in a unit square

What is the area of the equilateral triangle?

Pause the video, if you’d like to give this problem a try

and when you’re ready, keep watching to

learn how to solve this problem.

(music)
[音乐]

So, how can we solve this problem

Let’s write s for the length of the equilateral triangle side.

The square has a side length of 1.

Now, in this right triangle,

suppose the other leg is equal to X

by the Gogu theorem s^2 is equal to 1^2 plus x^2.

Now consider the following right triangle

Its hypotenuse is also equal to s and one of its legs is equal to 1

Therefore its other leg must also be equal to x

Finally we have one more triangle.

It must have legs of 1 minus x

because the entire length of the square side is equal to 1

We therefore have s squared

is equal to 1 minus x the quantity squared plus 1 minus x the quantity squared
S²＝（1－X）²+（1－X）²
Well simplify this a little bit and now we have two equations.

Well consider this system of equations

we’ll subtract the second equation from the first

and the s squared terms will cancel out.

We then get an equation in one variable X

we can routinely solve this and we get a quadratic equation.

We use the quadratic formula and then we simplify to get two possibilities

x is equal to 2 minus root 3 and x is equal to 2 plus root 3
X＝2－√3和X＝2＋√3
Now recall that X has to be less than the side of the square.

So X has to be between 0 & 1

Therefore we reject 2 plus root 3 and we accept 2 minus root 3

once we have the value of x we can go ahead and solve for s squared and

Then remember we’re trying to solve for the area of the equilateral triangle,

which is equal to s squared root 3 all over 4

we simplify this and we get 2 root 3 minus 3 ,

which is approximately 0.464

and that’s our answer.

Thanks for making mind your decisions one of the best channels on YouTube

as always thanks for watching and thanks for your support

Felicity