What is the method for determining the perimeter of a square inscribed in a circle, given the diameter of the circle (4 cm)?
The diameter of a circle with perimeter 4 is $\displaystyle \frac{4}{\pi}$ The diameter of the circle is the length from one corner of the square to the opposite corner, in other words it is the hypotenuse of a triangle formed by two sides of the square. The perimeter of the square is the length of a side times 4.. . let's write this out...Originally Posted by zoso
$\displaystyle P_{square}=4s$
$\displaystyle d^2=s^2+s^2$
$\displaystyle d=\frac{c}{\pi}$
therefore...
$\displaystyle d^2=2s^2\Longrightarrow\sqrt{\frac{d^2}{2}=s$
substitute: $\displaystyle \sqrt{\left(\frac{c}{\pi}\right)^2\div2}}=s$
$\displaystyle \sqrt{\frac{c^2}{\pi^2}\times \frac{1}{2}}=s$
$\displaystyle \sqrt{\frac{c^2}{2\pi^2}}=s$
$\displaystyle \frac{c}{\pi\sqrt{2}}=s$
therefore
$\displaystyle \frac{4c}{\pi\sqrt{2}}=4s=P_{square}$
rationalize the denominator:$\displaystyle \frac{4c\sqrt{2}}{2\pi}=P_{square}$
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Just for better understanding of the problem(for Zoso),The diameter of the circle is the length from one corner of the square to the opposite corner,
Why the above condition has to be true?
How many squares of different area can be inscribed in the circle? Why?
How many squares of different perimeter can be inscribed in the circle? Why?
Keep Smiling
Malay
Right, for Zoso , the above condition is only true for the largest possible square you can inscribe in a circle. If that doesn't help I'll show Zoso a diagram.Originally Posted by malaygoel
It's possible to have practically infinite squares inscribed in a circle, maybe if you set up a size restriction I could help you out more?How many squares of different area can be inscribed in the circle? Why?
How many squares of different perimeter can be inscribed in the circle? Why?
Keep Smiling
Malay
Yes, it is possible have infinite squares, but all of them will have the same area.It's possible to have practically infinite squares inscribed in a circle, maybe if you set up a size restriction I could help you out more?
I think are misinterpreting the word 'inscribe'. I means that the vertices of the square lie on the circumference of the circle.
There is no largest square, all of the inscibed squares have the same area.Right, for Zoso , the above condition is only true for the largest possible square you can inscribe in a circle. If that doesn't help I'll show Zoso a diagram.
The above condition is true because the vertex angle of square is 90 degrees.
Keep Smiling
Malay
I said practically infinite.Originally Posted by malaygoel
There is but one possible size of a square per circle that has corners that touch the outside of the circle. I've drawn one below (pay no attention to the fact it's a graph program and looks oval)There is no largest square, all of the inscibed squares have the same area.
The above condition is true because the vertex angle of square is 90 degrees.
Keep Smiling
Malay
as you can clearly see in the diagram, the diameter of the circle is the distance between two opposite corners of the square.