# Circles in a square and diameter of the circle

• MHB
• Wilmer
In summary, the formula for calculating the diameter of a circle is d = 2r, a circle can fit perfectly inside a square if the diameter is equal to the length of one side, the area of a circle inscribed in a square can be found using A = πr^2, the diameter of a circle is directly related to its circumference, and the diameter of a circle inscribed in a square cannot be longer than the side length of the square.
Wilmer
A circle is inscribed in a square with sides = 40.

A smaller (of course!) circle tangent to the above
circle and 2 sides of the square is inscribed in
one of the corners of the square.

What is the diameter of this circle?

$20(\sqrt2-1)$, easy.

No.
diameter = 40(3-2√2) = ~6.86
Not as easy as it appears...

You’re right. I overlooked the teeny bit in the extreme corner.

## 1. What is the formula for finding the diameter of a circle in a square?

The diameter of a circle in a square can be found by dividing the length of one side of the square by √2. This can also be expressed as d = s/√2, where d is the diameter and s is the length of one side of the square.

## 2. How do you find the area of a circle in a square?

The area of a circle in a square can be found by first finding the diameter using the formula d = s/√2. Then, the area can be calculated using the formula A = π(d/2)^2, where A is the area and d is the diameter.

## 3. Can a circle be inscribed in a square with a different shape?

Yes, a circle can be inscribed in a square with a different shape. The only requirement is that the circle's diameter is equal to the length of one side of the square. This means that the square does not have to be a perfect square, as long as the diameter of the circle is equal to the length of one side.

## 4. What is the relationship between the diameter of a circle in a square and the length of the square's diagonal?

The diameter of a circle in a square is equal to the length of the square's diagonal multiplied by √2. This can be expressed as d = √2 * diagonal, where d is the diameter and diagonal is the length of the square's diagonal.

## 5. How does the diameter of a circle in a square change if the length of the square's side is doubled?

If the length of the square's side is doubled, the diameter of the circle in the square will also double. This is because the diameter is directly proportional to the length of the square's side. So, if the length of the square's side is multiplied by 2, the diameter will also be multiplied by 2.

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