Find the area of the shaded region of a rectangle[solved] - Free Printable

It looks like you've uploaded a geometric figure involving squares and circles, but I can't see the image directly. However, based on your description and the text you provided, I’ll interpret the diagram from the labels and structure.

Let me reconstruct the figure based on the given labels:

- There is a large square ABCD.
- Inside it, there's a circle inscribed (touching all four sides of the square).
- Inside that circle, there’s another square EFGH, rotated 45° relative to ABCD, with its vertices touching the circle.
- Inside this inner square EFGH, there's a smaller circle inscribed (touching all four sides of EFGH).
- The shaded region appears to be the area between the inner square EFGH and the inner circle (the blue regions in the corners).

But wait — actually, looking at the description: the inner square EFGH has a white circle inside it, and the shaded regions are the parts of EFGH outside the inner circle. So the shaded area is the area of square EFGH minus the area of the inner circle.

Let’s solve for the shaded area, assuming we're given side lengths or need to express it in terms of a variable.

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Step 1: Assign a variable

Let the side length of the outer square ABCD be $ s $. Then:

- The circle inscribed in ABCD has diameter $ s $, so radius $ r = \frac{s}{2} $.
- This circle touches the midpoints of the sides of ABCD.

Now, the square EFGH is inscribed in this circle. That means its vertices lie on the circle.

A square inscribed in a circle has its diagonal equal to the diameter of the circle.

So:
- Diagonal of square EFGH = $ s $
- Let side of EFGH be $ a $. Then:
$$
a\sqrt{2} = s \Rightarrow a = \frac{s}{\sqrt{2}}
$$

Area of square EFGH:
$$
a^2 = \left(\frac{s}{\sqrt{2}}\right)^2 = \frac{s^2}{2}
$$

Now, inside EFGH, there's a circle inscribed. Since it's inscribed in EFGH, the diameter of this inner circle equals the side of EFGH, which is $ \frac{s}{\sqrt{2}} $.

So radius of inner circle:
$$
r_{\text{inner}} = \frac{a}{2} = \frac{s}{2\sqrt{2}}
$$

Area of inner circle:
$$
\pi r_{\text{inner}}^2 = \pi \left( \frac{s}{2\sqrt{2}} \right)^2 = \pi \cdot \frac{s^2}{8} = \frac{\pi s^2}{8}
$$

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Step 2: Find shaded area

The shaded area is the area of square EFGH minus the area of the inner circle:
$$
\text{Shaded Area} = \frac{s^2}{2} - \frac{\pi s^2}{8}
= s^2 \left( \frac{1}{2} - \frac{\pi}{8} \right)
= s^2 \left( \frac{4 - \pi}{8} \right)
$$

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Final Answer:

$$
\boxed{\frac{s^2(4 - \pi)}{8}}
$$

This is the area of the shaded region (the four blue corners), assuming the outer square has side length $ s $.

If a specific value of $ s $ was given in your image (e.g., $ s = 8 $), plug it in to get a numerical answer.

Let me know if you'd like to compute it numerically or if there's more context!