In the figure shown, a circle is inscribed in square . What is the area of the shaded region?
When a shaded region is formed by an inscribed shape, first decide whether you should add or subtract areas (here: square minus circle). Then use the key geometric relationship shown in the diagram—in this case, the square’s diagonal—to find the side length, since both the square’s area and the circle’s radius depend on it.
Hints
Identify what is shaded
The shading is everything inside the square that is not inside the circle.
Use the labeled diagonal
The figure gives the length of a diagonal of the square. For a square, diagonal .
Connect the circle to the square
An inscribed circle has diameter equal to the square’s side length, so the radius is half the side length.
Desmos Guide
Compute the side length from the diagonal
Enter s = 10/sqrt(2) to represent the square’s side length.
Compute square and circle areas
Enter As = s^2 and r = s/2, then enter Ac = pi*r^2.
Compute the shaded area and match to a choice
Enter As - Ac and compare the exact form (or a decimal approximation) to the answer choices to find the matching expression.
Step-by-step Explanation
Use the diagonal to find the side length
From the figure, diagonal . For a square with side length , the diagonal is , so
Find the areas of the square and the circle
Square area:
Because the circle is inscribed, its diameter equals the side length of the square, so the radius is
Circle area:
Subtract to get the shaded area
The shaded region is inside the square but outside the circle, so
Therefore, the correct choice is .