For a square with an inscribed circle, the shaded corners are the square’s area minus the circle’s area. A diagonal forms a right triangle with two equal sides, so you can use it to find the square’s area. Then use the square’s side as the circle’s diameter. The tempting slip is to treat the diagonal as a side or diameter.
Hints
- Hint 1
The shading covers the corners outside the circle. Find the shaded area by subtracting the unshaded circle from the whole square. Which area must you find first?
- Hint 2
A square’s diagonal is the long side of a right triangle whose other two sides are equal. Don’t use 10 as a square side: by the Pythagorean theorem, the diagonal’s square is twice a side’s square.
- Hint 3
An inscribed circle touches all four sides of its square. Its diameter equals a square side, so its radius is half that side. What happens to the side’s square when you halve the side?
Step-by-step
Subtract the circle from the square
Step 1Identify the area to subtract
The four shaded corners fill the square except for the circle. Write the shaded area as whole minus unshaded part:
- Step 2
Find the square’s area from its diagonal
The figure labels diagonal as , not a side. If each side is , the diagonal forms a right triangle. The Pythagorean theorem gives:
Divide by to get the square’s area, :
Type in Desmos. It shows square units.
- Step 3
Find the circle’s radius squared
The inscribed circle’s diameter is the square’s side, not its diagonal. The radius is half that side. Square this relationship:
Type on the next Desmos line. It shows ; the fraction button gives . That’s , not yet the circle’s area.
- Step 4
Subtract and keep the exact area
The circle area is . Subtract it from the square by typing on the next Desmos line. Desmos shows about , but the choices keep exact. The shaded area is square units. Choice A.