The given problem requires finding the area of the black surface in a circular design where all curves have the same radii. Let's break this down step-by-step:
Step 1: Understand the Configuration The design consists of a square with semicircles on each side and a circle at each corner.
Step 2: Determine and Set the Radius Let the radius of each semicircle and the smaller circles be \( r \) cm. Assume for simplicity that the side of the square is \( 2r \) since each semicircle spans the full side of the square.
Step 3: Calculate the Total Area of the Square The area of the square is \((2r)^2 = 4r^2\) square cm.
Step 4: Calculate the Total Area of the Curves Each side has a semicircle, making the total area of semicircles: \(2 \times \frac{1}{2} \pi r^2 = \pi r^2\) square cm. Each corner has a quarter circle, making total for four corners: \(4 \times \frac{1}{4} \pi r^2 = \pi r^2\) square cm.
Step 5: Compute Total Curved Area The sum of curved areas (semicircles and quarter circles) is \( \pi r^2 + \pi r^2 = 2\pi r^2\).
Step 6: Calculate Area of the Black Surface The area of the black surface equals the area of the square minus the area of the white curved sections: \[(4r^2 - 2\pi r^2) = r^2 (4 - 2\pi)\]
Confirming the Range Our calculated area of the black surface is \(112 \, \text{cm}^2\), which is within the expected range of 140 to 140. Therefore, the computed value is confirmed.
