Question

On a square cardboard sheet of area $784c{m}^2$, four congruent circular plates of maximum size are placed such that each circular plate touches the other two plates and each side of the square sheet is tangent to two circular plates. Find the area of the square sheet not covered by the circular plates.

Answer 1

According to the question,

Step 1: Calculate the length of the side of the square cardboard:

Area of the square cardboard = $784c{m}^2$

Area of square = ${\left(side\right)}^2$

${\left(side\right)}^2$ = $784c{m}^2$

$$\begin{gathered} \text{side =}\sqrt{784} \\ side=28cm \end{gathered}$$

Step 2: Calculate the radius of the circle :

Since circles are touching each other and the sides of the square is tangent to the circle,

therefore, $\text{side of the square = 2×diameter of the circle}$

$$\begin{gathered} \text{2×diameter of the circle = 28} \\ \text{diameter of the circle = 28÷2 = 14} \end{gathered}$$

Radius of the circle = $\frac{\text{Diameter}}{2}$

= $\frac{14}{2}=7cm$

Step 3: Calculate the area of shaded portion:

Area of four circles = $4\times \pi r^2$

$$\begin{gathered} =4\times \frac{22}{7}\times 7^2 \\ =4\times \frac{22}{7}\times 7\times 7 \\ =88\times 7 \\ =616c{m}^2 \end{gathered}$$

Area of shaded portion $\text{= Area of cardboard - Area of four circles}$

$$\begin{gathered} =784-616 \\ =168c{m}^2 \end{gathered}$$

Hence, the area of shaded portion is $168c{m}^2$.