A square with sides of length 6 cm is given. The boundary of the shaded region is defined by two semi-circles whose diameters are the sides of the square, as shown.
The area of the shaded region is __________ $cm^2$.
1. Identify the Given Information
- The square has a side length of \( s = 6 \) cm.
- Two semi-circles are drawn inside the square. Their diameters are the left and bottom sides of the square.
- The radius of each semi-circle is half the side of the square: $ r = \frac{6}{2} = 3 \text{ cm} $
2. Analyze the Shaded Region
By observing the image, we can see that the shaded region consists of two identical semi-circles, but the area where they overlap (the "leaf" shape in the bottom-left corner) is unshaded (white). Thus, the total shaded area is the sum of the areas of the two semi-circles minus twice the area of their overlap.
Let's divide the \( 6 \times 6 \) square into four smaller \( 3 \times 3 \) squares. The intersection of the two semi-circles happens entirely within the bottom-left \( 3 \times 3 \) square.
3. Mathematical Calculation
- Area of one semi-circle: $ A_{\text{semi}} = \frac{1}{2} \pi r^2 = \frac{1}{2} \pi (3)^2 = 4.5\pi \text{ cm}^2 $
- Area of the overlap (Leaf shape): The overlap is formed by two quarter-circles inside a \( 3 \times 3 \) square. The area of one quarter-circle is \( A_{\text{qc}} = \frac{1}{4} \pi (3)^2 = 2.25\pi \). The area of the overlap is: $ A_{\text{overlap}} = 2 \times A_{\text{qc}} - \text{Area of } 3\times3 \text{ square} $ $ A_{\text{overlap}} = 2(2.25\pi) - 3^2 = 4.5\pi - 9 $
- Total Shaded Area: According to the visual, the shaded area is the union of the two semi-circles, excluding the overlap twice (since it's white in both): $ \text{Area}_{\text{shaded}} = A_{\text{semi 1}} + A_{\text{semi 2}} - 2 \times A_{\text{overlap}} $ Substituting the values: $ \text{Area}_{\text{shaded}} = 4.5\pi + 4.5\pi - 2(4.5\pi - 9) $ $ \text{Area}_{\text{shaded}} = 9\pi - (9\pi - 18) $ $ \text{Area}_{\text{shaded}} = 18 \text{ cm}^2 $
Important Questions from Mensuration and Geometry
The city of Atlantis was crafted by the God of the seas, Poseidon. It was made of alternating concentric circular rings of land (shaded) and water (not shaded) as represented in the figure (not to scale). The radius of Inner Island was 2.5 stades (a unit of length used in ancient Greece). The water surrounding Inner Island was one stade wide (length AB). This was surrounded by two pairs of alternating rings of land and water. The first pair of land and water was two stades wide each (lengths BC and CD), and the outer pair is three stades wide each (lengths DE and EF).
The ratio of the surface area of the land to that of the water in the city of Atlantis is _________ (round off to two decimal places).
In the given figure, $P, Q$, and $R$ are three points on a circle of radius 10 cm with $O$ as its center, $\overline{PQ} = \overline{RQ}$, and $\angle PQR = 45^\circ$. The figure is representative.
The area of the shaded region $PQRO$ is ______________ cm$^2$.
- A straight line $y = x - 1$ intersects a circle with center at $x = 1, y = 1$ and radius of magnitude 1 at two points. The length of the chord formed by this intersection is _______. (rounded off to three decimal places)
- The shell of a hollow spherical nanoparticle has a uniform thickness of 3 nanometers (nm). The outer radius of the nanoparticle is 5 nm. The ratio of the volume of the shell to the volume of the hollow core is ________
(Round off to one decimal place) - The volume of a sphere of diameter 1 unit is ______ than the volume of a cube of side 1 unit.
