Question

\(A B C D\) is a square, 4 cqual circles are just touching cach other whose centres are the vertices \(A, B, C, D\) of the square. What is the ratio of the shaded to the unshaded area within square? (b) \(\frac{3}{11}\) (a) \(\frac{8}{11}\) (c) \(\frac{5}{11}\) (d) \(\frac{6}{11}\)

Short answer

Answer: The approximate ratio of the shaded area to the unshaded area within the square is \(\frac{6}{11}\).

Step-by-step solution

Find the Area of the Square

Let's denote the side length of square ABCD as "s" and the radius of each circle as "r". Since the circles touch each other and the vertices of the square, we can see that the side of the square is equal to the sum of the diameters of two touching circles. So, we have: s = 2r + 2r s = 4r

Calculate the Area of Circles

Since all the circles have the same radius, their areas are equal as well. The area of one circle is given by the formula: Area of circle = πr^2 Since there are four circles, the total area of the circles is: Total area of circles = 4 × πr^2

Find the Shaded and Unshaded Areas

To find the shaded and unshaded areas, we first need to find the area of the square and subtract the total area of the circles from it: Area of square = s^2 = (4r)^2 = 16r^2 Now, subtract the total area of circles from the Area of the square: Unshaded Area = Area of square - Total area of circles = 16r^2 - 4πr^2 Shaded Area = Total area of circles = 4πr^2

Calculate the Ratio of Shaded to Unshaded Area

Now, we can calculate the ratio of the shaded area to the unshaded area: Ratio = (Shaded Area) / (Unshaded Area) Ratio = (4πr^2) / (16r^2 - 4πr^2) Upon simplification, we get: Ratio = 4π / (16 - 4π)

Find the Correct Option

Now we need to compare the given options with the calculated ratio: (a) 8/11 ≈ 0.7273 (b) 3/11 ≈ 0.2727 (c) 5/11 ≈ 0.4545 (d) 6/11 ≈ 0.5455 The calculated ratio is: Ratio = 4π / (16 - 4π) ≈ (4*3.1416) / (16 - 4*3.1416) ≈ 0.5455 Comparing the results, we can find that option (d) \(\frac{6}{11}\) is the closest to the calculated ratio. Therefore, the ratio of the shaded area to the unshaded area within the square is approximately \(\frac{6}{11}\).

Key concepts

Area of a Square

The area of a square is a fundamental concept in geometry, central to understanding various problems, including those involving circles within squares. Simplified, the area is calculated by squaring the length of one of its sides. If we denote the side length by 's', the formula is:
\[ \text{Area of square} = s^2 \]
Considering a square where each side measures '4r', the calculation would be \( 16r^2 \).
This formula is a starting point for solving many geometric puzzles, like determining how much space is enclosed within the boundaries of a square, or in more complex situations, such as our problem, finding the proportion of a square occupied by other shapes.

Area of a Circle

Another crucial concept in geometry is the area of a circle. It's key not just in pure geometry, but also in practical applications like calculating how much material is needed for a round tablecloth. The formula relies on the circle's radius (r) and the constant \( \pi \), which is approximately 3.14159.
\[ \text{Area of circle} = \pi r^2 \]
In the context of our problem, with circles fitting snugly inside a square, understanding this formula allows students to unravel the part of the square they occupy. When four identical circles are placed at the corners of a square, their combined area is \( 4 \pi r^2 \), an essential step toward finding our ratio of shaded to unshaded area.

Ratio of Areas

Ratios are powerful in geometry, offering a way to compare different areas irrespective of the actual size of the shapes involved. It's about the relationship between two numbers or, in our case, two areas. When we talk about the ratio of shaded to unshaded area:
\[ \text{Ratio} = \frac{\text{Shaded Area}}{\text{Unshaded Area}} \]
The concept becomes particularly interesting when applied to complex figures, where shapes overlap or fit within each other, like circles within a square. Calculating the ratio requires finding the individual areas first and then dividing them to get a fraction that represents how much larger or smaller one is compared to the other. This fraction is a pure number, devoid of units, making it universally applicable.

Circle Geometry

Circle geometry encompasses various aspects, including chords, tangents, and area calculations. A tangent is a straight line that just touches the circumference of the circle without cutting it. In our problem with circles just touching each other, it presents a scenario where the circles' tangents are also the sides of the square.
As circles touch at single points, these tangents create intersections that reveal the relationship between the square and the circles. Understanding these geometric relationships helps in visualizing and tackling problems that involve both square and circular shapes inside a given space. Circle geometry isn't just about the shapes themselves, but also about their positions, interactions, and the resulting figures and areas they create.