Math Problem Statement

The diagram shown below has four circles of 7 cm radius with centres at A, B, C, and D. If the quadrilateral ABCD represents a square, then find the area of the shaded region.

Solution

We are given a diagram with four circles, each of radius 7cm7 \, \text{cm}, centered at points AA, BB, CC, and DD, forming a square. The task is to find the area of the shaded region, which consists of the overlapping portions of the circles inside the square.

Key Observations:

  1. Square and Circles Configuration:

    • The square ABCDABCD has its vertices at the centers of the circles.
    • The side length of the square is equal to the diameter of one circle. Since the radius is 7cm7 \, \text{cm}, the diameter is 14cm14 \, \text{cm}. Therefore, the side of the square is 14cm14 \, \text{cm}.
  2. Shaded Region:

    • The shaded region consists of the area of the square minus the areas of the segments of the circles that lie outside the square.
    • Each corner of the square cuts a quarter of each circle.

Steps to Calculate the Area of the Shaded Region:

1. Area of the Square:

The area of square ABCDABCD is: Area of square=side2=142=196cm2\text{Area of square} = \text{side}^2 = 14^2 = 196 \, \text{cm}^2

2. Area of One Circle:

The area of one circle is: Area of one circle=πr2=π(7)2=49πcm2\text{Area of one circle} = \pi r^2 = \pi (7)^2 = 49\pi \, \text{cm}^2

3. Area of the Four Circular Segments Inside the Square:

Each quarter of a circle is one-fourth the area of the circle, and there are four such quarters (one from each circle), which together form a full circle. Thus, the total area of the four quarter-circles is equivalent to the area of one full circle: Total area of four segments=49πcm2\text{Total area of four segments} = 49\pi \, \text{cm}^2

4. Shaded Area:

The shaded area is the area of the square minus the area of the four quarter-circles: Shaded area=Area of squareArea of four segments=19649πcm2\text{Shaded area} = \text{Area of square} - \text{Area of four segments} = 196 - 49\pi \, \text{cm}^2

Using π3.1416\pi \approx 3.1416: 49π49×3.1416=153.94cm249\pi \approx 49 \times 3.1416 = 153.94 \, \text{cm}^2 Shaded area196153.94=42.06cm2\text{Shaded area} \approx 196 - 153.94 = 42.06 \, \text{cm}^2

Thus, the area of the shaded region is approximately 42.06 cm².

Would you like more details or have any questions?

Here are 5 related questions:

  1. What is the formula for the area of a circle segment?
  2. How would the shaded area change if the radius of the circles were doubled?
  3. What if the quadrilateral was not a square, but a rectangle—how would that affect the calculation?
  4. Could this problem be solved using polar coordinates? If so, how?
  5. How do we calculate the perimeter of the square formed by the centers of the circles?

Tip: When solving geometric problems involving circles and squares, always look for symmetry to simplify the area calculations.

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Geometry
Circles
Area of Square
Area of Circular Segments

Formulas

Area of square = side^2
Area of a circle = πr^2
Area of circular segment (quarter circle) = (1/4)πr^2

Theorems

Basic Geometry Theorems related to areas of squares and circles

Suitable Grade Level

Grades 9-10