The diagram shown above has four circles of 7 cm radius with centres at A, B, C and D. If the q

step 1 In this casein diagram is given 4 circle is given A P and C are the centers of the circle and he

The diagram shown above has four circles of 7 cm radius...

The diagram shown above has four circles of $7 \mathrm{~cm}$ radius with centres at $\mathrm{A}, \mathrm{B}$, $\mathrm{C}$ and $\mathrm{D}$. If the quadrilateral $\mathrm{ABCD}$ represents a square, then find the area of the shaded region. (1) $42 \mathrm{~cm}^{2}$ (2) $21 \mathrm{~cm}^{2}$ (3) $63 \mathrm{~cm}^{2}$ (4) $84 \mathrm{~cm}^{2}$

The diagram shown above has four circles of $7 \mathrm{~cm}$ radius with centres at $\mathrm{A}, \mathrm{B}$, $\mathrm{C}$ and $\mathrm{D}$. If the quadrilateral $\mathrm{ABCD}$ represents a square, then find the area of the shaded region.
(1) $42 \mathrm{~cm}^{2}$
(2) $21 \mathrm{~cm}^{2}$
(3) $63 \mathrm{~cm}^{2}$
(4) $84 \mathrm{~cm}^{2}$

Key Concepts

Circle Geometry

This concept involves understanding the properties of circles, specifically the formula for the area of a circle (?r²). It is crucial in problems where parts of circles appear in composite figures, such as when only a segment or a fraction (like a quarter) of a full circle is used. In many geometric puzzles, knowing how to compute a circle’s area allows one to determine the area of its portions accurately.

Square Geometry

Square geometry is central to solving problems involving squares because it provides formulas for calculating the area (side²) and understanding properties such as equal sides and right angles. When a square interacts with circular regions, knowing its side length and area is essential for setting up the problem and determining how much of the contained or surrounding space is occupied.

Composite Area Calculation

This concept pertains to finding the area of a complex figure by breaking it down into simpler parts whose areas can be calculated individually. For instance, when a region is defined by subtracting circular segments from a square, one calculates the area of the square and then subtracts the area contributed by the circle portions. Mastery of this technique is vital in many geometry problems where shapes overlap or portions are removed.

Portions of Circles (Quarter Circles)

Understanding how to work with portions of a circle, such as a quarter circle, is key in many geometric configurations. Since a quarter circle represents one?fourth of a circle's area, its area is computed as (1/4)?r². This concept is frequently used when circles are positioned in relation to a square so that only segments of each circle lie within the square's boundaries.

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