In the figure below, PQRS is a square of diagonal 7 √(2) cm . With P as the centre, the arc QS

step 1 We have this diagram PQ orus which is a square with a diagonal 7 root 2 cm and with the p as the

In the figure below, PQRS is a square of diagonal 7 √(2) cm...

In the figure below, $\mathrm{PQRS}$ is a square of diagonal $7 \sqrt{2} \mathrm{~cm} .$ With $\mathrm{P}$ as the centre, the arc $\mathrm{QS}$ is drawn. Find the area of the shaded region (in $\mathrm{cm}^{2}$.

Key Concepts

Composite Area Calculation

Often, complex figures are composed of simple shapes whose areas can be computed separately. By either adding or subtracting these individual areas, one can calculate the area of a composite region. This technique is widely used in problems that involve overlapping or inscribed figures and is essential when dealing with segments, sectors, and polygons in a unified figure.

Circle Sector Area

A circle sector is a portion of a circle bounded by two radii and the connecting arc. Its area is calculated as a fraction of the whole circle’s area based on its central angle. Using the formula (central angle/360) × ? × r², one can determine the area of a sector, which is often needed when circles or arcs overlap with other geometric figures.

Diagonal?Side Relationship

In a square, the diagonal and side length are related by the Pythagorean theorem. Specifically, the diagonal is equal to the side length multiplied by the square root of 2. This relationship helps in determining one measure from the other and is particularly useful when only one of those lengths is provided.

Square Properties

A square is a regular quadrilateral with all sides equal and every angle measuring 90 degrees. Its symmetry and constant side lengths allow for straightforward calculations of area and perimeter. Understanding these properties is crucial when the square serves as the backdrop for more complex geometric constructions.

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