Perimeter and Area | A

Question 5

Try using different shapes (triangle and rectangle) to fill the given space (without overlaps and gaps) and find out the merits associated with using a square shape to find the area rather than another shape. List out the points that make a square the best shape to use to measure area.

Solution

Understand the Question

Area is the measure of the total flat surface covered by a closed figure.
To accurately measure area, the measuring unit shape must tile a region completely without overlaps and without gaps.
While shapes such as rectangles and triangles can tile flat regions, the unit square is the most convenient and standardized shape for measuring area due to its equal side lengths and right-angled corners.

Step 1 · Filling Space with Different Shapes

A flat space can be filled using different geometric shapes:

Rectangles: Smaller identical rectangular tiles placed side-by-side completely cover a square area without any gaps or overlaps.
Triangles: Right-angled triangles (obtained by splitting squares or rectangles diagonally) can also be fitted together edge-to-edge to cover the region completely.
Circles: Circular shapes leave empty gaps between them when packed together, making them unsuitable for measuring area.

Step 2 · Merits of Using a Square to Measure Area

The square is the ideal shape for measuring area due to the following reasons:

No Gaps or Overlaps: Squares fit together perfectly along all edges, tessellating flat surfaces completely.
Equal Dimensions: All four sides are equal in length with $90^\circ$ corners, avoiding direction-dependent measurements (unlike rectangles which have differing lengths and widths).
Grid Counting: Unit squares naturally arrange into neat rows and columns, allowing the total area to be found simply by multiplying $\text{number of rows} \times \text{number of columns}$ .
Simple Formula: The area of a square is straightforward: $\text{Area} = \text{side} \times \text{side}$
Standard Measurement Units: Standard units of area are defined as square units, such as square centimetres ( $\text{cm}^2$ ) and square metres ( $\text{m}^2$ ).

Answer

A square is the best shape for measuring area because:

It tiles flat surfaces perfectly without gaps or overlaps.
It has equal sides and right angles, making counting in rows and columns easy.
It simplifies area calculations ( $\text{side} \times \text{side}$ ).
It serves as the standard unit of measurement (such as $\text{cm}^2$ and $\text{m}^2$ ).

Common Mistakes

Using Shapes that Leave Gaps: Assuming all shapes can measure area equally well; curved shapes like circles leave empty gaps between boundaries when placed next to each other.
Confusing Linear and Area Units: Writing area in linear units (such as $\text{cm}$ or $\text{m}$ ) instead of standard square units (such as $\text{cm}^2$ or $\text{m}^2$ ).

Question 5

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