Draw a circle on a graph sheet with diameter (breadth) of length 3. Count the squares and use them to estimate the area of the circular region.

Question

Homi · Accepted Answer

We can find the area of a circle by drawing it on a graph sheet and counting the squares inside it.

Step 1 — Draw the Circle

We need to draw a circle on a graph sheet. The diameter of the circle is 3 units. The radius is half of the diameter. Radius = $3 \div 2 = \mathbf{1.5}$ units. Let us place the center at $(1.5, 1.5)$ . The circle extends from $x=0$ to $x=3$ . It also extends from $y=0$ to $y=3$ .

Diagram 1

Step 2 — Count Full Squares

We look for squares completely inside the circle. A square is a $1 \times 1$ unit area. The square at $(1,1)$ is fully inside. Let us count this square. There is 1 full square.

Step 3 — Count Partial Squares

Now, we count squares partly inside the circle. These squares are not fully inside or outside. There are 8 such squares around the full square. These squares are $(0,0), (1,0), (2,0)$ . Also $(0,1), (2,1), (0,2), (1,2), (2,2)$ . We count each partial square as 0.5 unit. Number of partial squares = 8.

Step 4 — Estimate the Area

The area from full squares is 1 square unit. Area from partial squares is $8 \times 0.5$ . $\text{Area from partial squares} = 8 \times 0.5$ $= \mathbf{4} \text{ square units}$ The total estimated area is the sum of these. $\text{Total estimated area} = 1 + 4$

$\boxed{5 \text{ square units}}$

Answer

(i) The estimated area of the circular region is 5 square units.

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