Question

sketch a polygon inside the circle and estimate the area of the circle. explain your thinking.

Explanation:

Step1: Identify circle radius

From the grid, the circle's radius $r = 3$ (distance from center to edge along axes).

Step2: Calculate circle area formula

Use the circle area formula: $A = \pi r^2$

Step3: Substitute radius and estimate

Substitute $r=3$: $A = \pi (3)^2 = 9\pi$. Estimate $\pi \approx 3.14$, so $9\times3.14=28.26$.

Step4: Inscribe polygon (square example)

An inscribed square has diagonal equal to $2r=6$. Side length $s = \frac{6}{\sqrt{2}} = 3\sqrt{2}$. Area of square: $s^2=(3\sqrt{2})^2=18$, which is less than the circle's area, confirming the circle's area is larger than this inscribed polygon's area.

Answer:

The estimated area of the circle is approximately 28.3 (or exactly $9\pi$). We can inscribe a square (or other polygon like an octagon) inside the circle: for an inscribed square, its diagonal matches the circle's diameter (6 units), so its area is 18, which is smaller than the circle's area. Using the circle area formula with radius 3, we calculate $A=\pi r^2=9\pi\approx28.3$.