Math Labs with Activity – Derive a Formula for Finding the Area of a Circle (Method 1)
To derive a formula for finding the area of a circle (Method 1).
- A sheet of white paper
- A long nylon thread of uniform thickness
- A pair of scissors
- A tube of glue
- A geometry box
The geometrical formula for finding the area (A) of a circle of radius r is given by A = πr².
Step 1: Draw any circle on the sheet of white paper. Let its radius be r units.
Step 2: Cut off various lengths of the nylon thread. Fill up the circle with circular loops of this thread, without leaving any gap between the loops as shown in Figure 32.1.
Step 3: Observe that the lengths of the thread pieces used in the loops increase from the centre towards the periphery.
Step 4: Straighten up all the thread pieces and paste them on the sheet of white paper, arranging from the smallest to the longest, forming a triangle as shown in Figure 32.2.
Step 5: Record your observations.
Observations and Calculations
We observe that the base of the triangle formed in Figure 32.2 is equal to the circumference of the circle, i.e., 2πr and its height is equal to the radius of the circle, i.e., r.
Now, area of the circle (Figure 32.1) = area of the triangle (Figure 32.2)
The area (A) of a circle of radius r is given by A = πr².
Remarks: From the above discussion it is clear that the area of a circle is equal to half of the product of its circumference and radius.