Answer to Question #222750 in Geometry for Sphesihle Zuma

Question

1.design an activity that will enable learners to discover the formula for the area of a circle.
2.in the figure is circle circumscribed to square. The side of the square is 5 units. Calculate the area of the shaded part of the figure.

Answer 1

Materials Required

1. Two sheets of white paper
2. A geometry box
3. A pair of scissors
4. A tube of glue

Theory

The geometrical formula to determine the area (A) of a circle of radius r is given by A = πr².

Procedure

Step 1: Draw any circle on a sheet of white paper. Let its radius be r units.

Step 2: Draw eight diameters dividing the circle into 16 equal parts. Shade the alternate parts as shown in Figure 33.1.

Step 3: Cut these 16 parts of the circle and place them on the other sheet of white paper as shown in Figure 33.2. Here, AB = the circumference of the circle (Figure 33.1) = 2πr.

Step 4: Now, paste the parts on the white paper to form a geometric figure closely resembling a parallelogram by arranging them in such a way that the shaded and unshaded parts appear alternately—the shaded parts pointing downwards while the unshaded parts pointing upwards as shown in Figure 33.3.

Observations

We observe that the approximate length of the base of the parallelogram formed (in Figure 33.3) is πr units (i.e., half of the circumference of the circle in Figure 33.1) and the approximate height of this parallelogram is r units (i.e., equal to the radius of the circle in Figure 33.1).

∴ area of the circle (Figure 33.1) = area of the parallelogram (Figure 33.3)

= πr x r = πr².

Result

The area (A) of a circle of radius r is given by A = πr².

Area of square = 5 × 5 = 25

Half length of square = $\frac{5}{2}=2.5$

Radius of circle$=\sqrt{2.5^2+2.5^2}=3.54$

Area of circle = $\frac{22}{7}×3.54×3.54=39.39$

Shaded part = $39.39-25=14.39$