Question

Carl knows that the area of a given circle is 400 cm2. He wants to defend an informal argument that the area of a circle can be approximated by dividing the circle into congruent segments, rearranging the segments to resemble a parallelogram, and replacing the dimensions of the parallelogram with appropriate values from the circle. Carl divides the circle into 6 congruent segments and makes his calculations, but the area he calculates for the circle is only 350 cm2. How could Carl defend his informal argument?

A. Carl could divide the circle into a larger even number of congruent sectors. Then each sector would be smaller, and the approximation of the circle’s area will be closer to the actual area of the circle.

B. There is too great a difference between Carl’s calculated value using the informal argument approach and the actual area of the circle. Carl should realize that his informal argument cannot be defended .

C. Carl could divide the circle into a smaller even number of congruent sectors. Then, each sector would be larger, and the approximation of the circle’s area will be closer to the actual area of the circle.

D.Carl could arrange the sectors of the circle into the shape of a trapezoid instead of into the shape of a parallelogram, and replace the dimensions of the trapezoid with appropriate values from the circle.

Answer 1

A. Carl could divide the circle into a larger even number of congruent sectors. Then each sector would be smaller, and the approximation of the circle’s area will be closer to the actual area of the circle.

STEP BY STEP EXPLANATION :

Area of a Circle by Cutting into Sectors:

1. Cut a circle into equal sectors and the more we divided the circle up, the closer we get to being exactly right.

2.Rearrange the sectors, which resembles a parallelogram.

What are the (approximate) height and width of the parallelogram?

The height is the circle's radius.

The width (actually one "bumpy" edge) is half of the curved parts around the circle. In other words it is about half the circumference of the circle.

We know that:

Circumference = 2 × π × radius

And so the width is about:

Half the Circumference = π × radius

ow we just multply the width by the height to find the area of the rectangle:

Area = (π × radius) × (radius)

= π × radius2

Conclusion

Area of Circle = π r2