Question

Proofs for showing area of circle is equal to area of rectangle by cutting sectors

Answer 1

Answer:

Step-by-step explanation:

Here is a way to find the formula for the area of a circle:

 

Cut a circle into equal sectors (12 in this example)

Divide just one of the sectors into two equal parts. We now have thirteen sectors – number them 1 to 13:

circle 13 including 2 half slices

Rearrange the 13 sectors

sectors laid out like rectangle

Which resembles a rectangle:

The height is the circle's radius: just look at sectors 1 and 13 above. When they were in the circle they were "radius" high.

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

And so we have (approximately):

rectangle is (pi x radius) by radius radius

π × radius  

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

Area = (π × radius) × (radius)

= π × radius2

Note: The rectangle and the "bumpy edged shape" made by the sectors are not an exact match.

But we could get a better result if we divided the circle into 25 sectors (23 with an angle of 15° and 2 with an angle of 7.5°).

And the more we divided the circle up, the closer we get to being exactly right.

Conclusion

Area of Circle = π r2