Slices into a rectangle

About this video

How much paint does a round table top need? Area means counting squares, but squares don't fit a curved edge: around the rim they are always cut. This animation finds the area of a circle another way.

It cuts the circle into slices, like a pizza, and lays them in a row, one pointing up and the next pointing down. With 8 slices the row is bumpy. With 16 it is flatter, and with 32 it is almost a rectangle, and the more slices you cut, the closer it gets. The rectangle is as tall as one slice is long, which is the radius. Its length is the curved edges of half the slices, half of the way round the circle, and since the way round is pi times the diameter, half of it is pi times the radius. A rectangle's area is length times height, so the circle's area is pi times the radius times the radius, about 3.14 × radius × radius.

To check it, the video draws a circle of radius 10 inside a square 20 wide. That square is made of four small squares of 10 by 10, each 100. The circle is about 3.14 × 10 × 10 = 314: just over three of the small squares, and less than all four, which is what you can see. This video goes with The Area of a Circle lesson of Geometry Level 1 on Knaph.

Transcript

1. 0:00How much paint does a round table need?
2. 0:06Area means counting squares. But squares don't fit a curved edge.
3. 0:14So cut the circle into slices, like a pizza. Lay them in a row, one up, one down.
4. 0:26More slices make flatter edges. With thirty-two, it's almost a rectangle.
5. 0:37Its height is one slice, the radius.
6. 0:44Its length is half the way round. That's pi times the radius.
7. 0:50So the area is pi, times radius, times radius.
8. 0:58Try a radius of ten. A ten by ten square is a hundred. The circle is about three hundred and fourteen, just over three of those squares.
9. 1:12A circle's area is pi times the radius, times the radius.