Estimating area of a circle using polygons: high estimate
About this video
Mina estimates a circle's area using a tilted square drawn outside it on a grid. By cutting the square into pieces that combine into two 3 by 3 squares, the video finds the square's area is 18 square units, so the circle's area must be less than 18. The best choice is 'slightly less than 18 square units'.
You'll be able to
- find the area of a tilted square on a grid by rearranging its pieces.
- explain why a square outside a circle gives an overestimate of the circle's area.
- choose a reasonable estimate for a circle's area from an upper bound.
Key words
- area
- The amount of flat space a shape covers, measured in square units.
- estimate
- A sensible, close guess at a value, based on what you know.
- overestimate
- An estimate that is bigger than the real value.
Transcript
- [Instructor] We're told the image shows a circle and a square. We see that, a circle and a square. Each grid, each grid square has an area of one square unit. So they're talking about one of these grid squares right over here. Mina uses the square outside the circle, so this square over here, to estimate the area of the circle.
Based on Mina's strategy, what is a reasonable estimate for the area of the circle? All right, pause this video and have a go at that before we do this together. Okay, now let's do it together. So what I'm gonna try to do is figure out the area of the square, and if I can figure out the area of the square, we know that the area of the circle is going to be less than that, and maybe that'll help us with these choices.
So the area of the square, we could just count these squares, and then we could use these little triangular sections right over here as a half. Or we could maybe divide it into quadrants like this. So like that and like this. Actually, this is probably sufficient because, actually, let me do it like this.
Let me cross it like that and then like that. Because what we could see here is if I were to combine this part right over here and this part right over here, those two combined would make a square of area. They'd be equivalent to a three by three square. So this plus this is nine square units.
Once again, another way to think about it is if I were to flip this over, over on this side, it would fill up this entire three by three square. So that's nine units up here, and then there's nine units down here. So the area of this entire square is 18, 18 square units.
And you could also try to count each of these individually. That's another way to do it. And then these triangular portions would be halves, and you should still get to 18. So given that we know that the square is 18 square units, this circle has to be, the area of that has to be less than 18 square units because it's not taking up, it's not taking up this part right over here, this part over here, this part over here, and that part over there.
So let's see which of these choices are consistent with that? The area of the circle is slightly less than 18 square units. Yeah, I mean, depending on how you define slightly, it looks like it would be at least, I don't know, four or more units, less than 18 square units.
So I would approximate maybe around 14 something, but yeah, let me go with this for now. Let me read the other ones. The area of the circle is slightly greater than 18 square units. No, definitely not. That would imply that it had, yeah, don't like that one. I'm gonna cross it out.
The area of the circle is slightly less than nine square units. No, nine square units would be a lot smaller than that. We know it's not that small. The area of the circle is about 24 square units. No, once again, it can't be larger than 18. So we're gonna rule that one out as well, so.
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