Estimating area of a circle using polygons: low estimate
About this video
Aisha estimates a circle's area using a square drawn inside it on a grid. The square is 6 by 6, so its area is 36 square units. The circle covers the square plus extra space, so its area is more than 36; the best answer is 'slightly greater than 36 square units'.
You'll be able to
- find the area of a square by multiplying a side length by itself.
- explain why a square inside a circle gives an underestimate of the circle's area.
- choose a reasonable estimate from a lower bound.
Key words
- area
- The amount of flat space a shape covers, measured in square units.
- underestimate
- An estimate that is smaller than the real value.
- square unit
- The area of one 1 by 1 grid square.
Transcript
- [Instructor] We're told the image shows a circle in a square. Yeah, we can see that. Each grid square has an area of one square unit. So they're talking about one of these right over here. Aisha uses the square inside the circle to estimate the area of the circle.
So, that's definitely going to be an underestimate, because it's fully inside of the circle. It's gonna miss out on all of this area out here. So, let's see what they ask us. Based on Aisha's strategy, what is a reasonable estimate for the area of the circle? So pause this video, have a go at that before we do it together.
Okay, now we know how to find the area of a square. It's just going to be one side. Well, it's gonna be one side times itself, 'cause all the sides have the same length. And we could see that it is one, two, three, four, five, six in each dimension. So, it's a six by six square.
It's six this way. It's six this way. And so, it's area of the square, the area of the square is equal to six times six, which is equal to 36. Now, and we could count that. We could go one, two, three, four and get all the way to 36. It's six rows of six. Now, the circle is going to be larger than that.
So, the area is more than 36. So, let's see. The area of the circle is slightly greater than 36 square units. Yeah, I mean, depending on how you define slightly, maybe. Let's see, the area of the circle is slightly greater than six square units. No, that's not right.
It's definitely more than 36. I'll rule that one out. The area of the circle is about 24. No, we said it has to be greater than 36. The area of the circle is slightly less than 36. No, we know it's greater than 36. So yeah, I might say it's, you know, depends how you define slightly here, 'cause we're losing a lot of area here.
I mean, we could even try to approximate it. There's almost one, two, three, four. Yeah, it's a good bit larger than 36 here. But we could go with 36, saying slightly greater for now. It's definitely the most correct of all the choices.
Video © Khan Academy, used under CC BY-NC-SA. Watch on Khan Academy. Puzzicle is not affiliated with Khan Academy.