Estimating area of a circle with upper and lower bounds
About this video
Ruby estimates a circle's area using a square inside it and a square outside it on a grid. The outer square is 10 by 10, giving an upper bound of 100 square units. The tilted inner square is cut and rearranged into two 5 by 5 squares, giving a lower bound of 50, so the circle's area is between 50 and 100.
You'll be able to
- explain why a shape inside a circle gives a lower bound and a shape outside gives an upper bound for its area.
- find the area of a tilted square by cutting and rearranging pieces.
- state two values a circle's area lies between.
Key words
- area
- The amount of flat space a shape covers, measured in square units.
- lower bound
- A value the answer must be bigger than.
- upper bound
- A value the answer must be smaller than.
- square unit
- The area of one grid square that is 1 unit by 1 unit.
Transcript
- [Instructor] The image shows a circle drawn on a grid. One square is inside the circle and another square is outside the circle. We can see all of that. Each square grid has an area of one square unit, so that's each of these little squares there. Ruby uses the squares to estimate the circle's area.
She finds that the areas between two values, what two values is the area of the circle between? So pause this video, have a go at that before we do it together. Okay, so to do this together, we have to realize that if I can figure out the area of this smaller square, the circle has a larger area than that.
So this would be the lower value of our two. And then if I figure out the area of the larger square, that is going to be the upper value. So let's see if we can do that. Well, the outer square is actually quite easy to figure out the area, reasonably straightforward.
We just have to figure out the height and the width. Well, it's actually a square so it's going to be the same. So this is one, two, three, four, five, six, seven, eight, nine, 10 units tall. It's also going to be 10 units wide 'cause it's a square, it's 10 by 10.
So this outer one is 100 square units. So that's going to be the upper bound. And actually, if we look at the choices, we can already see that there's one that has 100 as the upper bound. So we actually already know that A is the choice. But let's see what the area of the smaller square is.
I'm guessing it is going to be 50, but let's validate that. And since this one's at a diagonal, it's not quite as easy to count the dimensions or to figure out the dimensions. But we could break up this square into, let's do it into some quadrants right over here.
So if I were to divide it like that and then I were to divide it like that, you might notice that if I were to take this piece right over here and flip it over right over there, you would get a square right over there like that. And this is a five by five square.
So that has 25 square units. That looks messy, let me just write that a little bit neater. 25 square units. And then similarly, I would do the same thing down here. If I took this part right over here and flipped it around right over there, that would turn into another five by five square, which would be another 25 square units for a total of 50 square units.
So that would be the lower bound, 50. And the upper bound is 100, and we're done.
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