About this video
The video finds the shaded area in a 5 cm square ABCD with two quarter circles (quadrants) centred at opposite corners A and C. The two shaded corner pieces are equal, and each is the square's area minus a quarter circle: 25 - 25 pi/4. Doubling gives 50 - 25 pi/2 (or 50 - 12.5 pi) square centimetres.
You'll be able to
- find the area of a quarter circle as (1/4) pi r^2.
- use symmetry to find one piece of a shaded region and double it.
- find a shaded area by subtracting a curved area from a square.
Key words
- quadrant
- A quarter of a circle, made by two radii at right angles.
- symmetric
- Having matching parts, so one piece is an exact copy of another.
- radius
- The distance from the centre of a circle to its edge; here it equals the side of the square.
- exact area
- An area written with pi left in, such as 50 - 25 pi/2.
Transcript
- [Instructor] We are told the given figure has two quadrants centered at A and C. So they're really talking about quadrants of a circle. They overlap each other inside the square, ABCD, of side five centimeters. So all the sides here are five centimeters, it's a square.
Find the exact area of the shaded region. So like always pause the video, try it yourself, and then we will do it together. Okay, now let's do it together. Now the key here is to just realize that each of these areas are going to be the same. They are symmetric.
So what we could do is just focus on finding one of them and then just double it. And the way that my brain likes to approach it is, let me redraw this square. So, and I'm just hand drawing it, it's not perfect. So if I just do it there, let's just focus on this area right over here.
So if I were to draw the quadrant centered at C, remember this is C, D, A, and B. What we need to do to figure out, let me make it a little bit, make it look little bit more like a quarter circle. When they're talking about a quadrant, they're talking about a quarter circle there.
So to figure out this area, we need to figure out the area of the entire square, which is going to be five centimeter squared. So it's gonna be 25 square centimeters minus the area of this quadrant. Now how do we figure that out? Well, we know how to figure out the area of a circle if we know its radius.
The radius here is gonna be the length of one side of this square. That's a radius, that would also be the radius. That would also be the radius. And so it's going to be pi times R squared. So it's going to be 25 pi. Now you might be saying, wait, wait, this doesn't make sense.
This is bigger than that. We have to remember we're only doing a quadrant. This right over here is the area if we were to do the entire circle, if we were to do the entire circle. So what we want to do is divide this by four, so minus 25 pi over four. So this quantity in square centimeters is this area.
Now if we want the total area, we just multiply this by two. So if we multiply this by two, we multiply all of this business by two and we just distribute the two. We get 50 minus 25 pi over two, and we're done. We can say 50 minus 12.5 pi. However we wanna express it, square centimeters, and we are done.
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