 In this video we want to find the circumference and area of this circle. In order to find the circumference and area we need to know the radius or the diameter. Now the radius or diameter would be the hypotenuse of this right triangle. Also we know that these two legs of the right triangle are congruent. And so that means we have a 45-45-90. In a 45-45-90 the two legs are let's call those the n side, the hypotenuse if you remember is the n root 2. And so that means this diameter is the n root 2 side. And we know that n is equal to 5 root 3. And so the diameter is equal to the n root 2 side. n is 5 root 3 root 2. So the diameter would be 5 root 2 times root 3. And those square roots will multiply together to give us 5 root 6. 5 root 6 is the diameter. And now if 5 root 6 is the diameter that means the radius is half as big. So the radius is 5 root 6 over 2. And so we can use this information to help us find the circumference and the area. Let's find circumference first, which we know to be pi times diameter. And in the previous slide we knew that the diameter is 5 root 6. And so our circumference will be pi times 5 root 6. Which we can rewrite in a number of different ways. My personal preference would be 5 pi root 6. Leave the square root until the very end. And then of course we'd give this units. And if we were asked to approximate this, you could type 5 times pi times root 6 into your calculator. And that will give you 38.476 units. So there's our circumference. Now let's take a look at area. So in order to find the area, we'll need to use pi times radius squared. And of course we know the radius from the previous slide. So we have 5 root 6 over 2 squared. And now dealing with such a complex thing squared is going to take a little bit of work. So let's go through it step by step. So we have 5 times. Now if we're squaring a fraction, we'd square the top part. 5 root 6. Oh that 5 looks kind of cruddy doesn't it? 5 root 6 squared over 2 squared. And now we'll deal with this numerator piece a little bit more. If I have 5 times root 6 squared, that's the same as 5 squared times root 6 squared. Which simplifies one step further to pi times 25 times 6 all over 4. Now I'm running out of room. Hold on just a sec. And 25 times 6 divided by 4. Well 25 times 6 I know is 150 over 4. But 150 over 4 can reduce to 75 over 2. So our exact area would be 75 over 2 pi square units. So that would be an exact answer. And if I did ask you for an approximate answer, then you'd type that into your calculator and you would get approximately 117.810 square units. Again both of these answers are correct. It just depends on whether you're asked for an exact answer, which would be this one, or an approximate answer, which would be that last one.