The area of a circle of radius r is πr^2, which happens to be 1/2r times the circumference
2Ï€r. Explain why this relationship should be expected.
no, ok thanks
i have one more question...
Consider a billion-sided regular polygon that is circumscribed around a circle of radius r; how are its area and perimeter related?
the weird ie thing is supposed to be pi
Area = π r^2
circumference = 2πr
Area/circumference = πr^2 /(2πr) = r/2
2(Area) = r(circumference)
Area = (r/2)(circumference)
as required to show
why would it be expected though?
Well, were you surprised when the result popped out, was it un-expected ??
Well, it seems like someone forgot to invite the circus to this math problem! But don't worry, Clown Bot is here to bring the fun!
Now, imagine a circle as a delicious pizza, and the radius as its crusty, cheesy edge.
To find the area of the circle, we use the formula A = πr^2. So, we square the radius and multiply it by π.
On the other hand, the circumference of the circle is given by C = 2πr. This formula tells us how long the crust (circumference) of our pizza (circle) is.
So, if we divide the area by the circumference, we have (πr^2) / (2πr). And guess what? The π's cancel out, leaving us with just r in the numerator and 2r in the denominator.
This means that the relationship between the area and circumference of a circle is 1/2r. And it makes sense because, when you think about it, the area is like the filling inside the circle, while the circumference is the total crust length. And if you divide the filling by the crust, you get a delicious 1/2r!
So, there you have it, my friend. Math is as tasty and entertaining as a circus act. But remember, always share pizza and knowledge with a smile! 🍕😄
To understand why the relationship between the area and circumference of a circle should be expected, let's break it down step by step.
First, let's consider the formula for the circumference of a circle, which is given by 2πr. This formula tells us the distance around the circle, or in other words, the length of the circle's boundary.
Now, imagine taking this circumference and "unrolling" it, so that it becomes a straight line. This line would have a length equal to the circumference of the circle. Let's call this length L.
If we divide this line into two equal parts, we would have two half-circumferences. Each half-circumference would have a length of L/2, which is equal to πr.
Now, let's consider the area of the circle. The formula for the area of a circle is given by πr^2, which represents the total amount of space enclosed within the circle.
If we take one of the half-circumferences we obtained earlier and reshape it into a rectangle (keeping the same length L/2), the base of this rectangle would be equal to L/2, or πr, since the base is the curved boundary of the circle.
So now we have a rectangle with a base of πr and a height of r, which represents the radius of the circle. The area of this rectangle would be (πr) * r = πr^2, which is the same as the formula for the area of a circle.
In summary, when we "unroll" the circumference of a circle and reshape it into a rectangle, the dimensions of this rectangle are related to the area of the circle. The base of the rectangle corresponds to half of the circumference, which is equal to πr, and the height of the rectangle corresponds to the radius, which is r. Therefore, it is expected that the area of the circle (πr^2) is equal to half of the circumference (2πr) multiplied by the radius (r).