The Diameter, Radius, and Circumference of Circles

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If we were to measure the distance running from the center of a circle to the outer edge of said circle, we would be finding the radius. Think of a clock; if one of the hands was long enough to reach to the edge of the clock, this hand could be considered the radius of the clock – no matter what time it is!

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While the radius of a circle runs from its center to its edge, the diameter runs from edge to edge and cuts through the center. A circle’s diameter essentially splits the shape in half. Radius and diameter are close friends – a circle’s radius is half the length of its diameter (or: a circle’s diameter is twice the length of its radius).

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A radius is a line segment. Hence, it has two endpoints: the point at the center of the circle and the point along the circle’s edge which we’ve connected it to. With all of this in mind, we know to name a radius in the same way that we do all line segments: the name of both endpoints listed side-by-side (often with a bar above the two letters).

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Yes! If you remember only one fact about circles, let this one be it. Drill it into your mind! A radius is half the length of the diameter.

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If we only know the radius of a circle, we’ll just multiply that value by 2 in order to get the diameter \((d=2r)\). Similarly, if we only know the diameter of a circle, we divide by 2 and out pops the radius \((r=\frac{d}{2})\)!

But, what if we aren’t given either of these values? In order to solve for either the radius or diameter of a circle, we need to know either its circumference or its area.

Say that we were given the circumference. Since the equation for finding circumference looks like \(C=2πr\), we’ll just rearrange the equation and find:\(r=\frac{C}{2π}\).

If we were given the area, we’d rearrange its equation, \(A=πr^2\), to be: \(r=\sqrt{\frac{A}{π}}\)

Here’s an example: Given that the area of a circle is \(9π\text{ in}^2\), find its diameter. We know that the area equation is \(A=π×r^2\), but notice that there is not a ‘diameter’ variable in this formula. To solve this problem, we’ll either need to: (a.) replace \(\frac{d}{2}\) for r in the area equation, or (b.) find the value of r and then multiply that by 2. Here we’re going to use option b, but both methods are valid choices!

\(A=9π=πr^2\)

\(9=r^2\)

\(±3=r\)

(Notice that the square root of 9 can be either 3 or -3. Since we’re dealing with a real circle, we’ll simply use \(d=2r=6\text{ inches}\)

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In order to calculate a circle’s circumference, we need to know either its diameter or its radius. We then use the appropriate value in this equation: \(C=2πr\) (where “r” represents radius, of course). Here’s an example: Find the circumference of this circle: \(r=\frac{10\text{ units}}{2}=5\text{ units}\)

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We know that circumference is the length of the entire outer edge of a circle. (We could think of it this way: circumference is to circles what perimeter is to triangles, rectangles, pentagons, and so on!) With this in mind, let’s rearrange the variables in the equation \(C=2πr\) to get \(C=2r\timesπ\). Remember that \(2r=d\) (where d represents diameter), so we could rewrite this equation yet again: \(C=d\timesπ\). In other words, we can wrap a string (which is the same length as the diameter) around the circle 3.1415926… times.

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The applications of circumference in everyday life are truly endless! One example, though, is determining how large of a tire someone needs for a bike or for a car. Another example would be finding how much wood is in a tree: with a very, very old tree, it would be pretty difficult to measure the diameter of the tree’s base; but it’d be straightforward to wrap a rope around the base and measure the circumference. Then, you could use this circumference measurement and ‘reverse-engineer’ the circumference equation to determine the tree’s diameter. With this measurement (and the height of the tree), we could find the volume of wood within this tree. Again, the list of examples could go on and on forever, so keep an eye out for other ways that you use circumference throughout your life!

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Circumference is the length of one complete ‘lap’ around a circle, and diameter is the length of the line segment that cuts a circle in half. Think of circumference as an outer measurement and diameter as an inner measurement of the circle!

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No! Remember the equation \(C=2πr\) can be rewritten: \(C=2r\timesπ\), \(C=d\timesπ\), or \(C=πd\). So, perhaps we could say that diameter is 3.1415926…^th of the circumference…

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If length is defined as the distance between two points, then yes, diameter is a length. The diameter of a circle measures the distance between the two furthest points on a circle.