How to Calculate Diameter From Circumference

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A circle is a geometric form of which every point on the outside of the circle is the same distance away from the center. The distance around the edge of the circle is called the circumference. The distance from one side of the circle to the other, going through the center of the circle, is the diameter. The constant pi, designated by the Greek letter π, is the ratio of the circumference to the diameter of a circle. For any circle, if you divide the circumference by the diameter you get the mathematical constant pi, an irrational number usually rounded to 3.14. Circles are incredibly important throughout graphing, trigonometry, high school mathematics, and beyond.

TL;DR (Too Long; Didn't Read)

The equation for finding the diameter from the circumference formula can be found by solving the regular equation for the circumference of a circle to get the diameter formula:

diameter = circumference / pi.

Setting Up the Formula for Circumference of a Circle

The equation for finding the circumference of a circle is used to solve for diameter in terms circumference:

C = {\pi}d

where C = circumference, π = 3.14 (approximated to two decimal places) and d = diameter.

Plug the numerical value for your circle's circumference into the formula; for instance, 12 units. You should replace the symbol C with the measurement of the circumference of the circle. In this example, input the variables:

Example pt 1

12 = 3.14 * d

Solve the equation for the diameter of the circle,

d= \frac{C}{π}

Example pt 2

In the example, we are given a circumference of 12 units:

d \approx{\frac{12}{3.14}} \approx{3.82}

Extension: How Circumference Relates to Area of a Circle

Now that we have a formula for diameter in terms of circumference:

d = \frac{C}{\pi}

We can use the formula for diameter in terms of radius to then find the area of circles in terms of circumference. The following equations will set up the final substitution for this relationship.

The equation relating radius of a circle and diameter (see calculation here):

r = \frac{d}{2}

The formula for area in terms of radius of the circle:

A = {\pi}{r}^2

Substituting the formula for radius in terms of diameter and diameter in terms of circumference we get:

A = {\pi} \left( \frac{d}{2} \right)^2 = \ \pi \left( \frac{C}{2\pi} \right)^2 = \ \pi \left( \frac{C^2}{4\pi^2}\right) = \ \frac{C^2}{4\pi}

When to start with circumference in the real world

When trying to find circumference in the real world, scientists and engineers will often turn to measuring a line segment across a circle, to find the diameter, however there are also many situations where perhaps circumference would be a more readily accessible measurement. Perhaps measuring a straight line through the center of a circle would be difficult because of scale or accuracy, but the circumference could be a known value or easier to measure precisely. In these scenarios, we can use these equivalencies to then find diameter, radius, and area.

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