By Andrea Coventry, Feb 15, 2023 11:59 am EST
When we talk about the perimeter of a shape, we mean the total length around its edge. For polygons, the perimeter is simply the sum of its side lengths. A circle, however, has a continuous curved boundary, so we use a special term: the circumference.
The circumference of a circle can be found with either the radius or the diameter:
• C = 2 π r (using radius)
• C = π d (using diameter)
Circles are defined by the mathematical constant π (pi), the ratio of a circle’s circumference to its diameter. The value of pi is approximately 3.141592653589793, and is usually rounded to 3.14 for everyday calculations.
Starting from the definition of pi:
\(\displaystyle \pi = \frac{\text{Circumference}}{\text{Diameter}}\)
Multiplying both sides by the diameter gives:
\(\displaystyle \text{Circumference} = \pi \times \text{Diameter}\)
Using the common shorthand, let C denote circumference and d the diameter:
\(\displaystyle C = \pi d\)
The diameter is twice the radius (d = 2r). Substituting this into the formula yields:
\(\displaystyle C = 2\pi r\)
With the circumference formula in hand, you can easily determine the perimeter of a semicircle, the length of an arc, or other circular measurements. For example, a circle with a radius of 5 cm has a circumference of:
\(\displaystyle C = 2\pi(5) \approx 31.42\,\text{cm}\)
If you know the circumference, you can find the diameter or radius:
\(\displaystyle d = \frac{C}{\pi}\quad\text{and}\quad r = \frac{C}{2\pi}\)
These inverses are useful when calculating the area, surface area, or volume of circular objects.
Only the first 39 digits of pi are required to compute the circumference of the observable universe to within the width of a hydrogen atom—illustrating pi’s extraordinary precision.
For more detailed derivations of arc length and other advanced topics, you can consult mathematical texts or reputable online resources.
