The circumference of a circle is the length of its boundary. You need it when buying edging for a flower bed, wrapping a ribbon round a cake or setting the wheel size on a bike computer. The other way round, the circumference is often the only thing you can measure easily – a tree trunk, for example. Then you work back from the circumference to the diameter.
The circumference formula
The circumference is π times the diameter or 2π times the radius. A circle with a 10 cm diameter has a circumference of π × 10 = 31.42 cm. Rule of thumb: the circumference is a little more than three times the diameter.
Diameter from the circumference
The other way round, divide the circumference by π:
| Circumference | Diameter | Radius |
|---|---|---|
| 50 cm | 15.92 cm | 7.96 cm |
| 80 cm | 25.46 cm | 12.73 cm |
| 100 cm | 31.83 cm | 15.92 cm |
| 120 cm | 38.20 cm | 19.10 cm |
| 157 cm | 49.97 cm | 24.99 cm |
| 200 cm | 63.66 cm | 31.83 cm |
Example: measuring a tree trunk
Without calipers a tree's diameter is hard to measure, but its girth takes any tape measure. With a girth of 157 cm the diameter is 157 ÷ π = 49.97 cm, roughly 50 cm.
This matters beyond curiosity. Many German municipal tree protection bylaws protect trees above a certain trunk circumference, usually measured 1 m above the ground. Foresters, by contrast, quote the diameter at breast height, measured at 1.30 m. Which value and measuring height apply to you is set out in your local bylaw.
Example: wheel circumference for a bike computer
Bike computers work out speed and distance from the wheel circumference. A 700C (28-inch) wheel has a rim diameter of 622 mm – the number in the tyre size, such as 37-622. The tyre adds to the top and bottom, so the outer diameter is about 622 + 2 × 37 = 696 mm. That gives:
For more accuracy, roll the wheel one full turn with the rider on board and measure the distance, because the tyre flattens slightly under load.
A puzzle: the rope around the Earth
A rope lies taut around the equator, about 40,030 km long. Now it is lengthened by exactly 1 m and lifted evenly all the way round. Could a cat crawl underneath? The surprising answer: the gap does not depend on the size of the Earth, because from C = 2πr the change is:
Lengthen a band around a football by 1 m and you get exactly the same gap.
Circumference from the area
If you know the area, first work out the radius with r = √(A ÷ π) and put it into C = 2πr. A flower bed of 4 m² has a 1.128 m radius and therefore a 7.09 m circumference – the length of edging you need to buy.
Typical mistakes
- Diameter times 2π: C = 2πr applies to the radius. Use the diameter and the circumference comes out twice as large as it should.
- Using 3 instead of π: that is 4.5% too little. For a 7 m bed edging you would be over 30 cm short.
- Measuring at the wrong height: for trees and tapering objects the circumference depends on where you measure.
In the circle calculator choose “Circumference” as the known value and get diameter, radius and area with the working. The background to π is in the article The number π.